Appendix A

KdV Solitons via Inverse Scattering by G. E. Sarty

The inverse scattering method for finding soliton solutions of the KdV equa• tion as presented in Section 4.2 can be automated with the symbolic manipula• tion program Mathematica (and probably by other such programs as well). At the end of this appendix is a Mathematica routine that will generate n-soliton solutions for the KdV equation

Ut - 6UUx + Uxxx = 0 subject to the initial condition of

U(x, t = 0) = -n(n + 1)sech2(x).

To use the program (assuming you have purchased Mathematica), type the code as given into a file called "invskat.m". Then start up Mathematica and type «invskat .m. This loads the soliton generation code. Then to generate, say, a 2-soliton solution type soliton [2] ; or soli ton [3] for a 3-soliton solu• tion, etc. The solution is then returned as the function u[x_. tJ. The code

277 278 Appendix A. KdV Solitons via Inverse Scattering has been tested for up to 6 solitons and no attempt has been made to simplify the answer to any extentj it is given in terms of exponential functions. The program of Appendix B simplifies answers in terms of hyperbolic trigonometric functions which gives a neater looking final result. The logic of the program follows that given in Chapter 4 and comments in the code explain the steps followed. We give a brief elaboration here of the steps. First we must use the result that the eigenfunctions of the linear Schrodinger equation at t = 0 for the initial data under our consideration are the associated Legendre functions in the variable e = tanh(z). Symbolically, we write the associated Legendre functions as p;:'(e) where n is the number of solitons sought and 1 ~ m ~ n. (In the code, LegendreP[n.m.Tanh[x]] is P;:'(tanh(z)).) It is known that

1 [pm(e)]2 ~ = .!. (n + m)! 1-1 n 1 - e2 m (n - m)! and this is used to normalize the eigenfunction

n K(z,Yjt) = L Lm(z,t)e-k ",11 m=l where K(z,Yjt) is K[x_.y_.tJ and Lm(z,t) is L[m] at this point in the pro• gram. Note that km = m in our case. The expression for K(z, Yj t) is sub• stituted into the Gelfand-Levitan equation ( GL in the code). Terms involving e-k ",11 for 1 ~ m ~ n are separated out. That being done, the coefficient functions Lm(z,t) and hence K(z,Yjt) can be solved for algebraically. Finally, the solution is calculated as

{) U(z, t) = -2 {)z K(z, Zj t).

The symbolic program written in Mathematica is given below: 279

( ...... ) (. . . .) (. • IIVSKAT." -- A "athematica soliton generator. This •• ) (.. package contains subroutines that generate n-soliton solutions •• ) (.. ~or the KdV equation: •• ) (. . . .) (. • U - 6 U U + U = 0 •• ) (. • t x xxx •• ) (. . ..) (. • subject to the initial condition •• ) (. • 2 •• ) (. • U(x,t=O) = -n(n+1) sech [x]. • .) (. . ..) (. • The inverse scattering method and the Gel~and-Levitan •• ) (. • equation is used. • .) (. • Programmed by: •• ) (. • G. Sarty, July 1992 •• ) (. . ..) ( ...... ) invskat: : usage = "The package invskat.m returns n soli ton solutions to the standard KdV equation subject to the initial condition U(x,t=O) = -n(n+1) sech[x]·2 using the inverse scattering method. For more in~ormation, type '?soliton." soliton: :usage = "soliton[n] will return an n soliton solution to the standard KdV equation where n is an integer." sOliton[n_Integer] := Block[{m,Phi,Co,Cm,B,K,L,GL,y,eqn,k}, I~[ n <= 0, Return["Use a positive integer."] ];

(. define normalized eigen~unctions: .)

Do[ Phi[m,x] = (-1)·m Sqrt[ m(n-m)!/(n+m)! ] LegendreP[n,m,Tanh[x]]; Phi[m,x] = Factor[Phi[m,x]]; Phi[m,x] = Phi[m,x] II. {(1 - u_) (1 + u_) -> (1 - u·2), (-1 + u_) (1 + u_) -> - (1 - u·2), (1 - u_)·k_ (1 + u_)·k_ -> (1 - u·2)·k, (-1 + u_)·k_ (1 + u_)·k_ -> - (1 - u·2)·k}; Phi[m,x] Phi[m,x] II. {(1 - Tanh[x_]·2) -> Sech[x]·2, (-1 + Tanh[x_]·2) -> -Sech[x]·2, (1 - Tanh[x_]·2)·(k_/2) -> Sech[x]·k, (-1 + Tanh[x_]·2)·(k_/2) -> -Sech[x]·k, (1 - Tanh[x_]·2)·k_ -> Sech[x]·(2 k), (-1 + Tanh[x_]·2)·k_ -> -Sech[x]·(2 k)}; Phi[m,x] = PowerExpand[Phi[m,x]], {m,1,n,1}] ;

(. de~ine Cm(O) .)

Do[ Co[m] Phi[m,x] E·(m x); Co[m] Co[m] II. Sech[x] -> 2/(E·x + E·(-x»; Co[m] Co[m] II. Tanh[x] -> 1; Co[m] = ExpandAII[Co[m]]; 280 Appendix A. KdV Solitons via Inverse Scattering

top = lumerator[Co[m]]/E-(m x); bot • ExpandAll[Denominator[Co[.]]/E-(m x)]; Co[m] = top/bot; Co[m] • Limit[ Co[m] , x -> In£inity], {m,l,n,H] ;

Do[ Cm[m,t] = Co[m] E-(4 m-3 t), {m,l,n,H] ;

(. de£ine B(y,t) .)

Do[ B[y_,t_] := Sum[Cm[m,t]-2 E-(-m y), {m,l,n}], {m,l,n,H] ;

(. using separation of variables, define l(x,y;t) .)

(. substitute into the Gelfand-Levitan equation .) (. first solving the integral of B[y+z,t] l[x,z,t] from x to in£inity, .) (. vrt z: .)

in£int = ExpandA11[B[y+z,t] l[x,z,t]] II. E-(b_ + c_ z) -> -E-(b + c x) / c; GL = l[x,y,t] + B[x+y,t] + infint;

(. pullout E-(-km y) terms: .)

GL = ExpandA11[GL]; GL = ExpandAll[E-«n+l) y) GL]; Do[ GL = GL II. {E-(h_ + m y) -> y[m] E-h, E-(m y) -> y[m]}, {m,l,n,H] ; Do[ eqn[m] = GL; Do[ I£[k == m,eqn[m]=eqn[m] II. y[k] -> l,eqn[m]=eqn[m] II. y[k] -> 0], {k,l,n,H] , {m,l,n ,H];

(. solve for L[m] .)

system = {eqn[l] == O}; vars = {L[1]}; Do[ system = Append [system , eqn[m] == 0]; vars = Append[vars, L[m]] , {m,2,n,l}] ; soln = Solve[system,vars];

(. use L[m] to reconstruct l[x,x,t] .)

Do[ l[m] = Together[Expand[Part[Part[Part[soln,l],m],2]]], {m,l,n,H]; 281 k[x_,t_] := Together[Expand[Sum[l[mJ E-(-m x), {m,1,n}]]];

(. finally, calculate the solution, u, of the initial value problem .)

Return[u[x_,t_] = Together[Expand[-2 D[k[x,t] ,x]]]] ] Appendix B

KdV Solitons via Backlund Transform by G. E. Sarty

B.1 Backlund Transform Program

Biicklund transform and the nonlinear superposition principle were described in section 4.4. In this subsection, we describe how to use a symbolic computer code to implement the nonlinear superposition principle automatically to find multiple soliton solutions of the initial value problems for the KdV equation. The code is written in Mathematica and given in the later part of this subsec• tion. Specifically, the code has been designed to calculate soliton solutions to the KdV equation using the initial condition U(z, t = 0) = -n(n+ 1) sech 2 (z). The results of 2-,3-, 4-, 5-, 6-, and 7-soltion solutions computed from this code are presented in sections B.2, B.3, B.4, B.5, B.6 and B.7 respectively. The computer program essentially has two parts. One part is a set of routines that enables specific hyperbolic trigonometric function simplifications to be done. This set of routines is called "hyppac.m" . The other part calculates

283 284 Appendix B. KdV Solitons via Backlund Transform the soliton solutions. These two parts have been put into a package called "solipac.m". It can be used by copying the code into a file called "solipac.m" and entering «solipac.m at the Mathematica prompt. Typing soliton[N] will cause the program to generate an N -soliton solution. One may like to test the code by typing in soliton[l] or soliton[2] to generate 1- and 2-soliton solutions. These calculations take only a few seconds on any workstation. If you desire only to calculate an expression for the soliton ladder, type ladder [N] . After the ladder calculation is completed, if one want to see the cor• responding soltion, simply type finish. For instance, the command ladder [2] generates a 2-soliton ladder. Then the command finish will generate a 2- soliton solution. The program has been tested up to 7 solitons (the 7-soliton solution took about a week's worth of CPU time on a SUN Sparcstation 1+ (1991)). The hyperbolic function routines are pretty much self-explanatory with the locally defined hyperbolic functions being designated with small first letters (e.g. sinh, cosh) to avoid confusion with Mathematica's definitions (Sinh, Cosh, etc.). The soliton calculation itself proceeds by calculation of the the func• tions Wen) via the nonlinear superposition principle derived using the Backlund transform in section 4.4. We use notations: (i) (n) == 12··· [n - l]k ( a per• mutation of 1,2,3,···, n - 1 and k) where k is a number greater than n - 1; (ii) (n - 1)' == 12· .. [n - 2]k (a permutation of 1, 2, 3, ... , n - 2 and k and still for k> [n - 1]). We have the nonlinear superposition principle as:

4(An -i - An) Wen) = W(n-2) - , (B.1.1) W(n-i) - W(n-i)' where Am == _m2 • Note that equation (B.1.1) is valid for each value of k such that [n - 1] > k ~ N where N is the number of solitons sought. These equations show the relationship between the various Wen). This relationship is shown schematically by the connecting arrows in Figure B.1 which is an example for N = 5. The tricky part of the program is to reindex the Wen) for any N universal way so that a one-dimensional array w[n] can be used to define the functions. In our program, we number the Wen) in the following way:

Wi = w[l], W2 = w[2], W3 = w[3], W4 = w[4], W5 = w[5], W12 = w[6], W13 = w[7], W14 = w[8], W15 = w[9], W123 = w[10], W124 = w[ll], W125 = w[12], W1234 = w[13], W1235 = w[14], W12345 = w[15] for the case of the 5-soliton solution. The functions in the first column of Figure B.1 are defined by

Wn(X, t) = -2k tanh(kx - 4k3t), when k is odd,

wn(X, t) = -2k coth(kx - 4k3t), when k is even. B.l. Backlund Transform Program 285

W;:---• N345 "

/ " "w:. " "w 1'25 3 " , , , , " , 'w 4

"w 5

Figure B.l: Diagram of the nonlinear superposition principle for 5 solitons.

The N-soliton solution is then found by differentiating W12 ... N (or w[(N-2 N)/2]+) with respect to x. That is enough about the ideas used in developing the computer code. The whole package of code solipac.m is given below: 286 Appendix B. KdV Solitons via Biicklund Transform

( ...... ) (. . . .) (. • SOLIPAC." -- A "athematica soliton generator. This •• ) (. * package contains subroutines that generate n-soliton solutions * *) (* * £or the IdV equation: * *) (* • * *) (* * U - 6 U U + U • 0 * *) (* * t x xxx * *) (* * * *) (* * subject to the initial condition * *) (* * 2 * .) (* * U(x,t&O) • -n(n+1) sech [x]. * *) (* * * *) (* * The package HYPPAC." is included (and described below) * *) (* * along with the routines soliton, ladder and £inish. * *) (* * * *) (* • Programmed by: G. Sarty, Jan. 26, 1993 * *) (* * * *) (. ** ••• ***.****.****.***** •• ********.**********.************************ *) solipac::usage = "Solipac.m contains routines £or calculating n - soliton solutions £or the IdV equation. For more in£ormation type ?hyppac, ?soliton, ?ladder and ?£inish. The commands soliton[n], ladder[n] and £inish use the routines in hyppac.m to simpli£y expressions containing the hyperbolic trig £unctions cosh and sinh."

(* ********************************************************************** *) (* * * .) (* * HYPPAC." -- A collection o£ hyperbolic trig £unction * .) (* * substitution routines £or "athematica. The routines are * *) (. * called: * *) (* * 1) hypdif£ * *) (* * 2) hypsubs * *) (* * 3) hyp£ac * *) (* * 4) hypconv * *) (* * 5) hypmath * *) (* * 6) hypunmath. * *) (* * * *) (* ********************************************************************** .) hyppac: : usage .. "The £ile hyppac. m is included in the £ile solipac. m and contains routines £or manipulating hyperbolic trig £unctions. The routines included are hypdi££, hypsubs, hyp£ac, hypconv, hypmath and hypunmath. (Type ?hypdi££, ?hypsubs, ?hyp£ac, ?hypconv, ?hypmath or ?hypunmath £or more in£o on these routines.) To use hyppac, use small letters to denote the trig £unctions. That is, denote the hyperbolic cosine by cosh instead o£ Cosh and denote the hyperbolic sine by sinh instead o£ Sinh. This will prevent "athematica £rom converting everything into exponential notation. The routines hypmath and hypunmath will convert between the two types." (. DIFFEREITIATIOI OF HYPERBOLIC TRIG FUICTIOIS *) hypdi££: : usage = "The routine hypdi££ lets "athematica know what the derivatives o£ the hyperbolic trig £unctions sinh and cosh are. The de£initions were made when the £ile hyppac.m was read in." sinh'[x_] := cosh[x] cosh'[x_] := sinh[x] (* HYPERBOLIC TRIG SUBSTITUTIOI RULES *) hypsubs: :usage .. "The routine hypsubs, applied thus: »» expr II. hypsubs ««, does hyperbolic trig £unction product to sum conversions in the expression expr. This command was written primarily £or use by the hypconv command." B.l. Backlund Transform Program 287 hypsubs = { sinh[u_] cosh[v_] -> (1/2) sinh[u + v] + (1/2) sinh[u - v], cosh[u_] sinh[v_] -> (1/2) sinh[u + v] - (1/2) sinh[u - v], cosh[u_] cosh[v_] -> (1/2) cosh[u + v] + (1/2) cosh[u - v], sinh[u_] sinh[v_] -> (1/2) cosh[u + v] - (1/2) cosh[u - v] , sinh[u_]-(n_Integer) -> sinh[u]-(n-2) (1/2) (cosh[2 u] - 1), cosh[u_]-(n_Integer) -> cosh[u]-(n-2) (1/2) (cosh[2 u] + 1), sinh[-u_ v_.] -> -sinh[u v], cosh[-u_ v_.] -> cosh[u v], sinh[u_Real v_.] -> If[ u < 0, -sinh[-u v], sinh[u v] ], cosh[u_Real v_.] -> If[ u < 0, cosh[-u v], cosh[u v] ], sinh[u_Integer v_.] -> If[ u < 0, -sinh[-u v], sinh[u v] ], cosh[u_Integer v_.] -> If[ u < 0, cosh[-u v], cosh[u v] ] } (. FACTORIIG THE ARGUMEIT .) hypfac::usage = "The routine hypfac, applied as »» expr //. hypfac ««, to an expression expr factors out common factors in the arguments of hyperbolic trig functions (in particular, -1 may be factored out and thus made visible to the hypsubs routine). This command vas vritten primarily for use by the hypconv command. " hypfac = { sinh[u_] :> sinh [Factor [u]], cosh[u_] :> cosh[Factor[u]] } (. HYPERBOLIC TRIG COIVERSIOI ROUTIIE .) hypconv: : usage = "The routine hypconv, implemented on an expression expr thus: »» hypconv[expr] ««, uses the routine hypsubs to COMPLETELY convert all products of hyperbolic trig functions in expr into sums of hyperbolic trig functions. It uses the commands hypfac and hypsubs in a loop." hypconv[f_] := Block[ {nevf,oldf,j}, j = 1; oldf = f; While [ j == 1, nevf oldf II. hypsubs; nevf = Expand[nevf]; nevf = nevf //. hypfac; If[ SameQ[nevf,oldf], j = 0, j 1]; oldf = nevf]; Return[nevf] ] (. COlVERT TO MATHEMATICA FORM .) hypmath: :usage = "The routine hypmath, implemented on an expression expr thus: »» hypmath[expr] ««, converts cosh to Cosh and sinh to Sinh." hypmath[f_] := f //. {cosh[u_] -> Cosh[u], sinh[u_] -> Sinh[u]} (. COlVERT TO HYPPAC ROUTIIE FORM .) hypunmath: : usage = "The routine hypunmath, implemented on an expression expr thus: »» hypunmath[expr] ««, converts Cosh to cosh and Sinh to sinh." hypunmath[f_] := f //. {Cosh[u_] -> cosh[u], Sinh[u_] -> sinh[u]} ; (. • •••••••••••••••••••••••• EID OF HYPPAC •••••••••••••••••••••••••• ..) (. COMPLETE SOLITOI LADDER AID MULTIPLE SOLITOI CALCULATIOI .) soli ton: : usage = "»» soli ton [n] ««, vhere n is an integer, viII give an n-soliton solution for the KdV equation Ut - 6 U Ux + Uxxx = 0 subject to the initial condition U(x,t=O) = -n(n+1) sech-2[x]. The solution is returned as U and the associated soliton ladder is returned as W. lormally, U is printed out upon completion of this command. To execute this same procedure in tvo steps, use the ladder and finish commands. For more information type ?ladder and ?finish." soliton[n_Integer] := Block[{1ambda,v,s}, If[ n <= 0, Return["Use a positive integer."] ]; Array [lambda ,n] ; Array [v, (n-2 + n) /2] ; Array[s,n]; Do[ lambda[k] -(k-2), {k,n} ]; Do[ v[k] -2 k sinh[k x - 4 k-3 t] / cosh[k x - 4 k-3 t], {k,1,n,2}]; Do[ v[k] = -2 k cosh[k x - 4 k-3 t] / sinh[k x - 4 k-3 t], {k,2,n,2}]; 288 Appendix B. KdV Solitons via Backlund Transform s [1] = 1; Do[ s[l] = s[l-l] + n - (1-2), {1,2,n,1}]; j = 2; Do[ v[s[j]+i] = -(4 (lambda[j-1]-lambda[j+i]»/(v[s[j-1]]-v[s[j-1]+i+1]), {i,O,(n-j),l} ]; Do[ Do[ v[s[l]+i] = v[s[1-2]]-«4 (lambda[l-l]-lambda[l+i]»/(v[s[l-l]] v[s[l-l]+i+1]», {i,O,(n-l),l}], {1,3,n,l}]; W = Together[v[(n-2 + n)/2]] ; top lumerator[W]; top = hypconv[top]; top hypmath[top]; top Simplify[top]; bot Denominator[W]; bot hypconv[bot]; bot hypmath[bot]; bot Simplify[bot]; W = Expandlumerator[Together[top/bot]]; U = D[W,x]; first = Take[U,l]; second = U - first; first = first Denominator[W]~2; second = second Denominator[W]~2; first = Expand[first]; second = Expand[second]; topu = first + second; botu = Denominator[W]~2; topu = hypconv[topu]; topu = hypmath[topu]; topu = Simplify[topu]; U = Expandlumerator[topu/botu] ] (. SOLITOI LADDER OILY CALCULATIOI .) ladder: : usage = "»» ladder[n] ««, vhere n is an integer, viII give an n-soliton ladder for the KdV equation Ut - 6 U Ux + Uxxx = ° subject to the initial condition U(x,t=O) = -n(n+1) sech~2[x]. The solution is returned as W. lormally, W is printed out upon completion of this command." ladder[n_Integer] := Block[{lambda,v,s}, If[ n <= 0, Return["Use a positive integer."] ]; Array[lambda,n]; Array[v,(n~2 + n)/2]; Array[s,n]; Do[ lambda[k] -(k~2), {k,n} ]; Do[ v[k] = -2 k sinh[k x - 4 k~3 t] / cosh[k x - 4 k~3 t], {k,1,n,2}]; Do[ v[k] = -2 k cosh[k x - 4 k~3 t] / sinh[k x - 4 k~3 t], {k,2,n,2}]; s [1] = 1; Do[ s[l] = s[l-l] + n - (1-2), {1,2,n,1} ]; j = 2; Do[ v[s[j]+i] = -(4 (lambda[j-l]-lambda[j+i]»/(v[s[j-1]] - v[s[j-l]+i+l]), {i,O,(n-j),l} ]; Do[ Do[ v[s[l]+i] = v[s[1-2]]-«4 (lambda[l-l]-lambda[l+i]»/(v[s[l-l]] - v[s[1-1]+i+1]», {i,O,(n-l),l}], {l,3,n,l}]; W = Together[v[(n~2 + n)/2]]; top lumerator[W]; top hypconv[top]; top hypmath[top]; top Simplify[top]; bot Denominator[W]; bot hypconv[bot]; bot hypmath[bot]; B.2. Two Solitons 289

bot = Simplify[bot]; W = Expandlumerator[Together[top/bot]] ] (. CALCULATE MULTIPLE SOLITOIS FROM THE LADDER .) finish: :usage = ,,»» finish ««. Typing this command after a ladder[n] command has been executed viII return the multiple soliton solution, U, associated to the previously generated ladder, W. lormally, U is printed out upon completion of this command." finish := ( U = D[W,x]; first = Take[U,l]; second = U - first; first = first Denominator[W]-2; second = second Denominator[W]-2; first = Expand[first]; second = Expand[second]; topu = first + second; botu = Denominator[W]-2; topu = hypconv[topu]; topu = hypmath[topu]; topu = Simplify[topu]; U = Expandlumerator[topu/botu]

The results of 1-, 2-, ..., and 7-soliton solutions are presented in the follow• ing sections

B.2 Two Solitons

The analytical expression for the two soliton ladder is: 6sinh[3(12t - x)] + 6sinh[28t - x] W12 = cosh[3(12t - x)] + 3 cosh [28t - x] The analytical expression for the two soliton solution is: -36 - 48 cosh [2 (4t - x)] - 12 cosh[4(16t - x)] U12 = [ cosh[3(12t - x)] + 3 cosh [28t - x] F B.3 Three Solitons

The analytical expression for the three soliton ladder is: 12 sinh[6(24t - x)] + 48 sinh[4(34t - x)] + 60 sinh[2( 40t - x)] W123 = 10 cosh[72t] + cosh[6(24t - x)] + 6 cosh[4(34t - x)] + 15 cosh[2( 40t - x)] The analytical expression for the three soliton solution is:

U123 = [-3024 -1200cosh[2(4t - x)]- 360cosh[6(12t - x)]- 1920cosh[4(16t - x)]- 3240cosh[2(28t - x)]- 240cosh[8(28t - x)]- 24cosh[10(28t - x)]- 720cosh[6(36t - x)]- 960cosh[4(52t - x)]- 600 cosh[2(76t - x)] ]

[ 10 cosh [72t] + cosh[6(24t - x)] + 6 cosh[4(34t - x )]+ 15cosh[2(40t - x)] F 290 Appendix B. KdV Solitons via. Backlund Transform B.4 Four Solitons

The analytical expression for the four soliton ladder is:

W1234 = 4 [ 5 sinh[10( 40t - x)] + 140 sinh[4( 46t - x)] + 40 sinh[8( 49t - x )]+ 135sinh[6(56t - x)] + 100sinh[4(82t - x)] + 175sinh[2(88t - x)]- 35sinh[2(56t + x)] ]

[ 126cosh[120t] + cosh[10(40t - x)] + 70cosh[4(46t - x)]+ 10 cosh[8( 49t - x)] + 45 cosh[6(56t - x)] + 50 cosh[4(82t - x )]+ 175cosh[2(88t - x)] + 35cosh[2(56t + x)] ]

The analytical expression for the four soliton solution is:

U1234 = [-650400 - 224000 cosh[144t] - 98000 cosh[288t]- 2520 cosh[2(292t - 7 x)] - 63000 cosh[2(284t - 5x )]- 9000 cosh [2 (332t - 5x)] - 686880cosh[2(4t - x)]- 19600 cosh[6(12t - x)] - 224640 cosh[4(16t - x )]- 2240cosh[12(24t - x)] - 176400cosh[2(28t - x)]- 50400 cosh [8 (28t - x)] - 25200cosh[10(28t - x)]- 80640 cosh [8 (34t - x)] - 399840 cosh [6 (36t - x )]- 504000cosh[4(40t - x)]- 40cosh[18(44t - x)]- 720cosh[16(46t - x)]- 22400cosh[12(48t - x)]- 201600cosh[4(52t - x)] - 25200cosh[10(52t - x)]- 3600cosh[14(52t - x)] - 49000cosh[6(60t - x)]- 8000cosh[12(60t - x)] - 211680cosh[8(64t - x)]- 630000 cosh[2(76t - x)] - 141120 cosh[4(76t - x )]- 204120 cosh[6(76t - x)] - 70000 cosh[6(84t - x )]- 201600 cosh[4(112t - x)] - 176400 cosh[2(148t - x)]- 63000cosh[2(220t - x)] -17640cosh[2(116t + x)] ]

[ 126cosh[120t] + cosh[1O(40t - x)] + 70cosh[4{46t - x)]+ 10 cosh [8(49t - x)] + 45cosh[6(56t - x)] + 50cosh[4{82t - x)]+ 175 cosh[2(88t - x)] + 35 cosh[2(56t + x)] ]2 B.5. Five Solitons 291

B.5 Five Solitons

The analytical expression for the five soliton ladder is:

W12345 = [390sinh[892t - 13x] + 4410sinh[388t - 7x]+ 14700sinh[676t -7x] + 30sinh[15(60t - x)]+ 1701Osinh[5(76t - x)] + 5040sinh[9(76t - x)]+ 2310 sinh[11 (76t - x)] + 3150sinh[9(92t - x)]+ 22050sinh[3(108t - x)] + 11760sinh[5(124t - x)]+ 3600sinh[172t - x] + 2940sinh[3(204t - x)]+ 4410 sinh[316t - x] - 1260 sinh[5(20t + x )]- 5040 sinh[3 (36t + x)] - 4860 sinh[164t + x] ]

[ 15cosh[892t - 13x] + 315cosh[388t - 7x] + 1050cosh[676t - 7x]+ cosh[15(60t - x)] + 1701 cosh [5 (76t - x)] + 280 cosh[9(76t - x )]+ 105 cosh[11 (76t - x)] + 175cosh[9(92t - x)] + 3675cosh[3(108t - x)]+ 1176cosh[5(124t - x)] + 1800cosh[172t - x] + 490cosh[3(204t - x)]+ 2205cosh[316t - x] + 126cosh[5(20t + x)] + 840cosh[3(36t + x)]+ 2430 cosh[164t + x] ]

The analytical expression for the five soliton solution is:

U12345 = [-640926000 - 31752000 cosh[144t]- 200037600 cosh[240t]- 129654000 cosh[288t] - 17496000 cosh[336t]- 111132000 cosh[432t] - 42865200 cosh[480t]- 29635200 cosh[720t] - 1680 cosh[2(868t - 13x )]- 20160cosh[2(644t - 11x)] -134400cosh[2(788t -11x)]- 42000cosh[2(860t - 11x)] - 1128960cosh[2(292t - 7x)]- 24696000cosh[2(580t - 7x)] - 9525600cosh[2(604t -7x)]- 5268480cosh[2(652t -7x)]- 6585600cosh[2(724t -7x)]- 28224000 cosh[2(284t - 5x)] - 54432000 cosh[2(332t - 5x )]- 18522000 cosh[2(356t - 5x)] - 567000 cosh[4(392t - 5x )]- 73500 cosh [4 ( 416t - 5x )]- 32256000 cosh[2( 428t - 5x )]- 24696000cosh[2(572t - 5x)]- 8232000 cosh [2 (644t - 5x)]- 1632960cosh[6(212t - 3x)]- 333090240cosh[2(4t - x)]- 67939200cosh[6(12t - x)]- 385188720cosh[4(16t - x)]- 317520cosh[12(24t - x)] - 611452800cosh[2(28t - x)]- 40000800cosh[8(28t - x)]- 6585600cosh[10(28t - x)]- 11430720cosh[8(34t - x)] -157288320cosh[6(36t - x)]- 71442000cosh[4(40t - x)]- 12600cosh[20(40t - x)]- 241920 cosh[18 (44t - x)]- 61740000cosh[8(46t - x)]- 952560 cosh[16( 46t - x )]- 16669800 cosh[12( 48t - x )]- 1260000 cosh[16( 49t - x)] - 160003200 cosh[4(52t - x )]- 77565600 cosh[10(52t - x)]- 11707200 cosh[14(52t - x)]- 25515000cosh[12(56t - x)]- 296352000cosh[6(60t - x)]- 10584000cosh[12(60t - x)]- 269090640cosh[8(64t - x)]- 2143260cosh[12(64t - x)]- 340200cosh[20(64t - x)]- 292 Appendix B. KdV Solitons via Biicklund Transform

60cosh[28(64t - x)] - 10080cosh[24(66t - x)]- 705600cosh[16(67t - x)] - 1058400cosh[18(68t - x)]- 20412000 cosh[8(70t - x)] - 12600 cosh[24(72t - x )]- 164640000cosh[2(76t - x)] - 186701760cosh[4(76t - x)]- 91445760cosh[6(76t - x)] - 12130560cosh[14(76t - x)]- 12594960 cosh[16(76t - x)] - 235200 cosh[20(76t - x )]- 31500000 cosh[8(82t - x)] - 215460000 cosh [6 (84t - x )]- 57697920 cosh[12(84t - x)] - 2963520 cosh[18(84t - x )]- 1176000 cosh[16(85t - x)] - 385875000 cosh[4(88t - x )]- 50009400 cosh[8(88t - x)] - 14288400 cosh[12(88t - x )]- 4445280 cosh[16(9lt - x)] - 97977600 cosh[6(92t - x )]- 1470000cosh[16(94t - x)] - 46305000 cosh[12 (96t - x)]- 163900800cosh[10(100t - x)] - 136080000cosh[8(106t - x)]- 9878400 cosh[12(108t - x)] - 150028200 cosh[4(112t - x )]- 60011280cosh[6(116t - x)] - 34574400cosh[8(118t - x)]- 3087000cosh[12(120t - x)] - 52920000cosh[4(124t - x)]- 129820320cosh[8(124t - x)] - 67737600cosh[6(132t - x)]- 148803480cosh[4(136t - x)] -163296000cosh[6(140t - x)]- 573652800 cosh[2(148t - x)] - 2304960 cosh[8(154t - x )]- 66044160 cosh[6(156t - x)] - 32413500 cosh[4(160t - x )]- 201519360cosh[4(196t - x)] - 28224000cosh[2(220t - x)]- 8643600 cosh [4 (232t - x)] - 240045120cosh[2(244t - x)]- 63221760 cosh[2(364t - x)] - 38102400 cosh[2(388t - x)]- 423360 cosh[8(26t + x)] - 4898880 cosh[6( 44t + x )]- 15435000cosh[4(56t + x)] - 16329600cosh[4(68t + x)]- 10001880 cosh[4(104t + x)] - 106686720 cosh[2(116t + x )]- 24192000 cosh[2(140t + x)] - 59270400 cosh[2(212t + x )]- 3951360 cosh[2(356t + x)] ]

[ 15cosh[892t - 13x] + 315cosh[388t - 7x] + 1050cosh[676t - 7x]+ cosh[15(60t - x)] + 1701 cosh[5(76t - x)] + 280 cosh[9(76t - x)]+ 105 cosh [11 (76t - x)] + 175cosh[9(92t - x)] + 3675cosh[3(108t - x)]+ 1176cosh[5(124t - x)] + 1800cosh[172t - x] + 490cosh[3(204t - x)]+ 2205 cosh[316t - x] + 126 cosh[5(20t + x)] + 840 cosh[3(36t + x )]+ 2430 cosh[164t + x] F B.6. Six Solitons 293 B.6 Six Solitons

The analytical solution for the six soliton ladder is:

W123456 = [798 sinh[1756t - 19x] + 40950 sinh[1252t - 13x]+ 114660 sinh[1540t - 13x] + 30492 sinh[764t - llx]+ 261954 sinh[1244t - llx] + 155232 sinh[1484t - llx]+ 432180sinh[1180t -7x] + 25200sinh[3{516t - 5x)]+ 727650sinh[548t - 5x] + 14700sinh[3{564t - 5x)]+ 554400 sinh[692t - 5x] + 510300 sinh[1028t - 5x]+ 69300sinh[7{4t - x)] + 8316sinh[9{4t - x)]+ 161700 sinh[3{84t - x)] + 232848 sinh[9{84t - x )]+ 42 sinh[21{84t - x)] + 748440sinh[7{100t - x)]+ 7140sinh[17{100t - x)] + 661500sinh[9{132t - x)]+ 378000sinh[7{148t - x)] + 74088sinh[9{164t - x)]+ 1018710sinh[3{180t - x)] + 181104sinh[244t - x]+ 151200 sinh[3{324t -x)] + 224532 sinh[484t - x]+ 10584sinh[964t - x] - 173250sinh[3{12t + x)]- 258720 sinh[3{60t + x)] - 231000 sinh[188t + x]- 58212 sinh[476t + x] - 242550 sinh[28t + 5x))

[ 21 cosh[1756t - 19x] + 1575 cosh[1252t - 13x]+ 441Ocosh[1540t -13x] + 1386cosh[764t - llx]+ 11907 cosh[1244t - llx] + 7056 cosh[1484t - llx]+ 30870cosh[1180t - 7x] + 840cosh[3{516t - 5x)]+ 72765 cosh[548t - 5x] + 490 cosh[3{564t - 5x)]+ 55440cosh[692t - 5x] + 51030cosh[1028t - 5x]+ 4950 cosh [7{4t - x)] + 462cosh[9{4t - x)]+ 26950 cosh [3 {84t - x)] + 12936 cosh[9{84t - x )]+ cosh[21{84t - x)] + 53460cosh[7{100t - x)]+ 210 cosh[17{100t - x)] + 36750 cosh[9{132t - x )]+ 27000 cosh[7{148t - x)] + 4116 cosh[9{164t - x)]+ 169785cosh[3{180t - x)] + 90552cosh[244t - x]+ 25200 cosh[3{324t - x)] + 112266 cosh[484t - x]+ 5292 cosh [964t - x] + 28875cosh[3{12t + x)]+ 43120cosh[3{60t + x)] + 115500cosh[188t + x]+ 29106 cosh[476t + x] + 24255 cosh [28t + 5x] ] 294 Appendix B. KdV Solitons via Backlund Thansform

The six soliton solution is:

U123456 = [-1990617855408 - 611966880000cosh[144t]- 81327286656 cosh[240t]- 259891443000 cosh[288t]- 565820640000 cosh[336t]- 391727952000 cosh[432t]- 705144542256 cosh[480t]- 352983015000 cosh[576t]- 103733784000 cosh[672t]- 458859582720 cosh[720t]- 13070456784 cosh[960t]- 52390800000 cosh[1008t]- 123773265000 cosh[1056t]- 77014476000 cosh[1152t]- 3360cosh[2(1732t -19x)]- 100800cosh[2(1508t -17x)]- 564480 cosh[2(1652t - 17x)]- 164640 cosh[2(1724t - 17x)]- 12418560 cosh[2(868t - 13x)] - 14968800 cosh[2(892t - 13x )]- 37255680cosh[2(1156t -13x)]- 360581760cosh[2(1228t -13x)]- 254741760cosh[2(1396t -13x)]- 493920000cosh[2(1444t -13x)]- 186701760cosh[2(1468t -13x)]- 94832640cosh[2(1516t -13x)]- 110638080cosh[2(1588t -13x)]- 23284800cosh[2(644t - 11x)]- 576723840 cosh[2(788t - 11x )]- 310464000 cosh[2(860t - 11x )]- 507091200 cosh[2(1004t - 11x )]- 5119329600 cosh[2(1124t - 11x )]- 1825528320 cosh[2(1148t - 11x )]- 3353011200 cosh[2(1196t - 11x )]- 926100000 cosh[2(1220t - 11x )]- 1451520000 cosh[2(1292t - 11x )]- 968083200cosh[2(1436t -l1x)]- 290424960cosh[2(1508t -l1x)]- 179292960cosh[2(4t - 7x)]- 5524989120cosh[2(292t -7x)]- 6601240800 cosh[2(388t - 7x)]- 2390572800 cosh[2(508t - 7x)]- 28523880000cosh[2(580t - 7x)]- 36966948480cosh[2(604t - 7x)]- 27165600cosh[4(628t - 7x)]- 69065821440cosh[2(652t - 7x)]- 73347120000 cosh[2(676t - 7x)]- 39734217600 cosh[2(724t - 7x)]- 176329440cosh[4(736t -7x)]- 41074387200cosh[2(748t -7x)]- 14817600cosh[4(772t - 7x)]- 53343360cosh[4(796t -7x)]- 17287200cosh[4(808t -7x)]-197743835520cosh[2(892t -7x)]- 50709120000cosh[2(940t - 7x)]- 159016556160cosh[2(1012t - 7x)]- 7302113280 cosh[2(1084t - 7 x )]- 50409475200 cosh[2(1108t - 7x )]- 11379916800cosh[2(1228t - 7x)]- 6721263360cosh[2(1252t - 7x)]- 138124728000cosh[2(284t - 5x)]- 37966320000cosh[2(332t - 5x)]- 136914624000 cosh[2(356t - 5x)]- 1571724000 cosh[4(392t - 5x)]- 2376990000cosh[4(416t - 5x)]- 37255680000cosh[2(428t - 5x)]- 46103904000cosh[2(476t - 5x)]- 4400088000cosh[4(488t - 5x)]- 10584000 cosh[6( 492t - 5x )]- 6112260000 cosh [4 (524t - 5x )]- 28523880000cosh[2(572t - 5x)]- 1530900000cosh[4(572t - 5x)]- 2716560000cosh[4(596t - 5x)]- 3500658000cosh[4(608t - 5x)]- 84135744000cosh[2(644t - 5x)]- 8573040000cosh[4(644t - 5x)]- 2074464000 cosh[4(668t - 5x)] - 441712656000 cosh[2(716t - 5x)]- 114095520000cosh[2(788t - 5x)]- 403558848000cosh[2(836t - 5x)]- 10886400000cosh[2(1004t - 5x)]- 24893568000cosh[2(1076t - 5x)]- 1886068800cosh[6(212t - 3x)]- 1185528960cosh[6(244t - 3x)]- 12070840320 cosh[6(364t - 3x)]- 10287648000 cosh[6(380t - 3x)]- 5881105440cosh[6(404t - 3x)]- 14402707200cosh[6(428t - 3x)]- B.6. Six Solitons 295

1411200cosh[16(203t - 2x)] - 1905176582400cosh[2(4t - x)]- 9147600cosh[16(4t - x)] - 86063160960cosh[6(12t - x)]- 760656711360 cosh[4(16t - x)] - 6119668800 cosh[12(24t - x )]- 774568448640cosh[2(28t - x)] - 156168260640cosh[8(28t - x)]- 70624689408 cosh[10(28t - x)] - 220308076800 cosh[8(34t - x )]- 1145413301760cosh[6(36t - x)] - 1376925480000cosh[4(40t - x)]- 5122656cosh[20(40t - x)] - 168739200cosh[18(44t - x)]- 25101014400cosh[8(46t - x)] - 1703585520cosh[16(46t - x)]- 52257653280cosh[12(48t - x)] - 512265600cosh[16(49t - x)]- 624673042560 cosh[4(52t - x)] - 84544653312 cosh[10(52t - x )]- 11410398720cosh[14(52t - x)] -10373378400cosh[12(56t - x)]- 206705520000cosh[6(60t - x)] - 18928728000cosh[12(60t - x)]- 66528 cosh [30 (60t - x)] - 536220594000cosh[8(64t - x)]- 69313028400 cosh[12(64t - x)] - 555043104 cosh[20(64t - x )]- 1940400 cosh[28 (64t - x)] - 27941760cosh[24(66t - x)]- 11670785280 cosh[16 (67t - x)] - 4107438720cosh[18(68t - x)]- 56582064000 cosh[8(70t - x)] - 221288760 cosh[24 (72t - x )]- 1765617235200cosh[2(76t - x)] - 333902761920cosh[4(76t - x)]- 447524118720 cosh[6(76t - x)] - 192492181728 cosh[10(76t - x )]- 102769793280 cosh[14(76t - x)] - 27938267280 cosh[16(76t - x )]- 5215795200cosh[18(76t - x)] - 3940823040cosh[20(76t - x)]- 733471200 cosh[22(76t - x)] - 138600 cosh [32(79t - x )]- 12806640000cosh[8(82t - x)] - 3630682440cosh[16(82t - x)]- 234846259200cosh[6(84t - x)] - 185533286400cosh[12(84t - x)]- 20925273600cosh[18(84t - x)] - 17463600cosh[24(84t - x)]- 3725568cosh[30(84t - x)] - 38031840000cosh[16(85t - x)]- 156881340000cosh[4(88t - x)] - 878295088440cosh[8(88t - x)]- 236333401920cosh[12(88t - x)] - 20956320 cosh[28 (88t - x)]- 84cosh[40(88t - x)] -17854784640cosh[16(9lt - x)]- 420431679360cosh[6(92t - x)] - 2037420000cosh[18(92t - x)]- 30240 cosh[36 (92t - x)] - 28426536600cosh[16(94t - x)]- 2381400 cosh[32 (94t - x)] - 75547533600 cosh[12(96t - x )]- 770144760cosh[24(96t - x)] - 35280cosh[36(96t - x)]- 15688134000 cosh[8(100t - x)] - 904089263616 cosh[1O(100t - x )]- 11209968000 cosh[18(100t - x)] - 1232231616 cosh[20(100t - x )]- 1122660000cosh[24(100t - x)] - 10584000cosh[28(100t - x)]- 16003008cosh[30(100t - x)] - 456382080cosh[24(102t - x)]- 3333960 cosh[32(103t - x)] - 31120135200 cosh[12(104t - x )]- 75014100cosh[24(104t - x)] - 222017241600cosh[8(106t - x)]- 411600cosh[32(106t - x)] - 898502220000cosh[6(108t - x)]- 252558583200 cosh[12(108t - x)] - 24222401280 cosh[18(108t - x )]- 24893568 cosh[30(108t - x)] - 470318879520 cosh[4(112t - x )]- 88978594320 cosh[16(112t - x)] - 24863975136 cosh[20(112t - x )]- 1730937600 cosh[24(114t - x)] - 443603381760 cosh[6(116t - x )]- 11855289600 cosh[12(116t - x)] - 41074387200 cosh[18(116t - x )]- 420078960cosh[24(116t - x)] - 913242274560cosh[8(118t - x)]- 296 Appendix B. KdV Solitons via Biicklund Transform

15717240000 cosh[16(118t - x)] - 79866864000 cosh[12(120t - x )]- 1296540000 cosh[24(120t - x)] - 34496689920 cosh[16(12lt - x )]- 880310323200 cosh[4(124t - x)] - 360972612000 cosh[8(124t - x )]- 335988487296 cosh[10(124t - x)] - 107820266400 cosh[12(124t - x )]- 109919779200cosh[14(124t - x)] - 27871905600cosh[18(124t - x)]- 6883833600cosh[22(124t - x)] - 248935680cosh[24(126t - x)]- 36966948480cosh[16(127t - x)] - 74875185000cosh[16(130t - x)]- 430694288640 cosh[6(132t - x)] - 15717240000 cosh[12(132t - x )]- 72606240 cosh[24(132t - x)] - 412483246560 cosh[4(136t - x )]- 49417622100cosh[8(136t - x)] - 27725211360cosh[16(136t - x)]- 7633434816 cosh[20(136t - x)] - 7938000000 cosh[16(139t - x )]- 188606880000cosh[6(140t - x)] - 6096384000cosh[18(140t - x)]- 21874111560cosh[16(142t - x)] - 592215624000cosh[12(144t - x)]- 559109537280 cosh[2(148t - x)] - 395176320000 cosh[6(148t - x)]- 216062652960 cosh[8(148t - x)] - 4537890000 cosh[16(148t - x )]- 5547709440cosh[18(148t - x)] - 116169984cosh[20(148t - x)]- 149845449600 cosh[8(154t - x)] - 720170922240 cosh[6(156t - x )]- 13691462400cosh[12(156t - x)] - 25391439360cosh[16(157t - x)]- 1048252590000cosh[4(160t - x)] -173282571000cosh[8(160t - x)]- 79214889600 cosh[12(164t - x)] - 1016487360 cosh[16(166t - x )]- 50316124320 cosh[12(168t - x)] - 215550720000 cosh[2(172t - x )]- 130704567840cosh[6(172t - x)] - 424462500000cosh[8(172t - x)]- 215442767232cosh[1O(172t - x)] - 5511240000cosh[12{172t - x)]- 201264497280 cosh[8(178t - x)] - 33339600000 cosh[12(180t - x )]- 471247761600 cosh[4(184t - x)] - 12602368800 cosh[12(184t - x )]- 158403168000 cosh[8(190t - x)] - 570007247040 cosh[4(196t - x )]- 199168865280 cosh[6(196t - x)] - 69313028400 cosh[8(196t - x )]- 59147117568cosh[10(196t - x)] - 371482372800cosh[6(204t - x)]- 7468070400 cosh[12(204t - x)] - 355914377280 cosh[8(208t - x)]- 147867793920 cosh[6(212t - x)] - 187928328000 cosh[2(220t - x )]- 461039040000cosh[4(220t - x)] - 456382080000cosh[6(228t - x)]- 142967119680cosh[4(232t - x)] - 1694947161600cosh[2(244t - x)]- 11980029600cosh[8(244t - x)] - 1394039808cosh[1O(244t - x)]- 10287648000 cosh[8(250t - x)] - 218001611520 cosh[6(252t - x )]- 152488662480cosh[4(256t - x)] -77962500000cosh[4(268t - x)]- 5881105440 cosh[8(268t - x)] - 87886149120 cosh[6(276t - x )]- 134469720000cosh[2(292t - x)] - 58090919040cosh[4(292t - x)]- 621599378400 cosh[4(304t - x)] - 323460799200 cosh[2(316t - x )]- 4320812160cosh[6(332t - x)] - 133111440000cosh[4(340t - x)]- 905964998400cosh[2(364t - x)] - 22632825600cosh[4(364t - x)]- 77617552320 cosh[4(376t - x)] - 147867793920 cosh[2(388t - x)]- 152996659200cosh[2(436t - x)] - 533433600cosh[4(484t - x)]- 79068195360 cosh[2(508t - x)] - 188606880000 cosh[2(532t - x )]- 93139200000cosh[2(580t - x)] - 250847150400cosh[2(604t - x)]- 18255283200 cosh[2(724t - x)] - 960498000 cosh[12x]- 2801452500cosh[8(8t + x)] - 717171840cosh[12(12t + x)]- B.7. Seven Solitons 297

20582100000 cosh[8(20t + x)] - 1056198528 cosh[10(20t + x )]- 22058467200cosh[8(26t + x)] - 122245200000cosh[6(36t + x)]- 36212520960cosh[6(44t + x)) - 860606208 cosh[10(44t + x)]- 32955753600 cosh[4(56t + x)] - 10373378400 cosh[8(56t + x )]- 365858428320 cosh[4(68t + x)] - 35565868800 cosh[6(76t + x )]- 22590912960 cosh[6(84t + x)] - 19921440000 cosh[4(92t + x )]- 27725211360cosh[4(104t + x)] - 74413987200cosh[2(116t + x)]- 244490400 cosh[8(116t + x)] - 192022760160 cosh[4(128t + x )]- 262497312000 cosh[2(140t + x)] - 3353011200 cosh[6(156t + x )]- 392841187200 cosh[2(164t + x)] - 10040405760 cosh[4(164t + x )]- 316808849280 cosh[2(212t + x)) - 12474000000 cosh[4(236t + x))- 9241737120 cosh[4(248t + x)] - 103733784000 cosh[2(260t + x )]- 263560651200cosh[2(284t + x)] - 123295011840cosh[2(332t + x)]- 4563820800 cosh[2(356t + x)] - 81130896000 cosh[2(500t + x )]- 7302113280 cosh[2(572t + x)] - 1232231616 cosh[1440t]- 2286900000 cosh[2( 4t + 5x)) - 6830208000 cosh[2(76t + 5x)] ]

[ 21 cosh[1756t - 19x] + 1575 cosh[1252t - 13x]+ 4410cosh[1540t - 13x] + 1386cosh[764t - 1b]+ 11907 cosh[1244t - 11x] + 7056 cosh[1484t - 11x]+ 30870cosh[1180t - 7x] + 840cosh[3(516t - 5x)]+ 72765 cosh [548t - 5x] + 490cosh[3(564t - 5x)]+ 55440 cosh[692t - 5x] + 51030 cosh[1028t - 5x]+ 4950 cosh[7( 4t - x)] + 462 cosh[9( 4t - x )]+ 26950 cosh [3 (84t - x)] + 12936 cosh [9 (84t - x)]+ cosh[21(84t - x)] + 53460 cosh[7(100t - x )]+ 210cosh[17(100t - x)] + 36750cosh[9(132t - x))+ 27000cosh[7(148t - x)] + 4116cosh[9(164t - x)]+ 169785cosh[3(180t - x)] + 90552cosh[244t - x]+ 25200 cosh[3(324t - x)) + 112266 cosh[484t - x]+ 5292cosh[964t - x] + 28875cosh[3(12t + x)] +43120 cosh[3(60t + x)] + 115500 cosh[188t + x]+ 29106 cosh [476t + x] + 24255cosh[28t + 5x] J2

B.7 Seven Solitons

The analytical expression for the seven soliton ladder is:

W1234567 = [1456sinh[2(1564t -13x)] + 92400sinh[2(1460t - 11x)]+ 51744sinh[2(1532t - 11x)] + 7606368sinh[2(1276t - 7x))+ 4004000sinh[2(164t - 5x)] + 28028000sinh[2(596t - 5x)]+ 231000sinh[4(656t - 5x)] + 18018000sinh[2(668t - 5x)]+ 588000sinh[4(728t - 5x)) + 11858000sinh[2(812t - 5x)]+ 67914000sinh[2(956t - 5x)] + 8316000sinh[2(1172t - 5x)]+ 299376 sinh[6(356t - 3x)] + 2095632 sinh[6( 436t - 3x )]+ 1143072sinh[6(476t - 3x)] + 9225216sinh[8(22t - x)]+ 298 Appendix B. KdV Solitons via Backlund Transform

21525504sinh[4(28t - z)] + 27243216sinh[6(28t - z)]+ 48048 sinh[14(28t - z)] + 648648 sinh[12(32t - z )]+ 5605600sinh[8(40t - z)] + 4708704sinh[2(52t - z)]+ 4953312 sinh[6(68t - z)] + 192192 sinh[16(88t - z )]+ 20966400 sinh[4(100t - z)] + 2402400sinh[14(100t - z)]+ 22198176 sinh[8(112t - z)] + 13621608 sinh[12(112t - z)]+ 56 sinh[28(112t - z)] + 18144sinh[24(128t - z)]+ 3311616 sinh[16(133t - z)] + 85621536 sinh[6(148t - z )]+ 80080000 sinh[8(148t - z)] + 16166304 sinh[14(148t - z)]+ 26195400 sinh[12(160t - z)] + 7761600 sinh[16(160t - z )]+ 28658448sinh[2(172t - z)] + 19160064sinh[12(172t - z)]+ 6930000sinh[14(172t - z)] + 790272sinh[16(178t - z)]+ 63567504sinh[6(188t - z)] + 14968800sinh[12(200t - z)]+ 42499072sinh[8(202t - z)] + 116540424sinh[4(208t - z)]+ 48498912sinh[8(232t - z)] + 54885600sinh[6(260t - z)]+ 16816800sinh[4(280t - z)] + 1862784sinh[8(292t - z)]+ 12573792 sinh[6(308t - z)] + 26426400 sinh[2(340t - z )]+ 12705000 sinh[4(352t - z)] + 18213888 sinh[4(388t - z )]+ 33297264sinh[2(412t - z)] + 7114800sinh[2(700t - z)]- 8845200sinh[6(20t + z)] - 33434856sinh[4(32t + z)]- 38220000 sinh[2(92t + z)] - 3171168 sinh[4(152t + z )]- 11099088 sinh[2(308t + z)] - 243936 sinh[2(668t + z)] ]

[ 6019650 cosh[192t] + 5054400 cosh[336t] + 8494200 cosh[672t] + 490050 cosh[1344t] + 28 cosh[2(1564t - 13z )]+ 2100cosh[2(1460t - lIz)] + 1176cosh[2(1532t -l1z)]+ 271656cosh[2(1276t -7z)] + 200200cosh[2(164t - 5z)]+ 1401400 cosh[2(596t - 5z)] + 5775 cosh[4(656t - 5z)]+ 900900 cosh [2 (668t - 5z)] + 14700cosh[4(728t - 5z)]+ 592900cosh[2(812t - 5z)] + 3395700cosh[2(956t - 5z)]+ 415800 cosh[2(1172t - 5z)] + 8316 cosh[6(356t - 3z)]+ 58212cosh[6(436t - 3z)] + 31752 cosh [6 (476t - 3z)]+ 576576 cosh[8(22t - z)] + 2690688 cosh[4(28t - z )]+ 2270268 cosh [6 (28t - z)] + 1716cosh[14(28t - z)]+ 27027 cosh[12(32t - z)] + 350350 cosh [8 (40t - z )]+ 1177176 cosh [2 (52t - z)] + 412776 cosh[6(68t - z )]+ 6006cosh[16(88t - z)] + 2620800cosh[4(100t - z)]+ 85800 cosh[14(100t - z)] + 1387386 cosh[8(112t - z )]+ 567567 cosh[12(112t - z)] + cosh[28(112t - z )]+ 378 cosh[24(128t - z)] + 103488 cosh[16(133t - z )]+ 7135128 cosh[6(148t - z)] + 5005000 cosh[8(148t - z )]+ 577368cosh[14(148t - z)] + 1091475cosh[12(160t - z)]+ 242550 cosh[16(160t - z)] + 7164612 cosh[2(172t - z )]+ 798336cosh[12(172t - z)] + 247500cosh[14(172t - z)]+ 24696 cosh[16(178t - z)] + 5297292 cosh[6(188t - z )]+ 623700cosh[12(200t - z)] + 2656192cosh[8(202t - z)]+ B.7. Seven Solitons 299

14567553cosh[4(208t - x)] + 3031182cosh[8(232t - x)]+ 4573800 cosh[6(260t - x)] + 2102100 cosh[4(280t - x )]+ 116424 cosh[8(292t - x)] + 1047816 cosh[6(308t - x )]+ 6606600 cosh[2(340t - x)] + 1588125 cosh[4(352t - x )]+ 2276736 cosh[4(388t - x)] + 8324316 cosh[2( 412t - x )]+ 1778700 cosh [2 (700t - x)] + 737100 cosh [6 (20t + x )]+ 4179357 cosh[4(32t + x)] + 9555000 cosh[2(92t + x )]+ 396396 cosh[4(152t + x)] + 2774772 cosh[2(308t + x)]+ 60984cosh[2(668t + x)] ] The expression for the seven soliton solution is:

U1234S61 = [-27553846057384192 - 2222678091110400 cosh[144t]- 8495891967467520 cosh[240t] - 6538746445862400 cosh[288t]- 1156175262950400 cosh[336t] - 10989038170880000 cosh[432t]- 2090436576205824 cosh[480t] - 1840147482700800 cosh[528t]- 3546915573801600 cosh[576t] - 8622020177971200 cosh[672t]- 12392672682120192 cosh[720t] - 1571137568000000 cosh[864t]- 8108388570522240 cosh[960t] - 3702318344524800 cosh[1008t]- 407798251660800 cosh[1056t] - 1065086243020800 cosh[1152t]- 1124534572761600 cosh[1248t] - 886004184665600 cosh[1296t]- 1495260256089600 cosh[1392t] - 1628197592845824 cosh[1440t]- 713646940406400 cosh[1536t] - 543855312000000 cosh[1584t]- 1904612024899584 cosh[1680t] - 53328898022400 cosh[1728t]- 111217704998400 cosh[1968t] - 179485571865600 cosh[2016t]- 83431755703296 cosh[2160t] - 1735555852800 cosh[2736t]- 75600 cosh[2(3028t - 25x)] - 831600 cosh[2(2636t - 23x )]- 11642400 cosh[2(2876t - 23x)] - 6350400 cosh[2(2996t - 23x )]- 14817600 cosh[2(3020t - 23x)] - 1778112 cosh[2(3068t - 23x )]- 129729600 cosh[2(1732t - 19x)] - 148324176 cosh[2(1756t - 19x )]- 908107200 cosh[2(2164t - 19x)] - 6904209312 cosh[2(2236t - 19x )]- 384199200 cosh[2(2380t - 19x)] - 15402895200 cosh[2(2524t - 19x )]- 21824510400 cosh[2(2572t - 19x)] - 8762535936 cosh[2(2596t - 19x )]- 1344697200 cosh[2(2620t - 19x)] - 28560470400 cosh[2(2740t - 19x )]- 6845731200 cosh[2(2764t - 19x)] - 3734035200 cosh[2(2884t - 19x )]- 1045529856 cosh[2(2956t - 19x)] - 713512800 cosh[2(1508t - 17x )]- 12913284384 cosh[2(1652t - 17x)] - 6356750400 cosh[2(1724t -17x)]- 399567168 cosh[2(1772t - 17x)] - 35276072832 cosh[2(2012t - 17x )]- 229785117408 cosh[2(2132t - 17x)] - 90810720000 cosh[2(2156t - 17x )]- 133491758400 cosh[2(2204t - 17x)] - 174243484800 cosh[2(2348t - 17x )]- 48193947648 cosh[2(2372t - 17x)] - 229209750000 cosh[2(2420t - 17x )]- 895103104896 cosh[2(2492t - 17x)] - 93884313600 cosh[2(2564t - 17x )]- 112954564800 cosh[2(2588t - 17x)] - 261954000000 cosh[2(2660t - 17x )]- 61611580800 cosh[2(2708t - 17x)] - 223914087936 cosh[2(2732t - 17x )]- 3136589568 cosh[2(2852t - 17x)] - 185513328 cosh[2(388t - 13x )]- 59940197856 cosh[2(868t - 13x)] - 18551332800 cosh[2(892t - 13x )]- 764172208800 cosh[2(1156t - 13x)] - 1930518285696 cosh[2(1228t - 13x )]- 834323490000 cosh[2(1252t - 13x)] - 389577988800 cosh[2(1372t - 13x )]- 11026603224000 cosh[2(1396t - 13x)] - 3496212720000 cosh[2(1444t - 13x )]- 3775200863616 cosh[2(1468t - 13x)] - 5411713507200 cosh[2(1516t - 13x )]- 300 Appendix B. KdV Solitons via. Bii.cklund Transform

6541096161600 cosh[2(1540t - 13x)] - 6048 cosh[4(1552t - 13x )]- 2531003739264 cosh[2(1588t - 13x)] - 13586989852800 cosh[2(1660t - 13x )]- 2621560189248 cosh[2(1708t - 13x)] - 4724610508800 cosh[2(1732t - 13x )]- 16152502766400 cosh[2(1756t - 13x)] - 38830500878400 cosh[2(1876t - 13x )]- 50067668851200 cosh[2(1900t - 13x)] - 15556748118912 cosh[2(1948t - 13x )]- 5041461902400 cosh[2(1996t - 13x)] - 46481536886400 cosh[2(2020t - 13x)]- 1843734638592 cosh[2(2068t -13x)] -11956069540800 cosh[2(2092t - 13x)]- 15462224870400 cosh[2(2116t - 13x)] - 2161120500000 cosh[2(2164t - 13x )]- 1734982115328 cosh[2(2308t - 13x)] - 9624609086400 cosh[2(2356t - 13x)]- 6037934918400 cosh[2(2380t - 13x)] - 617463000000 cosh[2(2404t - 13x)]- 3630682440000 cosh[2(2452t -13x)] - 1355454777600 cosh[2(2476t -13x)]- 739338969600 cosh[2(2596t - 13x)] - 35618558976 cosh[2(284t - 11x )]- 43286443200 cosh[2(356t - 11x)] - 643727855232 cosh[2(644t -11x)]- 646100110656 cosh[2(764t - 11x)] - 4456344816000 cosh[2(788t -11x)]- 1498504946400 cosh[2(860t - 11x)] - 247913265600 cosh[2(908t - 11x )]- 24442044314688 cosh[2(1004t - 11x)] - 30667008468096 cosh[2(1124t - 11x )]- 864864 cosh[4(1136t - 11x)] - 46015153984800 cosh[2(1148t - 11x )]- 14444353123200 cosh[2(1196t - 11x)] - 35756721000000 cosh[2(1220t - 11x )]- 47684591018064 cosh[2(1244t - 11x)] - 14906579688000 cosh[2(1268t - 11x )]- 25734028320000 cosh[2(1292t - 11x)] - 29804544 cosh[4(1316t - 11x )]- 2045284441200 cosh[2(1340t - 11x)] - 69854400 cosh[4(1424t - 11x )]- 6852576931200 cosh[2(1436t - 11x)] - 104315904 cosh[4(1436t - 11x )]- 97140626751168 cosh[2(1484t - 11x)] - 123227725420800 cosh[2(1508t - 11x )]- 12236744520000 cosh[2(1556t - 11x)] - 24723218520000 cosh[2(1652t - 11x)]- 5753767219200 cosh[2(1700t - 11x)] - 368415513097632 cosh[2(1724t - 11x )]- 5177084220000 cosh[2(1772t - 11x)] - 88190182080000 cosh[2(1796t - 11x )]- 328945638155904 cosh[2(1844t - 11x)] - 137650889376000 cosh[2(1868t - 11x )]- 14825286630000 cosh[2(1916t - 11x)] - 63003869603136 cosh[2(1964t - 11x )]- 47851661088000 cosh[2(1988t - 11x)] - 46593757980000 cosh[2(2012t - 11x )]- 110937519000000 cosh[2(2060t - 11x)] - 51953075564544 cosh[2(2084t - 11x )]- 39204784080000 cosh[2(2132t - 11x)] - 84565256041728 cosh[2(2204t - 11x )]- 2587686393600 cosh[2(2348t -11x)] - 2074675680000 cosh[2(2372t -11x)]- 1138582013184 cosh[2(2444t - 11x)] - 6922501185600 cosh[2( 4t - 7x )]- 59003357212800 cosh[2(100t - 7x)] - 5235928169472 cosh[2(172t - 7x )]- 19392326553600 cosh[2(220t - 7x)] - 6363107150400 cosh[2(244t - 7x )]- 55214108078976 cosh[2(292t - 7x)] - 8181137764800 cosh[2(388t - 7x)]- 16921669564800 cosh[2(508t -7x)] - 788566622659200 cosh[2(580t - 7x)]- 343364762664000 cosh[2(604t -7x)] - 89503045632 cosh[4(628t - 7x)]- 881619665639328 cosh[2(652t - 7x)] - 90901530720000 cosh[2(676t - 7x )]- 54540918432 cosh[4(688t - 7x)] - 661400971862400 cosh[2(724t -7x)]- 2154745756800 cosh[4(736t -7x)] - 176943325759200 cosh[2(748t - 7x)]- 2440992153600 cosh[4(772t - 7x)] - 260308928880000 cosh[2(796t - 7x )]- 813664051200 cosh[4(796t - 7x)] -1454763518208 cosh[4(808t -7x)]- 1077372878400 cosh[4(832t -7x)] - 406639733900400 cosh[2(844t - 7x)]- 289766400 cosh[6(876t - 7x)] - 1153750197600 cosh[4(880t - 7x )]- 1455822468067968 cosh[2(892t - 7x)] - 2444204431468800 cosh[2(940t - 7x )]- 315555609600 cosh[4(940t - 7x)] - 1551699072 cosh[6(948t - 7x )]- 123480000 cosh[6(972t - 7x)] - 4405228027200 cosh[4(976t - 7x )]- 5638691874816 cosh[4(988t - 7x)] - 432081216 cosh[6(988t - 7x )]- 138297600 cosh[6(996t - 7x)] - 2106689124443904 cosh[2(1012t - 7x )]- B. 7. Seven Solitons 301

9310194432000 cosh[4(1012t -7x)]- 10594584000000 cosh[4(1024t - 7x)]- 5289520465152 cosh[4(1048t - 7x )]- 4417778534400 cosh[4(1060t - 7x )]- 1094699164759200 cosh[2(1084t - 7x)]- 1858472260444800 cosh[2(1108t -7x)]- 1485034405991136 cosh[2(1132t - 7x)]-19334952806400 cosh[4(1132t - 7x)]- 852864012000000 cosh[2(1156t - 7x)]- 4819394764800 cosh[4(1156t - 7x)]- 320513711918400 cosh[2(1180t - 7x)]-14311490054400 cosh[4(1192t -7x)]- 586419759525600 cosh[2(1228t - 7x)]- 630902587392 cosh[4(1228t - 7x)]- 4066364332800 cosh[4(1240t - 7x)]- 287525390390784 cosh[2(1252t - 7x)]- 1093890798000000 cosh[2(1300t - 7x)]- 844958822400 cosh[4(1300t - 7x)]- 492892646400 cosh[4(1312t - 7x)]- 1008129759436800 cosh[2(1444t - 7x)]- 128456449273152 cosh[2(1492t - 7x)]- 395625567330000 cosh[2(1516t - 7x)]- 533308702449600 cosh[2(1588t - 7x)]- 985948768861056 cosh[2(1612t - 7x)]- 388281316500000 cosh[2(1660t - 7x)]- 495724666719744 cosh[2(1732t - 7x)]- 12704483994048 cosh[2(1852t - 7x)]- 110937519000000 cosh[2(1900t - 7x)]- 67095011491200 cosh[2(1948t -7x)]- 975927439872 cosh[2(2092t - 7x)]- 241026786000000 cosh[2(68t - 5x)]- 80951270400 cosh[4(68t - 5x)]- 151502551200 cosh[4(176t - 5x )]- 14847250017600 cosh[2(212t - 5x )]- 1380352701974400 cosh[2(284t - 5x)]- 2268124008796800 cosh[2(332t - 5x)]- 660840681362400 cosh[2(356t - 5x )]- 17643451795200 cosh[4(392t - 5x )]- 4327213363200 cosh[2( 404t - 5x )]- 4857046286400 cosh[4( 416t - 5x )]- 1029964568371200 cosh[2( 428t - 5x )]- 4533271142400 cosh[4( 452t - 5x )]- 2061259200 cosh[6(468t - 5x )]- 932243478566400 cosh[2( 476t - 5x )]- 82176955660800 cosh[4( 488t - 5x )]- 214013654400 cosh[6( 492t - 5x )]- 237318681600 cosh[6(516t - 5x )]- 20138185267200 cosh[4(524t - 5x )]- 1780813429995600 cosh[2(548t - 5x )]- 80754273600 cosh[6(564t - 5x )]- 345945600 cosh[8(566t - 5x )]- 788566622659200 cosh[2(572t - 5x )]- 62531742873600 cosh[4(572t - 5x )]- 71034163200 cosh[6(588t - 5x )]- 8950304563200 cosh[4(596t - 5x)]- 315085047076800 cosh[4(608t - 5x)]- 115413160262400 cosh[4( 632t - 5x)] - 558835200 cosh[8( 632t - 5x )]- 1390768579200 cosh[6(636t - 5x)]- 920912547693600 cosh[2(644t - 5x)]- 80100999014400 cosh[4(644t - 5x)]- 172422809395200 cosh[4(668t - 5x)]- 326761419600 cosh[6(676t - 5x)]- 4120116000000 cosh[6(684t - 5x)]- 1877353928049600 cosh[2( 692t - 5x )]- 3113510400 cosh[8( 692t - 5x )]- 61535160086400 cosh[4(704t - 5x )]- 5867769600 cosh[8(704t - 5x )]- 2857318556400 cosh[6(708t - 5x )]- 2364884899977600 cosh[2(716t - 5x )]- 1066867200 cosh[8(722t - 5x )]- 6525096916800 cosh[6(756t - 5x )]- 3781810618185600 cosh[2(764t - 5x )]- 1120324867200 cosh[6(772t - 5x )]- 134358588000000 cosh[4(776t - 5x )]- 5829432527318400 cosh[2(788t - 5x )]- 2495269022400 cosh[6(796t - 5x )]- 4991679000000 cosh[6(804t - 5x )]- 146544065740800 cosh[4(812t - 5x )]- 244940382086400 cosh[4(824t - 5x )]- 480249000000 cosh[6(828t - 5x )]- 3345607551686400 cosh[2(836t - 5x )]- 1307045678400 cosh[6(836t - 5x )]- 508803837141600 cosh[4(848t - 5x )]- 263560651200 cosh[6(852t - 5x )]- 1396529164800 cosh[6(876t - 5x )]- 214807309516800 cosh[4(884t - 5x )]- 99891792000000 cosh[4(896t - 5x )]- 613328943427200 cosh[2(908t - 5x )]- 169009008537600 cosh[4(908t - 5x )]- 2380754376000000 cosh[2(932t - 5x )]- 52935349975200 cosh[4(944t - 5x )]- 136525305470400 cosh[4(992t - 5x)]- 2756456140838400 cosh[2(1004t - 5x)]- 53013342412800 cosh[4(1004t - 5x )]- 875839426862400 cosh[2(1028t - 5x )]- 159040027238400 cosh[4(1028t - 5x )]- 4088185101000000 cosh[2(1052t - 5x )]- 71940531801600 cosh[4(1064t - 5x )]- 569475841334400 cosh[2(1076t - 5x )]- 302 Appendix B. KdV Solitons via Biicklund Transform

3303876523147200 cosh[2(1124t - 5x)] - 1644025772188800 cosh[2(1148t - 5x )]- 1161818380800 cosh[4(1184t - 5x)] - 533032591291200 cosh[2(1196t - 5x )]- 2009885514436800 cosh[2(1268t - 5x)] - 4375563192000000 cosh[2(1292t - 5x )]- 75731041108800 cosh[2(1484t - 5x)] - 921952867651200 cosh[2(1508t - 5x )]- 41653340467200 cosh[2(1556t - 5x)] - 380205065116800 cosh[2(1628t - 5x )]- 20376279000000 cosh[2(1844t - 5x)] - 14910002553600 cosh[2(1868t - 5x )]- 1434355322400 cosh[6(44t - 3x)] - 2208907196496 cosh[6(92t - 3x)]- 52141956273792 cosh[6(212t ...:. 3x)] - 45773273145600 cosh[6(244t - 3x )]- 16868018591424 cosh[6(292t - 3x)] - 247591795651200 cosh[6(364t - 3x )]- 30893806944 cosh[12(368t - 3x)] - 235344607898400 cosh[6(380t - 3x )]- 32438497291200 cosh[6(388t - 3x)] - 227069481038400 cosh[6(404t - 3x)]- 430359242288976 cosh[6(412t - 3x)] - 138221899200 cosh[12(416t - 3x)]- 101949562915200 cosh[6( 428t - 3x)] - 86910050304 cosh[12( 428t - 3x )]- 1275914226787200 cosh[6(484t - 3x)] - 292999673584800 cosh[6(508t - 3x)]- 396196624107648 cosh[6(532t - 3x)] - 320412912019200 cosh[6(548t - 3x)]- 279921170390400 cosh[6(580t - 3x)] - 262902902169600 cosh[6(604t - 3x )]- 82343877739200 cosh[6(628t - 3x)] - 30114332430336 cosh[6(652t - 3x )]- 10082923804800 cosh[6(700t - 3x)] - 118756205568 cosh[16(203t - 2x )]- 17548876800 cosh[16(22lt - 2x)] - 159826867200 cosh[16(25lt - 2x )]- 423783360000 cosh[16(257t - 2x)] - 76822276608 cosh[16(263t - 2x )]- 152562009600 cosh[16(275t - 2x)] - 1278942787584 cosh[16(293t - 2x )]- 1337851468800 cosh[16(311t - 2x)] - 253018225152 cosh[16(323t - 2x )]- 251475840000 cosh[16(329t - 2x)] - 15951524583399552 cosh[2( 4t - x )]- 75906083661408 cosh[10(4t - x)] - 61374015166464 cosh[12(4t - x)]- 11641026196800 cosh[14( 4t - x)] - 1506939033600 cosh[16( 4t - x )]- 71788901184 cosh[18( 4t - x)] - 2706757871596800 cosh[6(12t - x )]- 122426304000000 cosh[12(12t - x)] - 4722157440000 cosh[16(13t - x )]- 158694639P0127488 cosh[4(16t - x)] - 134975000160000 cosh[8(16t - x )]- 46855607198400 cosh[12(16t - x)] - 7229150824896 cosh[16(16t - x )]- 47048958899712 cosh[12(24t - x)] - 24360820844371200 cosh[2(28t - x)]- 1700485353907200 cosh[8(28t - x)] - 406119170338560 cosh[10(28t - x )]- 923444121600 cosh[18(28t - x)] - 498659825664 cosh[20(28t - x )]- 5496691200 cosh[24(30t - x)] - 12217165728768 cosh[16(3lt - x )]- 842924192850432 cosh[8(34t - x)] - 7675481467506048 cosh[6(36t - x )]- 30165841305600 cosh[12(36t - x)] - 280560280000 cosh[18(36t - x )]- 5001025704998400 cosh[4( 40t - x)] - 518140820736 cosh [20( 40t - x )]- 10080551150208 cosh[18( 44t - x)] - 2538890021606400 cosh[8( 46t - x )]- 41584549450752 cosh[16( 46t - x)] - 758932057699200 cosh[12( 48t - x )]- 51814082073600 cosh[16( 49t - x)] - 6801941415628800 cosh[4(52t - x )]- 3253700893722048 cosh[10(52t - x)] - 493262525448000 cosh[14(52t - x )]- 1049235161990400 cosh[12(56t - x)] - 12348675159004800 cosh[6(60t - x )]- 462050549452800 cosh[12(60t - x)] - 82450368 cosh[30(60t - x )]- 11745902525220864 cosh[8(64t - x)] - 141631469711424 cosh[12(64t - x )]- 11120484380544 cosh[20(64t - x)] - 3964935744 cosh[28(64t - x )]- 313661365248 cosh[24(66t - x)] - 37404596275200 cosh[16(67t - x)]- 38151640296000 cosh[18(68t - x)] - 635164264627200 cosh[8(70t - x )]- 610295347200 cosh[24(72t - x)] - 8931262685587200 cosh[2(76t - x )]- 8181669932384256 cosh[4(76t - x)] - 4472342754397056 cosh[6(76t - x )]- 238561977221568 cosh[10(76t - x)] - 500067319175424 cosh[14(76t - x )]- 421634418847488 cosh[16(76t - x)] - 6464108851200 cosh[18(76t - x)]- B.7. Seven Solitons 303

23284561563648 cosh[20(76t - x)] - 909015307200 cosh[22(76t - x)]- 456648192 cosh[32(79t - x)] - 1295352051840000 cosh[8(82t - x )]- 55380009484800 cosh[16(82t - x)] - 9282339715756800 cosh[6(84t - x )]- 2164981130188800 cosh[12(84t - x)] - 164404944265408 cosh[18(84t - x)]- 1487143257600 cosh[24(84t - x)] - 16049281248 cosh[30(84t - x)]- 336336 cosh[42(84t - x)] - 77712740582400 cosh[16(85t - x)]- 15868062635040000 cosh[4(88t - x)] - 2422262233036800 cosh[8(88t - x )]- 757443074572800 cosh[12(88t - x)] - 173145772800 cosh[28(88t - x )]- 13837824 cosh[40(88t - x)] - 636287642597376 cosh[16(9lt - x )]- 3248675370864000 cosh[6(92t - x)] - 2525042520000 cosh[18(92t - x )]- 461260800 cosh[36(92t - x)] - 47318078126592 cosh[8(94t - x )]- 85673348006400 cosh[16(94t - x)] - 22250277504 cosh[32(94t - x )]- 1513621485129600 cosh[12(96t - x)] - 11184151410432 cosh[24(96t - x )]- 3175460288 cosh[36(96t - x)] - 89451613742400 cosh[6(100t - x )]- 1303947496051200 cosh[8(100t - x)] - 6562938297928320 cosh[10(100t - x)]- 961238736057600 cosh[14(100t - x)] - 18510857164800 cosh[16(100t - x )]- 102771371102400 cosh[18(100t - x)] - 104932828256256 cosh[20(100t - x )]- 3698850355200 cosh[24(100t - x)] - 303005102400 cosh[26(100t - x )]- 879708211200 cosh[28(100t - x)] - 113277292128 cosh[30(100t - x )]- 14832417600 cosh[34(100t - x)] - 3770730163200 cosh[24(102t - x )]- 50854003200 cosh[32(103t - x)] - 2618841534236928 cosh[12(104t - x )]- 12357522777600 cosh[24(104t - x)] - 4448193752217600 cosh[8(106t - x )]- 494300911104 cosh[24(106t - x)] - 67805337600 cosh[32(106t - x )]- 1113543751320000 cosh[6(108t - x)] - 1775022616012800 cosh[12(108t - x )]- 805305653152000 cosh[18(108t - x)] - 3847683840000 cosh[24(108t - x )]- 917764079232 cosh[30(108t - x)] - 33633600 cosh[42(108t - x )]- 197440148505600 cosh[16(109t - x)] - 6830388519292800 cosh[4(112t - x )]- 898012680364800 cosh[16(112t - x)] - 258664201443648 cosh[20(112t - x )]- 1109908800 cosh[36(112t - x)] - 290594304 cosh[40(112t - x )]- 27468783014400 cosh[24(114t - x)] - 2141123807614176 cosh[6(116t - x )]- 1952992987545600 cosh[12(116t - x)] - 192497587682400 cosh[18(116t - x )]- 34915618902528 cosh[24(116t - x)] - 239183407008 cosh[30(116t - x )]- 112 cosh[54(116t - x)] - 6004413002649600 cosh[8(118t - x )]- 1338428931840000 cosh[16(118t - x)] - 685235381107200 cosh[12(120t - x )]- 12114039974400 cosh[24(120t - x)] - 24991366400 cosh[36(120t - x )]- 369600 cosh[48(120t - x)] - 644267332380672 cosh[16(12lt - x )]- 4188857283072000 cosh[4(124t - x)] - 150216121036800 cosh[6(124t - x )]- 4343519961639936 cosh[8(124t - x)] - 9628480327834368 cosh[10(124t - x )]- 355237588113408 cosh[12(124t - x)] - 3026131633324800 cosh[14(124t - x )]- 452541072791808 cosh[18(124t - x)] - 30505427656704 cosh[20(124t - x )]- 181510631108352 cosh[22(124t - x)] - 2136026640384 cosh[28(124t - x )]- 352418242176 cosh[32(124t - x)] - 12915302400 cosh[36(124t - x )]- 226328256 cosh[42(124t - x)] - 84672 cosh[50(124t - x )]- 49337596019712 cosh[24(126t - x)] - 1881600 cosh[48(126t - x )]- 313260659712000 cosh[16(127t - x)] - 1144615547275200 cosh[12(128t - x )]- 526848 cosh[48(129t - x)] - 501365466201600 cosh[16(130t - x )]- 5192377344 cosh[40(130t - x)] - 9869105897481600 cosh[6(132t - x )]- 1758176259840000 cosh[12(132t - x)] - 482298817000000 cosh[18(132t - x )]- 1107487180800 cosh[24(132t - x)] - 3196137776832 cosh[30(132t - x )]- 14874675200 cosh[36(132t - x)] - 97020000 cosh[42(132t - x )]- 304 Appendix B. KdV Solitons via Backlund Transform

4630347489132288 cosh[4(136t - x )]- 8957763150336000 cosh[8(136t - x )]- 912587038114752 cosh[16(136t - x )]- 115683241250304 cosh[20(136t - x )]- 29675173312512 cosh[24(136t - x )]- 16328978265600 cosh[28(136t - x )]- 149361408 cosh[44(136t - x )]- 47883239040000 cosh[24(138t - x )]- 873700834406400 cosh[16(139t - x )]- 5214195627379200 cosh[6(140t - x )]- 826188665702400 cosh[12(140t - x)] - 165924550209600 cosh[18(140t - x )]- 1775545168320 cosh[30(140t - x)] - 48680755200 cosh[36(140t - x )]- 333653114995200 cosh[16(142t - x)] - 12793131187200 cosh[24(142t - x )]- 616548240000 cosh[32(142t - x)] - 2190633984 cosh[40(142t - x )]- 9186418297728672 cosh[12(144t - x)] - 43073340710400 cosh[24(144t - x )]- 22411620000 cosh[36(144t - x )]- 515540933107200 cosh[16(145t - x )]- 59810410243584 cosh[24(146t - x)] - 24391994032929600 cosh[2(148t - x )]- 504841559040000 cosh[4(148t - x)] - 2797255581120000 cosh[6(148t - x )]- 3295675666483200 cosh[8(148t - x)] - 530879469120000 cosh[14(148t - x )]- 747553847040000 cosh[16(148t - x)] - 240134914656000 cosh[18(148t - x)]- 805349068812288 cosh[20(148t - x)] - 250951237992000 cosh[22(148t - x )]- 51455250000 cosh[34(148t - x)] - 33264000000 cosh[36(148t - x )]- 20537193600 cosh[38(148t - x)] - 1194891264 cosh[40(148t - x )]- 11099088000000 cosh[24(150t - x)] - 595672948975104 cosh[16(15lt - x )]- 2053719360000 cosh[32(15lt - x)] - 151444339200 cosh[36(152t - x )]- 8434538836770816 cosh[8(154t - x)] - 420526282176000 cosh[16(154t - x )]- 10954287590400 cosh[24(154t - x)] - 586642839552 cosh[32(154t - x )]- 5970939347493120 cosh[6(156t - x)] - 1758463625318400 cosh[12(156t - x )]- 725159398163200 cosh[18(156t - x)] - 254721153426432 cosh[24(156t - x )]- 2428566970368 cosh[30(156t - x)] - 20445917184 cosh[36(156t - x )]- 414330679526400 cosh[16(157t - x)] - 461039040000 cosh[32(157t - x )]- 2141957412302400 cosh[4(160t - x)] - 2396444046796800 cosh[8(160t - x )]- 282745738207296 cosh[20(160t - x)] - 10882058140800 cosh[28(160t - x)]- 5808499200 cosh[36(160t - x)] - 7831516492800 cosh[24(162t - x )]- 717724922923008 cosh[12(164t - x)] - 577398790096128 cosh[18(164t - x )]- 1796593696128 cosh[30(164t - x)] - 84487176603648 cosh[16(166t - x )]- 76920826397184 cosh[24(166t - x)] - 586776960000 cosh[32(166t - x )]- 3735371428588800 cosh[12(168t - x)] - 20517013440000 cosh[24(168t - x )]- 219689293824 cosh[32(169t - x)] - 267139192320000 cosh[2(172t - x )]- 5046503364302400 cosh[6(172t - x)] - 5254913664000000 cosh[8(172t - x )]- 6429553414764480 cosh[10(172t - x)] - 907899632640000 cosh[12(172t - x )]- 1143953537126400 cosh[14(172t - x)] - 2451962261548800 cosh[16(172t - x )]- 135963828000000 cosh[18(172t - x)] - 248102442491904 cosh[20(172t - x )]- 78172462368000 cosh[26(172t - x)] - 85515570201600 cosh[24(174t - x )]- 984562561382400 cosh[16(175t - x )]- 946122837660000 cosh[12(176t - x )]- 1662892001971200 cosh[8(178t - x)] - 14106505301288400 cosh[6(180t - x )]- 988500114201600 cosh[12(180t - x)] - 690986069989600 cosh[18(180t - x )]- 20917512000000 cosh[24(180t - x)] - 53670532608 cosh[30(180t - x )]- 2181455406434304 cosh[16(18lt - x)] - 7353614629670400 cosh[4(184t - x )]- 1214497338582528 cosh[8(184t - x)] - 1060522604773632 cosh[12(184t - x)]- 355900254893568 cosh[20(184t - x)] - 71914186713600 cosh[24(184t - x )]- 37917927032832 cosh[24(186t - x)] - 4759411629772800 cosh[12(188t - x )]- 791472635553600 cosh[18(188t - x)] - 5942763873446400 cosh[8(190t - x)]- 654158158617600 cosh[16(190t - x)] - 530453163240000 cosh[12(192t - x )]- 235459224000000 cosh[16(193t - x)] - 18044209654732800 cosh[4(196t - x)]- B.7. Seven Solitons 305

5391238529188512 cosh[6(196t - x)] - 7410917562048000 cosh[8(196t - x)]- 2713122404280288 cosh[1O(196t - x)] - 6831800479760256 cosh[14(196t - x)]- 391770827078400 cosh[18(196t - x )]- 77436854820864 cosh[20(196t - x )]- 16943184720000 cosh[22(196t - x)] - 9581833046016 cosh[24(196t - x )]- 1646568000000 cosh[24(198t - x)] - 198530590809600 cosh[16(199t - x )]- 581055764889600 cosh[12(200t - x)] - 1777434590131200 cosh[16(202t - x)]- 4034067230793600 cosh[6(204t - x)] - 1476943753678848 cosh[12(204t - x)]- 98132630948352 cosh[18(204t - x)] - 3614546073600 cosh[24(204t - x )]- 3592050721459200 cosh[8(208t - x)] - 110937519000000 cosh[16(208t - x )]- 325634455770624 cosh[16(211t - x )]- 3032999496727200 cosh[6(212t - x )]- 39306093750000 cosh[18(212t - x )]- 950320962759168 cosh[8(214t - x )]- 4208054410560000 cosh[12(216t - x)] - 184013254656 cosh[24(216t - x )]- 730175189990400 cosh[16(217t - x)]- 3303269697974400 cosh[2(220t - x)]- 4307653724774400 cosh[4(220t - x)] - 2295924752721600 cosh[6(220t - x)]- 1563641491059648 cosh[10(220t - x)] - 824107284000000 cosh[14(220t - x )]- 140967296870400 cosh[16(220t - x)] - 300854099145600 cosh[18(220t - x )]- 27842332041216 cosh[20(220t - x )]- 1794365268119424 cosh[12(224t - x )]- 4320386598528000 cosh[8(226t - x)] - 116326419922944 cosh[16(226t - x )]- 6748314452880000 cosh[6(228t - x)] - 4359044020531200 cosh[12(228t - x)]- 61848899481600 cosh[18(228t - x)] - 458206304371200 cosh[4(232t - x )]- 2720862144000000 cosh[8(232t - x )]- 56929100659200 cosh[16(235t - x )]- 34151723020800 cosh[18(236t - x)] - 1048624075699200 cosh[8(238t - x )]- 126785736000000 cosh[16(238t - x )]- 11515760532593856 cosh[2(244t - x )]- 1897652091535200 cosh[6(244t - x )]- 2606082555428352 cosh[8(244t - x )]- 1130356339047936 cosh[10(244t - x)] - 110912161852800 cosh[14(244t - x)]- 60857153280000 cosh[16(244t - x)]- 161880027724800 cosh[16(247t - x)]- 156715857043200 cosh[12(248t - x )]- 925964219980800 cosh[8(250t - x )]- 8643047938324800 cosh[6(252t - x)]- 486606785280000 cosh[12(252t - x)]- 17329677589224000 cosh[4(256t - x)] - 2335571804966400 cosh[8(256t - x )]- 762887885760000 cosh[12(256t - x)] - 2181859353273600 cosh[6(260t - x )]- 193636396800 cosh[18(260t - x )]- 4672211544000000 cosh[8(262t - x )]- 10843638220800 cosh[16(262t - x )]- 576842316078336 cosh[12(264t - x )]- 345027653006400 cosh[2(268t - x )]- 6925450371840000 cosh[4(268t - x )]- 89706461644800 cosh[8(268t - x )]- 2349516450048768 cosh[10(268t - x )]- 47544651000000 cosh[14(268t - x)] - 154044669240000 cosh[12(272t - x )]- 2125029880157184 cosh[8(274t - x)]- 1114493329757808 cosh[6(276t - x)]- 386555621760000 cosh[12(276t - x )]- 1108239497164800 cosh[8(280t - x )]- 175769368879104 cosh[12(284t - x )]- 2085793892582400 cosh[8(286t - x )]- 7831516492800 cosh[12(288t - x )]- 1853346697020000 cosh[2(292t - x )]- 2718655011072000 cosh[4(292t - x)]- 2787430509864000 cosh[6(292t - x)]- 936425614924800 cosh[8(298t - x )]- 7534519696704000 cosh[6(300t - x )]- 5836657428201600 cosh[4(304t - x)] - 98168384160000 cosh[12(312t - x )]- 400875750475200 cosh[2(316t - x)] - 1645840454049792 cosh[6(316t - x)]- 4913823064550400 cosh[8(316t - x )]- 69778811950848 cosh[1O(316t - x )]- 353575170748800 cosh[6(324t - x )]- 8482134786048 cosh[12(324t - x )]- 6718449497760000 cosh[4(328t - x )]- 166826557497600 cosh[6(332t - x )]- 354442522097664 cosh[8(334t - x)] - 144851259571200 cosh[2(340t - x )]- 438564923596800 cosh[4(340t - x )]- 6165631520848800 cosh[6(340t - x )]- 19084803268608 cosh[10(340t - x )]- 83936853000000 cosh[6(348t - x )]- 25216214229581760 cosh[2(364t - x)] - 2956506448896000 cosh[4(364t - x )]- 306 Appendix B. KdV Solitons via Backlund Transform

381317845708800 cosh[6(364t - x)] - 384820065907200 cosh[8(370t - x)]- 3373899977448000 cosh[6(372t - x)] - 5804890554887424 cosh[4(376t - x )]- 126582878822400 cosh[8(376t - x)] - 3293299379462400 cosh[2(388t - x )]- 151618159300608 cosh[6(396t - x)] - 305918613000000 cosh[4( 400t - x )]- 95067567302400 cosh[8( 400t - x)] - 59640010214400 cosh[8( 406t - x )]- 1275914226787200 cosh[6(412t - x)] - 893282450860800 cosh[6(420t - x)]- 1401142209667200 cosh[2( 436t - x)] - 9477350553775104 cosh[4( 436t - x )]- 571381050240000 cosh[4(448t - x)] - 7302858393600 cosh[8(460t - x)]- 11299191750000 cosh[6(468t - x)] - 368241102028800 cosh[4(472t - x)]- 4239062826014016 cosh[2(484t - x)] - 87875717529600 cosh[4(484t - x)]- 80690064840000 cosh[6(484t - x)] - 97007053651200 cosh[6(492t - x)]- 1824248406236256 cosh[4( 496t - x)] - 16263969163041600 cosh[2(508t - x)] 1281520880640000 cosh[4(508t - x)] - 331783452000000 cosh[4(520t - x )]- 812494863180000 cosh[2(532t - x)] - 36970720617600 cosh[6(532t - x)]- 1173259064577600 cosh[4(544t - x)] - 709273278185472 cosh[4(556t - x)]- 2247025904323200 cosh[2(580t - x)] - 1502683414936704 cosh[2(604t - x )]- 710000121600 cosh[6(612t - x)] - 545363851608576 cosh[4(616t - x )]- 2213927440124400 cosh[2(628t - x)] - 1580764905720000 cosh[2(652t - x )]- 224471126880000 cosh[2(676t - x)] - 12452170500000 cosh[4(688t - x)]- 4954936218248016 cosh[2(724t - x)] - 35702863257600 cosh[4(724t - x)]- 6645601138176 cosh[4(736t - x)] - 565667239737600 cosh[2(748t - x)]- 1080329730480000 cosh[2(796t - x)] - 4089600700416 cosh[4(796t - x )]- 3284939753327232 cosh[2(844t - x)] - 2622379301942400 cosh[2(868t - x )]- 9419178472704 cosh[2(964t - x)] - 875839426862400 cosh[2(988t - x )]- 223311668580000 cosh[2(1012t - x)] - 120869068320000 cosh[2(1036t - x )]- 591849277987968 cosh[2(1084t - x)] - 63008109964800 cosh[2(1228t - x )]- 6973215480000 cosh[2(1372t - x)] - 4998400856064 cosh[2(1444t - x )]- 86451906340800 cosh[12x] - 921387359692800 cosh[8(2t + x )]- 766289987438976 cosh[6(4t + x)] - 461500079040000 cosh[8(8t + x)]- 486101585970000 cosh[6(12t + x)] - 10939261132800 cosh[12(12t + x )]- 68021553600 cosh[18(12t + x)] - 1371314520576 cosh[16(14t + x)]- 412527673958400 cosh[4(20t + x)] - 500457359001600 cosh[8(20t + x)]- 1308982042368 cosh[10(20t + x)] - 5713810502400 cosh[14(20t + x )]- 73735364102400 cosh[12(24t + x)] - 732082129422336 cosh[8(26t + x )]- 826188665702400 cosh[6(28t + x)] - 2834413319337600 cosh[6(36t + x )]- 7313677443072 cosh[12(36t + x)] - 112687848000000 cosh[8(38t + x )]- 56689500067200 cosh[2(44t + x)] -1040997718785024 cosh[4(44t + x)]- 174785616948096 cosh[6(44t + x)] - 380920700160000 cosh[8(44t + x)]- 119482370463456 cosh[10(44t + x)] - 319470049080000 cosh[6(52t + x )]- 3315151046592000 cosh[4(56t + x)] - 298505443332096 cosh[8(56t + x )]- 15069390336 cosh[16(59t + x)] - 856823325758400 cosh[6(60t + x)]- 558320911948800 cosh[8(62t + x)] - 1996996309862400 cosh[4(68t + x)]- 329642913600 cosh[14(68t + x)] - 385874538940800 cosh[6(76t + x)]- 402532261840800 cosh[4(80t + x)] - 1907219714400 cosh[12(80t + x )]- 2490573014721792 cosh[6(84t + x)] - 1562751590400 cosh[12(84t + x )]- 3281778339840000 cosh[4(92t + x)] - 50904857203200 cosh[8(92t + x )]- 311230489667328 cosh[4(104t + x)] - 479960082201600 cosh[6(108t + x)]- 5804960407424640 cosh[2(116t + x)] - 675972544665600 cosh[4(116t + x)]- 805527410688 cosh[8(116t + x)] - 2531657576448 cosh[10(116t + x )]- 92774102252832 cosh[6(124t + x)] - 4056375921998400 cosh[4(128t + x )]- B.7. Seven Solitons 307

21976194240000 cosh[8(128t + x)] - 327190543680000 cosh[6(132t + x )]- 51277786560000 cosh[8(134t + x)] - 2777210426944800 cosh[2(140t + x )]- 731367744307200 cosh[4(140t + x)] - 39190461133824 cosh[8(146t + x )]- 14444353123200 cosh[6(156t + x)] - 486861178003200 cosh[2(164t + x )]- 3253525211768832 cosh[4(164t + x)] - 124633509000000 cosh[6(180t + x )]- 1438441804800 cosh[8(182t + x)] - 4716877347000000 cosh[2(188t + x)]- 80103228004800 cosh[6(196t + x)] - 1350883319385600 cosh[4(200t + x)]- 199154946454464 cosh[6(204t + x)] - 4377355143638400 cosh[2(212t + x)]- 1077964967321424 cosh[2(236t + x)] - 641271056025600 cosh[4(236t + x)]- 25716257094528 cosh[6(244t + x)] -76357285804800 cosh[4(248t + x)]- 2483906162774400 cosh[2(260t + x)] - 80121541500000 cosh[4(272t + x)]- 3125509285298400 cosh[2(284t + x)] - 789970928947200 cosh[4(308t + x )]- 53873070451200 cosh[4(320t + x)] - 193390509312 cosh[6(324t + x )]- 8644100901836400 cosh[2(332t + x)] - 189030432374784 cosh[4(344t + x)]- 1738836535822848 cosh[2(356t + x)] - 65539004731200 cosh[4(368t + x )]- 140967296870400 cosh[4(380t + x)] - 1442910297628800 cosh[2(404t + x)]- 649296648000000 cosh[2( 428t + x)] - 3857812216562304 cosh[2( 476t + x )]- 4777681708800 cosh[4(488t + x)] - 434366614281600 cosh[2(500t + x )]- 149777752924800 cosh[2(572t + x)] - 839632436042400 cosh[2(620t + x)]- 533142472262400 cosh[2(644t + x)] - 1033760177049600 cosh[2(692t + x)]- 118790120344896 cosh[2(716t + x)] - 719130999787200 cosh[2(764t + x)]- 288389772582912 cosh[2(836t + x)] - 10878216148800 cosh[2(980t + x )]- 35729866972800 cosh[2(1004t + x)] - 119540836800 cosh[2(1340t + x )]- 397207256846400 cosh[2( 4t + 5x)] - 197868158888400 cosh[2(28t + 5x )]- 156250741046400 cosh[2(76t + 5x)] - 12322416178800 cosh[2(124t + 5x )]- 69994178654400 cosh[2(148t + 5x)] - 170053884000000 cosh[2(172t + 5x )]- 2337467932800 cosh[2(364t + 5x)] - 20885233168800 cosh[2(508t + 5x )]- 7499376284400 cosh[2(116t + 7x)] - 1632171340800 cosh[32(133t - x)] ]

[ 6019650 cosh[192t] + 5054400 cosh[336t] + 8494200 cosh [672t]+ 490050 cosh[1344t] + 28 cosh[2(1564t - 13x )]+ 2100 cosh[2(1460t - 11x)] + 1176 cosh[2(1532t - 11x )]+ 271656 cosh[2(1276t - 7x)] + 200200 cosh[2(164t - 5x )]+ 1401400 cosh[2(596t - 5x)] + 5775 cosh[4(656t - 5x )]+ 900900 cosh[2(668t - 5x)] + 14700 cosh[4(728t - 5x )]+ 592900 cosh[2(812t - 5x)] + 3395700 cosh[2(956t - 5x )]+ 415800 cosh[2(1172t - 5x)] + 8316 cosh[6(356t - 3x)]+ 58212 cosh[6(436t - 3x)] + 31752 cosh[6( 476t - 3x )]+ 576576 cosh[8(22t - x)] + 2690688 cosh[4(28t - x)]+ 2270268 cosh[6(28t - x)] + 1716 cosh[14(28t - x )]+ 27027 cosh[12(32t - x)] + 350350 cosh[8(40t - x )]+ 1177176 cosh[2(52t - x)] + 412776 cosh[6(68t - x )]+ 6006 cosh[16(88t - x)] + 2620800 cosh[4(100t - x )]+ 85800 cosh[14(100t - x)] + 1387386 cosh[8(112t - x )]+ 567567 cosh[12(112t - x)] + cosh[28(112t - x )]+ 378 cosh[24(128t - x)] + 103488 cosh[16(133t - x )]+ 7135128 cosh[6(148t - x)] + 5005000 cosh[8(148t - x )]+ 577368 cosh[14(148t - x)] + 1091475 cosh[12{160t - x )]+ 242550 cosh[16(160t - x)] + 7164612 cosh[2(172t - x )]+ 798336 cosh[12(172t - x)] + 247500 cosh[14(172t - x )]+ 308 Appendix B. KdV Solitons via. Backlund Transform

24696 cosh[16(178t - x)] + 5297292 cosh[6(188t - x )]+ 623700 cosh[12(200t - x)] + 2656192 cosh[8(202t - x )]+ 14567553 cosh[4(208t - x)] + 3031182 cosh[8(232t - x )]+ 4573800 cosh[6(260t - x)] + 2102100 cosh[4(280t - x )]+ 116424 cosh[8(292t - x)] + 1047816 cosh[6(308t - x )]+ 6606600 cosh[2(340t - x)] + 1588125 cosh[4(352t - x )]+ 2276736 cosh[4(388t - x)] + 8324316 cosh[2( 412t - x )]+ 1778700 cosh[2(700t - x)] + 737100 cosh[6(20t + x )]+ 4179357 cosh [4(32t + x)] + 9555000 cosh[2(92t + x )]+ 396396 cosh[4(152t + x)] + 2774772 cosh[2(308t + x)J+ 60984cosh[2(668t + x)] ]2 Appendix C

Derivation of the Stationary KdV by G. E. Sarty

Here we derive the stationary Korteweg - deVries equation

A/'1 - ~211 / /' - ~/'"61 = 0 (C.O.I) using an asymptotic analysis similar to the one used in Chapter 6. A Math• ematica program is used this time to help us do some relatively complicated substitutions and simplifications during the course of a manual derivation. In theory, a Mathematica program could be written to do all of the substitu• tions and transformations required in the derivation but since it only needs to be done once, there is no real saving of effort by doing this. In fact, it would take more effort because the program would need to be debugged and some experimentation would be required to get Mathematica to simplify cer• tain mathematical expressions in the appropriate form. However, writing such a program might be a useful exercise in learning how to apply Mathematica to the types of problems we've encountered in these appendices.

309 310 Appendix C. Derivation of the Stationary KdV

We begin with the long wave assumption by assuming that the number [= (H/L)2 is positive and much smaller than 1. Here, as in Chapter 6, His the height of the fluid stream and L is the typical wavelength. The coordinate system (x*, y*) is as defined by Figure 6.1. Since we are interested in the time• independent and forcing free scenario, O'*(x*) and p*(x*), shown in Figure 6.1, are both assumed to be zero. The fluid velocities in the x* and y* directions are given by u· and v* respectively. In the metric system, for example, the gravitational constant 9 has units of meters/second2 and H has units of meters. The quantity ..fill consequently has units of meters/second which are units of velocity. So we can use the quantity ..fill to nondimensionalize the velocities u* and v*. This is done by introducing the nondimensionalized velocities u and v as:

u* u=-- u* = vIfiii u (C.O.2) ..fill and v* v = [-1/2 __ v* = [1/2 vIfiii v. (C.O.3) ..fill This choice of non-dimensionalization is chosen because (as will be seen) it leads to a1/l a1/l u = uc ay and v = -uc ax where U c is the upstream fluid velocity and 1/1 is the normalized stream function which will be defined below. If we had non-dimensionalized u· and v* as u· u = .JgH u· = vlfiiiu and v = [1/2 __v* .JgH then we are be led to a1/l a1/l u=uc -ay and v = -[uc ax instead. Except for that difference, we would otherwise end up with the same results (i.e. with equations (C.O.4), (C.0.5) and (C.0.6) below). Our immediate goal is to define a change of variables (x, y) I-t (,1/1), where ( = x and 1/1 = 1/I(x, y), from the non-dimensionalized position variables x and y to variables defined by the normalized stream function 1/1. The inverse transformation is denoted in symbols by (,1/1) I-t (x, y) where x = ( and y = f(, 1/1). This will transform the fluid domain in the x, y plane into a strip no = R x [0, 1] in the (,1/1 plane. Then we will use the Euler equations and the boundary conditions to derive: 311

fIJI" - 2f/(/.pI(.p + (1 + f/[)I.p.p = 0 in no, } u~(1 + f/l) + [2(1 - 1) - u~JfJ = 0 on 'I/J = 1, (C.O.4) 1=0 on 'I/J = o. After deriving (C.O.4), let

I = 'I/J + f'I/J II (() + f2 h ((, 'I/J) + ... and Uc = c+ fA + .... Then we can get (C.0.5) and (C.0.6)

The stationary KdV equation (equation (C.O.l)) then follows from equation (C.0.6) by differentiating it with respect to (. We now define the change of variables (x, y) H- ((, 'IjJ). Recall that the partial derivative of the potential function <1>* with respect to x*, <1>;., is the velocity u* and that v* = ~.. The stream function \):1* is defined as the harmonic conjugate of the potential function <1>*. That is, <1>* and \):1* are the real and imaginary parts of an analytic function. So we have, as the Cauchy• Riemann equations: (C.0.7) and (C.0.8) Since the units of u* are the same as those of -JiiH (velocity), the units of \):1* / H are also the same as those of -JiiH. In other words, the units of \):1* are the same as those of H-JiiH and so we nondimensionalize the stream function to \):I: \):1* \):1= H-JiiH (C.0.9) The nondimensional coordinates are y* y = - ¢:} y* = H y (C.0.I0) H and fl/2 x= --x * (C.O.11) H We do not deal with time since we are considering a stationary process. Now we normalize \):I. Let 17*(X*) describe the free surface. The downstream condition at x = -00 is then given by 17* ( -00) = H with fluid velocity

u~ == u*(-oo,y*) = \):IZ.(-oo,y*) for all 0::; y* ::; H (C.0.12) 312 Appendix C. Derivation of the Stationary KdV

(note that the lower limit for y* follows from the assumption of a flat bottom, i.e. u*(x*) == 0.) Also, on the top and bottom surfaces, "1)* (x· , H) = C1 and "I). (x· ,0) = C2 respectively where C1 and C2 are constants. Integrating equation (C.0.12) with respect to y* gives:

"I). (-00, y*) = u:y* + K (C.0.13) where K is an arbitrary constant. Using equation (C.0.13):

at y* = H "1)·(-00, H) = u:H + K = C1

at = 0 "1)*(-00,0) = K = C2. So choose K = 0 and deduce that

0<_ "I). _< u· c H (C.O.14) between the top and bottom surfaces. Dividing equation (C.O.14) by H.../ill and using equation (C.O.9) gives

(C.O.15)

With equation (C.O.2), u; = VgH U c , then inequality (C.O.15) implies (C.O.16)

Defining (C.O.17) and dividing equation (C.O.16) by U c yield

05: "p 5: 1, so "p is the normalized "I). Using equation (C.O.17) in equation (C.O.9) gives "1)* "p = ---== (C.O.1S) ucHVgH Now, from equations (C.O.2), (C.O.7), (C.O.1S) and (C.O.10), we have

=> (C.O.19) From equations (C.O.3), (C.O.S), (C.O.1S) and (c.o.n) we have

1 1 (.1/2 V = _(.-1/2 J::lY "I);. = _(.-1/2 J::lYucH Viii"pzx z• = _(.-1/2uc H-"pz ygH ygu H 313

(C.0.20)

Now we can look at the transformation (x, y) I-t (, w) where ( = x and w = tP(x, y). Note that we have replaced the notation tP = tP(x, y) with the more precise w = tP(x, y). This will avoid confusion when we need to clearly separate the domain of the inverse transformation from the functions that define the transformation. The inverse transformation is given by (, w) I-t (x, y) where x = ( and y = f(', w). So w = tP(x, y) = tP(, f(, w)). Put

with respect to ,:

We will assume that f have second order continuous derivatives so that equa• tions (C.0.25) and (C.0.26) are the same. Since c)* is harmonic, .6.*c)* = c);oz:o + c)Zoyo = 0 in the fluid domain. V*C)* = 0 is Euler's equation for incompressible, irrotational fluid flow. Now

\);-3:- = (-v* )z:o (_f.l/2,fiH v)z:o (f.l/2,fiH uctPz:)z:o f.l/2JgH uctPz:z:xz:o

1/2,fiH £1/2 = f. 9 uc-U tP z:z: = f.Uc AtPz:z: where we have used equations (C.O.B), (C.0.3), (C.O.20) and (C.O.ll). Also,

WZoyo (u*)yo = (,fiH u)yo = (,fiH uctPy)yo = JgH uctPyyyyo = ucAtPyy where we have used equations (C.0.7), (C.0.2), (C.0.19) and (C.0.10). Hence

wc.j"ftPz:z: + uc~tPyy = 0, or

, or (C.0.27) 314 Appendix C. Derivation of the Stationary KdV

Substituting equation (C.0.27) into equation (C.0.22) yields

1 2 - -tPyy + 2tPxyf( + tPyyf( + tPyf" = o. (C.0.2S) f Equation (C.0.24) implies tP -- tPyy f~ (C.0.29) y - fww and equation (C.0.29) in equation (C.0.25) gives

tPxy = tPyy (fj~~w - f( ) . (C.0.30)

Substituting equation (C.0.29) and (C.0.30) into equation (C.0.2S), we get

1.1. 2.1. (fwf(w ., )., .1..,2 tPyyf~., 0 --o/yy + o/yy -.,-- - J( J( + o/yyJ( - --.,-J" = . f JWW Jww This can be simplified to

ff~f" - 2ffwf(wf( + (1 + ff{)fww = o. (C.0.31) This is the exact differential equation that f((, w) must satisfy. Now we deal with the boundary conditions. On the bottom w(= tP) = 0 and y = O. Since y = f((, w) we have f((,O) = 0, i.e. the lower boundary condition for equation (C.0.31) is

f = 0 on w = O. (C.0.32) On the top, w = 1 and y* = 7]*(x*) + H or using equation (C.O.lO), Hy = 7]* + H ~ y = ~ + 1. But f((, 1) = y so * f((, 1) = ~ + 1 ¢!? 7]* = H(J - 1) on w = 1. (C.0.33)

Also on the top, Bernoulli's equation must be used

(C.0.34)

Now, II 'V'*<)* 112= (<);.)2 + (<);.)2 = (W;.)2 + (W;.)2 by equations (C.0.7) and (C.O.S). Differentiate equation (C.0.1S)

with respect to y*: w;. = ucH../iiitPyyy· = uc../iii tPy where we have used equations (C.0.10) and (C.O.11). So

(W;.)2 = f.U~gH tP; (C.0.35) 315 and (W;.)2 = u~gH t/J~. (C.O.36) Substituting equations (C.O.33), (C.O.35) and (C.O.36) into equation (C.O.34) gives 1 1 2U~gH(ft/J; + t/J~) + g(H + H(I - 1)) = 2gHu~ + gH

¢? U~(ft/J; + t/J~) + 2(1 - 1) = u~

¢? u~ (f (- J~ )2 + I~) + 2(1 - 1) - u~ = 0

¢? u~(f/t + 1) + 1~2(1 - 1) - I~u~ = 0

¢? u~(1 + fin + [2(1 - 1) - u~]!~ = 0 on w = 1 (C.O.37) where we have used equations (C.O.21) and (C.O.23). This is the upper bound• ary condition for equation (C.O.31). Together, equations (C.O.31), (C.O.32) and (C.O.37) define the boundary value problem (C.O.4). Now, as promised, we let

(C.O.38) and (C.O.39) Then we substitute equations (C.O.38) and (C.O.39) into the expressions of (C.O.4). For this, the following Mathematica code is useful.

(* ******************************************************************* *) (* * * *) (* * skdv.m -- IIathematica code 1:or use in deriving * *) (* * the stationary KdV equation. lote that e is used for * *) (* * epsilon, z for zeta, 1 1:or lambda and u2 for u squared. * *) (* * c * *) (* * * *) (* * Programmed by: * *) (* * G. Sarty June 1992 * *) (* * * *) (* ******************************************************************* *) f[z_,v_] := v + e v 1:1[z] + e-2 1:2[z,v] + 0[e]-3 u2 = c-2 + 2 e c 1 + e-2 1-2 0[e]-3 pde := e D[f[z,v] ,v]-2 D[D[f[z,v] ,z] ,z] - 2 e D[f[z,v] ,z] D[f,v] D[D[f[z,v] ,z] ,v] + (1 + e D[1: [z ,v] ,z]-2) D[D[f[z,v] ,v] ,v] == 0 top := ExpandAll[ u2 (1 + e D[1: [z ,v] ,z]-2) + ( 2(f[z,v] - 1) - u2) D[1:[z.v] .v]-2 == 0] 316 Appendix C. Derivation of the Stationary KdV

Using the code we find that (C.O.4) becomes [w(ft},d() + (h)ww(, W)]f2 + 0(f3) = 0 in no

[ - 2 + 2w] + [6wft () - 4ft () - 2c2ft ()]f+ [- 4c,Xft() - 2ff() - c2(ft}2() + 6wff()+ on w = 1 (C.0.40) 212(, w) - 4(h)w(' w) - 2c2(h)w(, w) +4w(h)w(,w)]f2 + 0(f3) = 0

on w = 0

The fO terms from the top boundary condition give

- 2 + 2w = 0 on w = 1 which shows only that we have not made any mistakes here. The f1 terms from the top boundary condition give

which implies that c2 = 1 if ft # O. Using c = 1 and w = 1 in the f2 terms of that same top boundary condition we get 212(,1) - 2(h)w(' 1) + 3ff() - 4,Xft() = O. (C.0.41)

Finally, the f2 terms of the partial differential equation itself gives:

wf~'() + (h)ww('w) = 0 for (,w) E nO. (C.0.42) Integrate equation (C.0.42) with respect to w to get

(h)w(' w) = -~f~'()w2 + K 1 () (C.0.43) where K1 is an arbitrary function of (. Integrate equation (C.0.43) with respect to w again to get

(C.0.44) where K2 is another arbitrary function of (. The f2 terms of the bottom boundary condition give 12(,0) = 0 so K 2 () = 0 when this information is put into equation (C.0.44). Thus, equations (C.0.43) and (C.0.44) become

12(,1) = -!ff'() K 1 (), } + (C.0.45) (h)w(' 1) = -!ff'() + K1()' Substituting (C.0.45) into (C.0.41) gives:

-~ff'() + 2K1 () + ff'() - 2K1 () + 3ff() - 4,Xft () = 0 and this simplifies to equation (C.0.6). As was previously mentioned, equation (C.0.6) can be differentiated once to yield the stationary Korteweg-deVries equation (C.O.l) and our derivation is complete. Bibliography

[1] M. J. Ablowitz and H. Segur (1981), Solitons and Inverse Scat• tering Transform, SIAM, Philadelphia, Pennsylvania. [4] (This [4] means that this entry is cited in Chapter 4. The same rule applies to other entries in this bibliography.)

[2] T. R. Akylas (1984), On the excitation oflong nonlinear water waves by a moving pressure distribution, J. Fluid Mech. 141,455-466. [6]

[3] P. G. Baines (1987), Upstream blocking and airflow over mountains, Ann. Rev. Fluid. Mech. 19, 75-97. [6]

[4] A. Barone (1974), Josephson Effect: Achievements and Trends, World Scientific, Singapore. [7]

[5] A. Barone, F. Esposito, C. J. Magee and A. C. Scott (1971), Theory and applications of the sine-Gordon equation, Rivista Del Nuovo Cimento 1, 227-267. [7]

[6] T. B. Benjamin (1957), Wave formation in a laminar flow down an in• clined plate, J. Fluid Mech. 2, 554-574. [5]

[7] T. B. Benjamin (1967), Instability of periodic wavetrains in nonlinear dispersive systems, Proc. Roy. Soc. Lond. A 229, 59-75. [8]

[8] T. B. Benjamin and J. F. Feir (1967), The disintegration of wave trains on deep water, Part 1. Theory, J. Fluid Mech. 27,417 - 430. [8]

[9] A. Bishop and T. Schneider (1978), Solitons and Condensed Matter Physics, Springer-Verlag, New York. [7]

[10] G. Bricogne (1984), Maximum entropy and the foundations of direct methods, Acta Cryst. A 40, 410-445. [9]

[11] R. Camassa and T. Y. Wu (1991), Stability offorced solitary waves, Phil. Trans. R. Roy. Lond. A 337,429-466. [6]

[12] S. Chandrasekhar (1981), Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York. [8]

317 318 Bibliography

[13] H. H. Chen (1974), General derivation of Biicklund transformations from inverse scattering problems, Phys. Rev. Lett. 33, 925-928. [4] [14] J. D. Cole (1951), On a quasilinear parabolic equation occurring in aero• dynamics, Q. J. Math. 9, 225-236. [5] [15] D. Coles (1965), Transition in circular Couette flow, J. Fluid Mech. 21, 385-425. [8] [16] A. D. D. Craik (1985), Wave Interactions and Fluid Flows, Cambridge University Press, New York. [9] [17] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris (1982), Solitons and Nonlinear Wave Equations, Academic Press, New York. [4, 7] [18] P. G. Drazin (1983), Solitons, Cambridge University Press, London. [4] [19] P. G. Drazin and R. S. Johnson (1989), Solitons: An Introduction, Cam• bridge University Press, New York. [4, 7] [20] P. G. Drazin and W. H. Reid (1981), Hydrodynamic stability, Cambridge University Press, New York, Chapters 1, 2 and 3. [8] [21] A. Erdelyi (1956), Asymptotic Expansions, Dover, New York, Chapter 1. [1] [22] J. C. Gallop (1991), SQUIDS, the Josephson Effects and Superconducting Electronics, Adam Hilger, New York. [7] [23] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura (1974), Korteweg-de Vries equation and generalizations. VI. Methods for exact solution, Comm. Pure Appl. Math. 27, 97-133. [4] [24] I. M. Gel'fand and B. M. Levitan (1955), On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. Ser. 2, 253-304. [4] [25] R. Grimshaw (1987), Resonant forcing of barotropic coast ally trapped waves, J. Phys. Oceanagr. 17,53-65. [26] R. H. J. Grimshaw and N. Smyth (1986), Resonant flow of a stratified fluid over topography, J. Fluid Mech. 169,429-464. [6] [27] M. E. Gurtin (1975), On the breaking of water waves on a sloping beach of arbitrary shape, Q. Appl. Math. 33, 187-189. [3]

[28] J. 1. Hammack and H. Segur (1974), The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments, J. Fluid Mech. 65, 289-314. [3, 4] [29] H. Hasimoto and H. Ono (1972), Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33, 805-811. [7] Bibliography 319

[30] H. Hauptman (1975), A joint probability distribution of seven structure factors, Acta Cryst. A 31, 671-679. [9]

[31] H. Hauptman (1975), A new method in the probabilistic theory of the structure invariants, Acta Cryst. A 31, 680-687. [9]

[32] H. Hauptman (1983), The phase problem of x-ray crystallography, Proc. Indian Acad. Sci. (Chern. Sci.) 92, 291-321. [9]

[33] H. Hauptman (1985), Phase problems of X-ray crystallography, in "En• cyclopedia of Statistcal Sciences", Vol. 6, ed. by Kotz and Johnson, John Wiley & Sons, New York, pp. 702-709. [9]

[34] E. Hopf (1950), The partial differential equation Ut + UU", = jlU",,,,, Com• mun. Pure Appl. Math. 3, 201-230. [5]

[35] E. Infeld and G. Rowlands (1990), Nonlinear Waves, Solitons and Chaos, Cambridge University Press, New York, Chapter 1. [8]

[36] A. Jeffery and T. Kakutani (1970), Stability of the Burgers shock wave and the Korteweg-de Vries soliton, Indiana Univ. Math. J. 20, 463-468. [5] [37] R. S. Johnson (1973) , On the development of a solitary wave moving over an uneven bottom, Proc. Camb. Phil. Soc.73, 183-203. [4]

[38] D. J. Kaup and P. J. Hansen (1986), The forced nonlinear Schrodinger equation, Physica D 18, 77-84. [7]

[39] J. B. Keller and S. I. Rubinow (1960), Asymptotic solution of eigenvalue problems, Ann. Phys. 9, 24-75. [1]

[40] P. K. Kundu (1990), Fluid Mechanics, Academic Press, New York, Chap• ter 11. [8]

[41] T. Kuusela, J. Hietarinta, K. Kokko and R. Laiho (1987) , Soliton exper• iments in a nonlinear electrical transmission line, Eur. J. Phys. 8, 27-33. [4]

[42] G. L. Lamb (1980), Elements of Soliton Theory, John Wiley, New York. [4, 7]

[43] H. Lamb (1945), Hydrodynamics, 6th ed., Dover Publications, New York. [6, 8]

[44] L. Landau and E. Lifschitz (1958), Quantum Mechanics, Nonrelativistic Theory, Pergamon Press, New York. [4]

[45] P. D. Lax (1968), Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21, 467-490. [4] 320 Bibliography

[46] S. J. Lee, G. T. Yates and T. Y. Wu (1989), Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances, J. Fluid Mech. 199, 569-593. [6] [47] C. C. Lin (1955), The Theory of hydrodynamic stability, Cambridge Uni• versity Press, New York. [8] [48] W. F. Lucas (1978), Models in Applied Mathematics, Springer-Verlag, New York, Vol. 1. [2]

[49] D. Ludwig (1983), Parsimonious asymptotics, SIAM J. Appl. Math. 43, 664 - 672. [1] [50] J. Marsden and T. J. R. Hughs (1983), Mathematical Foundation of Elasticity, Prentice-Hall, New York, Chapter 2. [2]

[51] A. McPherson (1982), Preparation and Analysis of Protein Crystals, John Wiley & Sons, New York Chapters 5, 7, and 10. [9] [52] L. F. McGoldrick (1965), Resonant interactions among capillary-gravity waves, J. Fluid Mech. 21, 305-331. [9] [53] R. E. Meyer (1982), Introduction to Mathematical Fluid Dynamics, Dover, New York, Chapters 1 - 3, and Chapter 6. [2] [54] J. W. Miles (1986), Stationary, transcritical channel flow, J. Fluid Mech. 162, 489-499. [6]

[55] R. M. Miura (1976), The Korteweg-de Vries equation: A survey of results, SIAM Review 18, 412-459. [4]

[56] L. F. Mollenauer and R. H. Stolen (1982), Solitons in optical fibers, Fiberoptic Technology, April, 1982, 193-198. [7]

[57] G. H. Nayfeh (1973), Perturbation Methods, John Wiley, New York, Chapters 1 - 4. [1] [58] A. H. Nayfeh and D. T. Mook (1979), Nonlinear Oscillations, John Wiley, New York, Chapter 4. [1]

[59] A. C. Newell (1985), Solitons in Mathematics and Physics, SIAM, Philadelphia, Pennsylvania. [4]

[60] F. W. J. Olver (1974), Asymptotics and Special Functions, Academic Press, New York. [1]

[61] Shih- I Pai (1956), Viscous Flow Theory, I - Laminar Flow, D. van Nos• trand Co. Inc., New York. [5]

[62] R. D. Parmentier (1978), Fluxions in long Josephson junction, in "Soli• tons in Action" ed. by K. Lonngren and A. Scott, Academic Press, New York, pp. 173 - 199. [7] Bibliography 321

[63] O. M. Phillips (1967), Theoretical and experimental studies of gravity wave interactions, Proc. Roy. Soc. A 299, 104 - 119. [9]

[64] O. M. Phillips (1974), Wave interactions, in "Nonlinear Waves" ed. by S. Leibovich and A. R. Seebass, Cornell University Press, Ithaca, New York, pp. 188-211. [9]

[65] C. Rebbi and G. Soliani (1984), Solitons and Particles, World Scientific Publishing Co., Singapore. [4, 7]

[66] M. Sachs (1963), Solid State Theory, McGraw-Hill, New York, Chapters 2 and 4. [9]

[67] D. Saint-James, E. J. Thomas and G. Sarma (1969), Type II Supercon• ductivity, Pergamon Press, Toronto. [7]

[68] M. C. Shen and R. E. Meyer (1963), Climb of a bore on a beach, J. Fluid Mech. 16, 113-126. [3]

[69] M. C. Shen and S. M. Sun (1987), Critical viscous surface waves over an incline, Wave Motion 9, 323-332. [5]

[70] S. S. Shen (1989), Disturbed critical surface waves in a channel of arbi• trary cross section, J. Appl. Math. Phys. (ZAMP) 40, 216-229. [6]

[71] S. S. Shen (1990), Blocking of solitary pulses in a nonlinear fiber, Wave Motion 12, 551-557. [7]

[72] [14] S. S. Shen (1991), Locally forced critical surface waves in channels of arbitrary cross section, J. Appl. Math. Phys. (ZAMP) 42, 122-138. [6]

[73] S. S. Shen (1992), Forced solitary waves and hydraulic falls in two-layer flows, J. Fluid Mech. 234,583 - 612. [1,6]

[74] S. S. Shen and M. C. Shen (1990), A new equilibrium of subcritical flow over an obstruction in a channel of arbitrary cross section, Euro. J. Mech. B/Fluids 9,59-74. [6]

[75] J. Smoller (1983), Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, Chapters 15-18. [2]

[76] J. J. Stoker (1957), Water Waves: the Mathematical Theory with appli• cations, Interscience, New York. [3]

[77] M. Van Dyke (1982), An Album of Fluid Motion, The Parabolic Press, Stanford, California. [2, 8]

[78] H. D. Wahlquist and F. B. Estabrook (1973) , Backlund transformation from solitons of the Korteweg-de Vries equation, Phys. Rev. Lett. 31, 1386-1390. [4] 322 Bibliography

[79] G. B. Whitham (1974), Linear and Nonlinear Waves, John Wiley, New York, Part I. [2, 3, 5, 6, 7] [80] M. M. Woolfson (1987), Direct methods - from birth to maturity, Acta Cryst. A 43, 593-612. [9]

[81] T. Yao-Tsu Wu (1987), Generation of upstream advancing solitons by moving disturbances, J. Fluid Mech. 184, 75-100. [6]

[82] C. S. Yih (1963) , Stability of liquid flow down an inclined plate, Phys. Fluids 6, 321-334. [5] [83] C. S. Yih (1988), Fluid Mechanics, West River Press, Ann Arbor, Michi• gan, Chapter 9. [8]

[84] N. J. Zabusky and M. D. Kruskal (1965), Interaction of "solitons" in a collisionless plasma and the recurrence of the initial states, Phys. Rev. Lett., 15, 240-243. [4] Index

Ablowitz, 122, 317 Bragg, 262 Akylas, 186, 317 Bragg's law, 259 Ampere's law, 209 breaking time, 43 antisoliton, 200 breather solution, 202 asymptotic approximation, 5 Bricogne, 275, 317 best, 4 bump, 147 parsimonious, 4 buoyancy force, 231 asymptotic expansion, 1,7,127,210 Burgers equation, 129 matched, 10 Burgers shock wave, 131, 135, 141 singular, 7 uniform, 7 Camassa, 186 asymptotic sequence, 6 carrier wave, 206 singular, 7 Cauchy-Riemann equations, 106 uniform, 7 cell, 262 asymptotic solution, 10, 14 centrifugal force, 237 atomic number, 265 Chan-Kerkhoven scheme, 174 Chandrasekhar, 220, 241, 246, 317 Backlund transform, 105 characteristic method, 35, 39 Benard problem, 231 characteristics, 36, 39 Baines, 186, 317 Chen, 115, 121,318 Barone, 216, 217, 317 cnoidal wave, 164 basic state, 233 Cole, 145, 318 beach, 62 Cole-Hopf transformation, 131 bell shape, 202 Coles, 246, 318 Benjamin, 144, 243,246,317 collision process, 200 Benjamin-Feir instability, 255 commutator, 103 Bernoulli equation, 56, 149 complete elliptic integral, 167 Bessel function, 271 complex conjugate, 227 bifurcation, 239 conjugate equation, 107 bifurcation diagram, 19, 161 conservation Bishop, 216, 317 mass, 30 boundary conditions, 54 momentum, 31 dynamical, 54 conservation law, 25, 35 geometrical, 54 hyperbolic, 35 kinematic, 56 constitution relation, 32 boundary layer, 8, 125 continuity equation, 31 Boussinesq equation, 68, 251 Couette flow, 222

323 324 Index

Coulomb's law, 209 subcritical, 162 coupled pendulums, 190 supercritical, 153 Craik, 258, 275, 318 transcritical, 168 crystal, 258 Forsyth, 131 unit cells, 263 four-wave interactions, 255 crystal dislocation, 191 Fourier representation, 248 crystallography, 258 Fourier pair, 265 Fourier transform, 132, 174 damping force, 12 free surface, 34 decaying, 14 deformation rate, 32 Gallop, 217, 318 depression, 168 Gardner, 75, 121, 318 diffraction index, 209 Gaussian pulse, 206 diffraction pattern, 259 Gel'fand, 83, 115, 121,318 diffraction plane, 259 Gibbon, 121, 216, 318 dimension analysis, 137 Ginzburg-Landau, 76 direct method, 262 gratings, 263 dispersion relation, 65, 251 Greene, 75, 121, 318 dispersive waves, 64 Grimshaw, 186,318 Dodd, 76,121,216,318 group velocity, 208, 210 Drazin, 76, 121, 217, 220,226, 246, Gurtin, 74, 318 318 Hamilton-Jacobi, 65 eikonal equation, 65 Hamiltonian operator, 80 Eilbeck, 121, 216, 318 Hammack, 74, 121, 318 electromagnetic waves, 206 Hansen, 216, 319 electron density, 248 harmonic function, 55 energy transfer, 252 harmonics, 14 entropy, 46 Hasimoto, 216, 318 entropy condition, 48 Hauptman, 262, 270, 275, 319 Erdelyi, 23, 318 heat equation, 132 Esposito, 216, 317 Hietarinta, 122, 319 Estabrook, 105, 110, 121, 321 Hopf, 145, 319 Euler equations, 32, 54 Hughs, 51, 320 Eulerian, 29 hydraulic fall, 164, 167 evolution equation, 102 hydrostatic, 61 external pressure, 147 hyperbolic, 25 hyppac.m, 283 Farady's law, 209 Feir, 243, 246, 317 inductance, 195 Fermi-Pasta-Ulam,76 Infeld, 246, 319 fission condition, 120 inner solution, 9 flow instability, 219 intensity, 262 flux, 26 intensity data, 265 forced Korteweg-de Vries, 147 interaction equations, 252, 257 locally, 154 interface, 142, 143 non-locally, 154 inverse scattering, 82 Index 325

Jacobian, 29 Lipschitz condition, 166 Jeffery, 142, 145, 319 Lucas, 51, 320 Johnson, 76, 121,217,318,319 Ludwig, 23, 320 Josephson junction, 191, 194 Josephson penetration length, 195 Magee, 216, 317 Josephson plasma frequency, 195 magnetic monopoles, 209 marginal stability, 241 Kakutani, 142, 145, 319 Marsden, 51,320 Karle, 262, 270 mass conservation, 28 Kaup, 216, 319 Mathematica, 2, 115, 175,277 KdV hierarchy, 102, 105 Maxwell equations, 209 Keller, 4 McGoldrick, 254, 275, 320 Kelvin-Helmholtz instability, 225 McPherson, 275,320 kinematic condition, 34 Meyer, 51, 74,320, 321 kinematic viscosity, 124 Michell, 243 kink, 198 microwave frequency, 193 Kirchoff's law, 195 Miles, 156 Klein-Gordon, 65 Miller indices, 263 Kokko, 122, 319 minimization formulation, 267 Korteweg-de Vries, 65, 82 Miura, 75, 98, 121, 318, 320 equation, 69 modes, 249 Kruskal, 75, 76, 121, 318, 322 newer, 249 Kundu, 236, 246,319 primary, 249 Kuusela, 122, 319 modulated wave, 211 Mollenauer, 208,216,320 L-C circuits, 195 Mook, 23, 320 Lagrangian coordinates, 29 Morris, 121, 216, 318 multiple scales, 11 Laiho, 122, 319 Lamb, 78, 121, 186, 216, 246, 319 natural oscillation, 15 laminar flow, 223 Navier-Stokes equations, 32, 124, 239 Landau, 121, 319 Nayfeh, 23, 320 Laplace equation, 56, 106, 226 neighborhood principle, 270 lattice, 259 Newell, 76, 121, 320 reciprocal, 264 Newtonian fluid, 32, 143 Laue spots, 259 no slip condition, 124 Lax, 121, 319 nonlinear Schrodinger, 76, 189, 211 Lax equation, 103 forced, 212 Lax pair, 102 nonlinear Schrodinger, 65 Lee, 187, 320 normal mode, 223 Legendre equation, 89 numerical noise, 183, 220 Legendre polynomials, 89 Levitan, 83, 115, 121, 318 Olver, 23, 320 Lifschitz, 121, 319 one-step state, 200 Lin, 246, 320 Ono, 216, 318 linearized stability, 225 operating curve, 19 Liouville equation, 107 oscillator, 12 326 Index

harmonic, 249 Rubinow,4 nonlinear, 18 Russell,76 stationary, 19 outer solution, 9 Sachs, 275, 321 Saint-James, 217, 321 Pai, 145, 320 Sarma, 217, 321 paradox, 156 scales, 14 Parmentier, 217, 320 hidden, 15 partial sum, 3 spatial, 14 pat phenomenon, 11 temporal, 14 phase, 210 scattering, 81 phase difference, 194 scattering data, 78 phase problem, 262, 265 scattering method, 78 direct method, 267 scattering potential, 78 neighborhood principle, 270 Schneider, 216, 317 probabilistic distribution, 267 Schrodinger equation, 80 phase velocity, 65, 66 Scott, 216, 317 Phillips, 254, 275, 321 secular terms, 14, 252 Planck's constant, 80 Segur, 74, 121, 122, 317, 318 plane waves, 210, 247 self-focusing, 209 polarization vector, 209 seminvariant, 270 potential flow, 54 separation of variables, 14, 91 potential theory, 255 nonlinear, 14 Prandtl number, 234 shallow water equations, 60 probability distribution, 270 shallow water waves, 62 pulse broadening, 206 Shen, 4, 74, 144, 186,217,321 quantum mechanics, 79 shock, 45 quartet invariant, 273 rarefaction, 47 strength, 48 Rankine-Hugoniot condition, 34, 46, thickness, 51 49 shock path, 46 rarefaction, 48 shoreline, 63 Rayleigh, 230, 237 side band frequencies, 245 Rayleigh instability, 231 side band instability, 245 Rayleigh number, 232 signal broadening, 208 critical, 233 Silicon Graphics, 2 Rebbi, 76, 121, 216,321 similarity solution, 137 reflection coefficient, 77 simple wave, 47 Reid, 220, 226, 246, 318 sine-Gordon, 76, 107 resonance, 14 sine-Gordon equation, 189 resonance condition, 248, 250 sinusoidal, 20 Reynolds number, 125 Smoller, 51, 321 critical, 129 Smyth, 187,318 Riccati equation, 83, 132 Soliani, 76, 121, 216, 321 Rodriques formula, 89 solipac.m, 284 Rowlands, 246, 319 solitary wave, 153 Index 327

stability, 183 . traveling wave, 28, 36 soliton, 76 triangular wave, 131, 137, 140 N-soliton, 85 triplet invariant, 270 I-soliton, 87 tunneling current, 193, 194 2-soliton, 88 turbulent flow, 223 3-soliton, 93 two-step state, 200 fission, 119 KdV, 85 ultraviolet radiation frequency, 193 ladder, 110 uncertainty principle, 79 N-soliton, 97 van der Pol, 15 radiated, 177 Van Dyke, 45, 51, 246, 321 spectral scheme, 178 viscosity, 32, 124 spectrum, 75 vortex sheet, 229 discrete negative, 99 vorticity, 54 positive continuous, 99 vorticity equation, 55 SQUIDS, 193 stability, 20 Wahlquist, 105, 110, 121, 321 standard KdV, 82, 88, 102, 103 wake, 99, 168 state equation, 26 wave function, 79 Stoker, 74, 321 wave number, 227 Stokes waves, 242 white noise, 229 Stolen, 208, 216, 320 Whitham, 51, 74, 133, 145, 186,216, stratification, 55 322 stress tensor, 142, 143 Woolfson, 275, 322 structure factors, 265 Wu, 168, 186, 322 structure invariants, 269 conditions, 269 X-ray crystallography, 262 Sun, 144,321 SUN Sparcstation, 284 Yang-Mills, 76 superconductor layer, 194 Yates, 187,320 superposition principle, 110 Yih, 144, 246, 322 nonlinear, 108 surface tension, 143, 231 Zabusky, 76,121,322

tangent formula, 271 Taylor expansion, 60 Taylor number, 238 critical, 242 Taylor problem, 237 tension coefficient, 142 thermal instability, 231 Thomas, 217, 321 traffic flow, 26 transcritical flows, 167 transmission coefficient, 77 transport theorem, 29 Nonlinear Topics in the Mathematical Sciences An International Book Series dealing with Past, Current and Future Advances and Developments in the Mathematics ofNonlinear Science

1. M.S. Berger: Mathematical Structures of Nonlinear Science. An Introduction. 1990 ISBN 0-7923-0728-3 2. P.G. Bakker: Bifurcations in Flow Patterns. Some Applications of the Quantitative Theory of Differential Equations in Fluid Dynamics. 1991 ISBN 0-7923-1428-X 3. S.S. Shen: A Course on Nonlinear Waves. 1993 ISBN 0-7923-2292-4

KLUWER ACADEMIC PUBLISHERS - DORDRECHT I BOSTON I LONDON