IC/88/162 INTERNAL REPORT Nonreiativistic Limit (Limited Distribution) 1.

International Atomic Energy Agency There b no systematic method, up to now, of classifying an arbitrary nonlinear

and partial differential equation (NLPDE) as being CMntegrable or S-integrable (Calogero, Educational Scientific and Cultural Organitatioit 1988). If a given NLPDE is integrable, then we have a number of gyitematic methods, such JITTERNATIONAL CENTRE FOR THEORETICAL PHYSICS as the inverse tcattering tranifonn (1ST), Backlund transform (BT), Oarboux transform

(DT), to construct exact solutions. When we are not in a position to judge the Integrabtlity

then we can use other special techniques like Hirota's bilinear operator, similarity and

NONLINEAR DE BROGLIE WAVES symmetry analysis, to find the exact solutions. Meat of the NLPDE are not integrable and AND THE RELATION BETWEEN so their solutions are difficult to End. Hirota's method will not always give exact solutions, EELATIVISTIC AND NONRELATIVISTIC SOWTONS* for example, it fails for equation of motion of the \j* Lagrangian field theory. Similarity

A.O.Barai" and B.V. Baby analysis or group theoretical methods are also not always available. Even if we are able to

International Centre for Theoretical Physic*, Trieste, Italy. Gnd some symmetries, the reduced equations are nonlinear and difficult to solve in most

The purpose of this work is to try to relate ecpfe nonintegrable class of equation

to the well-known integrable class of NLPDEs using the nonrelativistic limiting procedure MIRAMARE-TRIESTE and so this method may give some information about the poisible structure of the exact July 1988 solutions for a nonintegrable class of equations if we know the respective integrable class

of NLPDEs.

In this paper we show that the well-known envelope soliton and kink solutions of

the nonlinear Schrodinger equations (NLSE) are the nonrelativistic (NR) limit of the corre-

sponding solutions of the nonlinear Klein-Gordon equation (NLKGE). To our knowledge,

To be submitted for publication. the NR limit of the nonlinear field equations and their solutions have not been discussed Permanent address: Department of Physics, University of Colorado, Boulder, CO 80309, U.S.A. before. As a result of this NR limiting process, we obtain new solutions for the 'double

2 ••mm;

nonlinear Schrodinger equation* (ONLSE)' and exact solution* for the perturbed NLSE or Another interesting fact is that the perturbed DNLSE or Generalised nonlinear

Generalised NLSE. We thall farther show that the dispersion relation* for tht phase of the Schrodinger equation (GNLSE) (Cowan et al., 1986; Tussynski et si., 1984; Gagnon and lolution differ* from the de Broglie relation by a negatire internal energy «o of the soliton Winternlti 1987 and 1988a,b)

•oration. In the limit *o -• 0, these folution* disappear and we have the usual plane ware*.

Conversely, we alto conjecture that corresponding to all the envelope solution* of NLSE*, where /8 is the perturbation coefficient, is the NR limit of the same KGB (Eq. (1.2)) but there are envelope solution* of NLKGEs, such that their NR limit exactly reduce to the with a alight modification in the coefficient of linear term* solution* of the NLSE*. This method provides new exact solutions for the NLKGE family. D * + (mV + 2m?P)+ + •ea^|*|a» + tc'4\t\*' - 0 (1.5) We further discuss the role of the internal degrees of freedom on the de Broglie phase. The above correspondence assure* us that we will be able to find exact solution* The NR limit (Burnt and Rac.ska, 1980; Barut and Van Hnele, 1985) or the of DNLSE and GNLSE if we know the solutions of NLKGEs. However it Is important to transformation for constructing an NR Schrodinger equation from an NLKGE of motion, note that all known solutions of NLKGEs are without an envelope phase factor except for b defined by some recent results (Barut and Rusu, 1987; Baby and Barut, 1987 and 1988), whereas the m ttwexP (l^) *(*, t) -* *% t) (1.1) NLSE has the well-known envelope type solutions! Moreover, if we take the NR limit of the

where *(£,<) i* the field variable of the NLKGE. known solution* of the NLKGs we get vacuum solutions of NLSEs. This suggests that we

hare to modify the structure of the solutions of the NLKO& so that envelope type solutions - 0 (1.2) are the relativtstic form of the corresponding nonrelativistic envelope solutions of NLSEs.

and v Is any real number, a, b, are real parameters and ifr(f, t) is the field variable for the For this purpose we introduce a phase factor exp(tf') into the solutions of NLKGEs, so

NLSE and that when we take the modulus of the Geld variable the phase factor disappears. This

property is analogous to L. de Broglie's earlier ideas on the 'double folution* of nonlinear

equations' (de Broglie, 1960; Barut and Rusn, 1987) Inserting Eq. (1.1) into Eq. (1.2) we get

2. Envelope Solutions of Nonlinear Klein-Cordon Equations

where A«V - $„ + tfw + i>,,. Eq. (1.9) i* an NLSE but with two nonlinear term*, and In this section we will describe the method of introducing de Broglie phase factor we shall denote it a* the double nonlinear Schrodinger equation (DNLSE). exp(tC) in the known solutions of NLKGE*. Throughout this paper our solutions will have

3 4 a complex phase factor of the form The absolute valued 7^* Lagrangiaiu gites the well-known NLKGE

• * 4 mV* - ac2W|to - ^W*0 = 0 (2.8)

for 9 •= 1. in a relathrbtic quantum mechanical notation of a 'de Brofli e phase'. Here fi — v/e,

7 - (1 - i?/c2)-1'* and *' is an arbitrary parameter and if we write f «Jf(* - v«) + «£, Let ^(1, t) be a eomplex^valued function and assume that Eq. (2,6) has a solution where Jf » ^, then the phase velocity is u. of the form

We introduce another exponential factor with a real exponent and with S ' ' ( »«,*.-!* u*w **1 constant where h(t, t) is abo a complex valued function. Then it can be shown that A(x, t) ha* to exp(») = exp [?£(*- ««) + *•]. (2.2) satisfy the following set of equations Oar toliton-Iike envelope solutions are expressible entirely in terms of the product of these a h = -mVfc, (a»fc)(5,.fc) = -mac*h2 two exponentials

3 ((*,()-exp(ir)exP(») (2.3) a A* - -mVA*, (a"&*)(3MA*) - -mVA* so that the envelope velocity is v, which is different from the phase velocity u. There is a *»-0,l (2.8) relation between the parameter w of the envelope with the parameter * of the de BrogUe where A* Is the complex conjugate of h(t,t). All solutions of Eq. (2.8) are the solutions phase of Eq. (2.0) via Eq. (2.7). For real valued fc(x, () and 4{x, t), this result is reported earlier a 3 « +« -mV (2.4) (Bart, 1978). It can easily be verified that h(i,t) = g(x,t) is a solution of Eq. (2.8). So where m is a parameter fixed by the nonlinear equation, whereas the parameters e and v the new exact envelope-type solution of the NLKGE (Eq. (2.0)) is given by (Baby and do not occur in the wave equations, they characterise the solutions. The known dispersion Barut, 19S7) relations are obtained by assigning i>0. We need also the combinations (2.9) ( 3 l 111 + •I»+li)w»J + 4(a*+iV! f - exp(2#) (2.5) For * - 0, this solution is known (Bantt and Rusu, 1987) and for tt m 1, it reduces where j* is the complex conjugate of f(*,t). to the sottton-like envelope solution of the well-known A^'-Lagrangian field theory. The

solution (2.9) is reported earlier when A(x, t) is a real valued function (Burt, 1078). 0 Of Eqs. (2.8) the first two are linear and so we may be able to use the linear that we will find the different values of the parameter combinations and their dispersions superpositions of solutions of the form relations under this limiting process.

Since t has the dimension of energy it can be written in the form (2.10)

e — me1"/ + -nv3 + r^*, B, tun are parameters. (J.1) when 2

(2.11) (Pi,* - Using Eq. (2.4), we get then la order to find an exact solution of Eq.(2.8) we need the following additional con- — *, ^-/i?! a * Bm'v1 Bmv3 2m *NR\ straints

m'e* We hare the following limits a3

Po,*»o<* ~ 0 /, fe - 1,2... AT (2.12) It -=.: -Aa (3.2)

Bat the wt of Eq. (2.12) i* an orerdetermined system and it b ralid only for j - it • 1. ($.3) r—oo hC h In (141) dimensions there exiit only one independent wave rector. The dimensionality (3.4) of space-time (d+1) and the maximum possible number of solutions N are related by

ff < 2d-1. Moreover, these JV-enrelope solutions are highly restricted by the constraints *)-*{*,*) (J.S) (2.10) and therefore they are not of the type of the well-known N-soliton solutions of the where integrable systems (Hlrota, 1972) and so we call them 'constrained N-envelope soliton* like solutions.

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S. NonwlatlTbtk redacUom (S.7)

such that In the fint section we obtained, as a result of the NR limiting process, DNLSB and pg - (3.8) GNLSE from the NLKGEs. In this section we will develop exact solutions of the DNLSEs

and GNLSEs using the NR limit from the solutions given In the previous sections. Before [9.9)

7 Consequently by oar limiting procedure we obtun the following new exact wiiton like When equality holds, the solution (Eq. (3.15)) vanishn identically. If 0 a lufficiently large envelope solution of the DNLSE (Eq. (1.3)) and negative, the solution can even be purely imaginary. Another interesting fact is that the perturbation never influences the 'internal frequency' of the de Broglie phase, which is (S.10) *(*,«) «± 1/a 3 I * constant under all type* of perturbations. When we assign b = 0 in Eq. (1.4), we get the

01 + well-known perturbed NLSE, which has been very extensively studied using approximate a 4(V+l)Jl»J + 4(3i+l)\' f For *0 - jjA , * » 0 this solution reduces to the well-known toliton envelope solutions solutions and Inverse Scattering transforms (1ST) perturbation methods (Zakharcrv and (Zakharov and Shabat, 19T2; Hirota, 1973) Synakh, 1970; Pereira and Stenflo, 1977; Davydov, 1970; Rabinovieh and FVbrikant 1979;

Basse, 1980; Kodama and Ablowiti, 19S0 and 1981; Scott, 19S2; Tusiynski, Paul and

Chatterjee, 1984; Klvshar and Malomed, 1980; Malomed, 1987a,b; see also Fushchich and which satisfies the NLSE Serov, 1087). Oar reeults are in agreement with those earlier results. In addition to that (3.12) we can find that If 0 is sufficiently Urge then A3 < 0 and in such situations amplitude of

When B - 1 and tWJt - 0, our rotation (3.10) vanishes Identically. Using the limiting the soliton grows Indefinitely and we get even imaginary solutions for some values of 0 and procew we can abo find the exact fofation of the GNLSE (Eq. (1.4)). The new dispersion relation b now

wa+ra«mV + 2mc30 (US) 4. Conclusion* and the corresponding change of the Eq. (3.2) becomes

It is well-known that the NLSE in (1+1) dimensions is an integrable system and

its 1ST Is well studied, whereas, the DNLSE and GNLSE have BO such property and so it is The new exact envelope tolutlon of the GNLSE ii difficult to find exact solutions for those equations. The method of constructing NR limit

(S.15) of a relativists equation has been used to find a number of solutions of the NR equations

like DNLSE and GNLSE from the solutions of the corresponding NLKGEa. Then converse Thit «howi that the perturbation lustiiiu the wliton envelope folution, or any other of this procedure, i.e. a systematic method of finding the solutions of relativistk equations •oltttion of the DNLSE, obtainable from the limiting procen, provided that like NLKGEs from the known solutions of NR equations like NLSEs Is abo interesting and

(3.16) will be reported separately.

10 For example, it is known that NLSE Eq. (J.12) has an envelope kink solutions Acknowledgments)

^(* - vt)] (4.1) One of the author* (B.V.B.) is extremely thankful to N.C. freeman, Martin Kruskai, A.S. Pokas, J. Hietarinta and Miki Wadati for several discussions. He would

The respective relativistic form of the kink envelope type Motion) and the \jr* Lagrangian also like to thank Professor Abdut Salam, the International Atomic ISnergy Agency and equations are respectively UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

and

- 0 (4.3)

Note that the known kink solution of (4.3) was without the de Broglie phase (Rajaraman, ~

1975, Bullough and Caudrey, 1085).

Oar study reveals thus that the envelope-type solutions of NLKGEs are the natu-

ral relativistic forms of the envelope solutions of the NR equation* like NLS Efe. In quantum

theory free particles are described by plane waves solutions of the linear Schrodinger equa-

tion with the usual probability interpretation. On the other hand, appropriate nonlinear

field equation could be used to desribe a localised single particle by their envelope soliton-

iike solution* (Klein, 197«; Em, 1985 and 1980; Barut and RUSH, 1987). The present

study continues the searches in this direction (Barut and Rusu, 1987; Baby and Bsrut,

1987; Barut and Baby, 1988). The final goal, which is also related to the original thoughts

of L. de Broglie (de Broglie, I960) is to find a satisfactory description of single events in

quantum phenomena.

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