International Journal of Bifurcation and Chaos, Vol. 4, No.5 (1994) 1095-1112 © World Scientific Publishing Company

INELASTIC INTERACTION OF BOUSSINESQ SOLITONS

C. 1. CHRISTOV* and M. G. VELARDE Instituto Pluridisciplinar, Universidad Complutense, Paseo Juan XXIII, No 1, Madrid, 28040, Spain

Received December 10, 1993; Revised March 21, 1994

Two improved versions of Boussinesq equation (Boussinesq paradigm) have been considered which are well-posed (correct in the sense of Hadamard) as an initial value problem: the Proper Boussinesq Equation (PBE) and the Regularized Long Wave Equation (RLWE). Fully implicit difference schemes have been developed strictly representing, on difference level, the conservation or balance laws for the mass, pseudoenergy or pseudomomentum of the wave system. Thresholds of possible nonlinear blow-up are identified for both PBE and RLWE. The head-on collisions of solitary waves of the sech type (Boussinesq solitons) have been investigated. They are subsonic and negative (surface depressions) for PBE and supersonic and positive (surface elevations) for RLWE. The numerically recovered sign and sizes of the phase shifts are in very good quantitative agreement with analytical results for the two-soliton solution of PBE. The subsonic surface elevations are found to be not permanent but to gradually transform into oscillatory pulses whose support increases and amplitude decreases with time although the 10 total pseudoenergy is conserved within 10- . The latter allows us to claim that the pulses are solitons despite their "aging" (which is felt on times several times the time-scale of collision). For supersonic phase speeds, the collision of Boussinesq solitons has inelastic character exhibiting not only a significant phase shift but also a residual signal of sizable amplitude but negligible pseudoenergy. The evolution of the residual signal is investigated numerically for very long times.

1. Introduction distance along a uniform canal with little or no change of shape. Russell's "solitary" type may be J. Scott Russell's discovery (August, 1834) around regarded as an extreme case of Stokes' oscillatory "Turning Point" in Union Canal near Edinburgh and later on laboratory systematic experimental in­ waves of a permanent type, the wavelength being vestigations on waves (Russell [1838, 1845, 1895], great compared with the depth of the canal, so that see also Bazin [1865], Daily & Stephan [1951]) led the widely separated elevations are practically in­ him to pay great attention to a particular type which dependent of one another [Stokes, 1847, 1849]. The he called the "solitary wave." This is a wave con­ methods of approximation employed by Stokes, sisting of a single elevation, of height not necessarily however, become unsuitable when the wavelength small compared with the depth of the fluid, which, much exceeds the depth [Lamb, 1945, art. 252, if properly started, may travel for a considerable p.423].

'On leave from National Institute of Meteorology and Hydrology, Sofia

1095 1096 C. 1. Christov & M. G. Velarde

Russell's observations and conclusions were at Grimshaw [1986], Sander & Hutter [1991], Ursell first sight in conflict with Airy's wave theory [Airy, [1953]). 1845] where "a wave of finite height of length great Motivated by the study of Fermi, Pasta & Ulam compared with the depth must inevitably suffer a [1955] on energy sharing in nonlinear discrete sys­ continual change of form as it advances, the changes tems, Zabusky & Kruskal [1965] investigated the being the more rapid the greater the elevation above interaction of nonlinear waves in the KdVE and the undisturbed level" [Lamb, 1945, art. 252]. It introduced the notion of "soliton" for waves that was not until the works of Boussinesq [1871a, 1871b, upon (overtaking) collision behave as particles. It 1872] and Lord Rayleigh [1876] (see also McCowan was in classical mechanics the opposite side of the [1895]) that due credit was given to the Russell's de Broglie wave-particle dichotomy in quantum me­ findings. Indeed, Boussinesq provided a theory (see chanics. The first numerical results [Zabusky & also [Keulegan & Patterson, 1940; Rayleigh, 1876]) Kruskal, 1965] showed that save the phase shift where higher-order linear dispersion was introduced they experienced in the course of collision the in­ in the long-wave limit and the crucial role of the teraction of the "solitary" particle-waves appeared balance between nonlinearity and dispersion was elastic (hereafter the coinage "solit-on"). Moreover, shown, albeit not clearly seen by later authors (for now we have laboratory experiments with waves an illuminating paper on the subject, see Ursell that we interpret as the result of particle collisions, [1953]). Particularly relevant is that Boussinesq bound-states, and so on [Weidman et at., 1992; found analytical expression of sech type for the per­ Linde et at., 1993a; Linde et at., 1993b; Velarde manent long-wavelength waves which are solutions et ai., 1994a, 1994b], while in quantum mechanics of the equation he derived. A precursor of Boussi­ the collision or scattering of two particles is inter­ nesq in the same problem was Lagrange, according preted by means of Schrodinger waves. to Lord Rayleigh who also says "I have lately seen a The problem with the KdVE is that it is an memoir by Boussinesq in which is contained a the­ equation for slow evolution of a wave system in ory of the solitary wave very similar to mine. So a frame moving with the characteristic speed, i.e. far as our results are common, the credit of priority when the differences between the phase speeds and belongs, of course, to Boussinesq" (p. 279). the characteristic speed are small (within the first Later on, exploring consistently the simplifica­ order of the appropriate small parameter). In other tions of the problem under the assumption of words, the KdVE is valid only for a wave system slow evolution in the moving frame (already in that is slowly evolving in the moving frame. The Boussinesq derivations) and quoting the work of one-way approximation completely rules out the Boussinesq, as well as acknowledging the great in­ possibility to investigate head-on collisions in the fluencl2 of Lord Rayleigh's work [Rayleigh, 1876], framework of KdVE. An equation describing two­ Kort~weg and de Vries [1895] derived an evolution way waves should contain second time-derivatives equation governing the wave profile, now called the (as apparently, albeit not in reality, does the Boussi­ Korteweg-de Vries equation (or KdVE). For waves nesq equation). slowly evolving in the moving frame (quasistation­ ary) the Boussinesq equation strictly reduces to KdVE in the right-moving coordinate frame. Nat­ 2. Boussinesq Equations: urally, the same analytical solution of sech type is Variations on a Theme then valid also for the KdVE, and that was perhaps the solution which has received the greatest atten­ 2.1. Original, proper, improper, tion during the last two decades. This bias is also and improved equations the result of great discoveries made by Zabusky & Kruskal [1965] on the one hand, and Gardner et at. The (1 + 2)D potential flow with free sufface is a [1967] on the other. well-posed initial-value problem for the longitudinal Korteweg and de Vries found also another velocity component u and the relative surface eleva­ solution - the cnoidat wave train - that at two tion h. This system can be reduced to (1 + l)D by appropriate limits reduces to the Boussinesq soli­ means of an approximate solution for the 2D bulk tary wave and to the harmonic wave train, respec­ flow under the assumption that the scale of mo­ tively. (For accounts of the approximations leading tion .A is long in comparison with the depth of layer to KdVE and related matters see Bullough [1988], H. If no additional simplifications are effected, the Inelastic Interaction oj Boussinesq Solitons 1097 reduced (1 + l)D system will also be well-posed as where the "primes" are omitted without fear of con­ an initial-value problem. fusion since in what follows we shall use only the From the approximate (but correct) system, dimensionless form. Here Q stands for the dimen­ Boussinesq succeeded to derive an ill-posed one­ sionless parameter of nonlinearity (amplitude pa­ equation model through mathematically illicit, rameter) and f3 is the dimensionless dispersion pa­ albeit physically relevant, assumptions about the rameter. These are supposed to be small quantities approximate relation between %t and,tx' where in order for Boussinesq derivation to be valid. , = V9H is the characteristic velocity of the sys­ Including the effect of surface tension as was tem (characteristic speed of gravity waves). The done by Korteweg & de Vries [1895] (see also Kaup said relation is physically pertinent only for a solu­ [1975] for the two-way equation) one arrives once tion that evolves slowly enough in the right-going again at the Boussinesq equation but with a modi­ frame. Thus he was the first to derive an equa­ fied coefficient of the fourth derivative: tion containing dispersion in the form of fourth spa­ tial derivative. The equation derived by Boussinesq himself [197la, 1971b, 1972] we shall refer to as the "Original Boussinesq Equation" (aBE) (Eq. (26) (3) Boussinesq [1872]): 2 [Ph = 8 [( H)h 39 h2 9H382h] (1) 2 2 2 8t 8x 9 + 2 + 3 8x . where a and p are the air-liquid surface tension and One should not be deceived by the presence of density of the fluid respectively. second time derivatives in (1). Strictly speaking, Equation (3) is correct in the sense of Hadamard this equation is a one-way wave equation due to the (well-posed as an initial-value problem) only when assumptions under which it was derived. In addi­ the surface tension is strong enough in order to have tion, the sign of the fourth spatial derivative in (1) f3 < O. When this is the case we call the above equa­ is positive which makes it incorrect in the sense of tion Proper Boussinesq Equation (or PBE). Hadamard. The aBE [Eq. (1)] has the same singu­ Expecting no confusion in the reader we shall larity as the heat equation with reversed direction call indistinctly aBE and PBE [Eqs. (2) and (3) re­ of time. The ill-posedness of (1) does not prevent spectively] Boussinesq equations (BEs). Other au­ it from having stationary solutions of physical in­ thors call it "good" Boussinesq equation [Manoran­ terest, i.e., its restriction to the sub-space of solu~ jan et at., 1984]. tions spanned by the stationary propagating waves The ill-posedness of aBE can be removed if the is mathematically correct. We shall address the Boussinesq assumption that %t ~ tx is used in "re­ problem of quasistationarity in the moving frame verse," i.e., when some spatial partial derivatives solution later on. are replaced by their temporal counterparts. In the The assumptions related to the moving frame case of one-equation model it amounts to replacing 8 2 8 2 led Boussinesq to establish the approximate rela­ ffX'l by W' namely, tion h = UJH9-1 between the surface elevation and the longitudinal velocity component of velocity. Then Eq. (1) holds also for the velocity component (4) u within the same order of approximation. Upon introducing the dimensionless variables For the two-equation model, it was first done ~ by Peregrine [1966] and proved crucial for the di­ x = AX', h = h* h', u = J9H u', t = j 9 t' , rect numerical solution for the undular bore. The detailed arguments and derivations for "reversion" one arrives at the following dimensionless form of of Boussinesq derivations can be found in Benjamin Eq. (1): et at. [1972], Peregrine [1966], Whitham [1974]. Equation (4) is called by some authors "improved" (IBE), but the latter coinage is too dangerously (2) close to the "improper" BE and we shun to use it. Following the established terminology we call Eq. (4) Regularized Long Wave Equation (RLWE). 1098 C. 1. Christov & M. G. Velarde

As argued by many authors (see, e.g., Bona scale being of the order of VlJ. The only "long" et at. [1980] for the KdVE), the original (with seches are those moving at phase velocities, c, suf­ purely spatial derivatives) and the "improved" ficiently close to the characteristic speed, namely, (RLWE-type) models should be closely interrelated. when c :::::: [1 + 0({3)]. But then their amplitude is We sustain the same thesis for the more general two­ 0({3), i.e., they are weakly nonlinear (see Sander way equations outlined above. & Hutter [1991], Ursell [1953], for detailed discus­ sions of these interrelated properties). Without the 2.2. The solitary wave weakly-nonlinear assumption (which must be con­ sidered as additional to the long-wave assumption), All versions of the Boussinesq equations considered one finds for the sech solutions that their support is here possess solitary-wave solutions of sech type. of order (3-1. Thus, on the sech solution, the higher­ For the BEs [Eqs. (2), (3)], the solution reads order derivatives neglected in Boussinesq's trunca­ tion procedure are of the same order of magnitude ~ 2 2 u= _ c - 1sech2 (x - ct Jc - 1). (5) as the retained fourth-order derivative. Then it is 2 O! 2 {3 clear that if one is to progress beyond the territory of weakly-nonlinear assumption, one must retain [Here the phase velocity is made dimensionless by higher-than-fourth order derivatives in the trunca­ means of'Y = J(gH).] tion process. Thus the simplified equations can­ The corresponding solution for Eq. (4) (RLWE) not be accepted as rigorously derived as far as the is solitary (permanent, nonlinear, etc.) waves of arbi­ ~ 2 2 trary wave speeds are concerned. On the other hand, u = _ c - 1sech2 (x - ct Jc - 1). (6) the crucial idea of incorporating (linear) dispersion 2 O! 2c {3 in the model (which was the point of Boussinesq The mathematically correct Boussinesq equa­ against Airy) was sound and led to the discovery tion (namely PBE) is encountered for {3 < 0, or, of seches which testified that a local balance be­ 2 what is the same, a > 1/3pgH , i.e., when one tween the linear dispersion and non-linear steepen­ is faced with capillary waves with surface tension ing of a wave is possible [Ursell, 1953]. There can being of prime importance in the problem. This be mentioned numerous works on the generalization case corresponds to subcritical bifurcation of the of KdVE or BEs, e.g., Hunter & Vanden Broeck surface equilibrium state (see Grimshaw, [1986], [1983], Kawahara [1972], Nagashima & Kawahara Hunter & Vanden Broeck [1983]) and then the shape [1981], Pomeau et al. [1988J which deal with incor­ of the surface is a solitary depression. The case of poration of the effects of higher-than-fourth-order supercritical bifurcation requires {3 > °(or a < terms for the dispersion, but we shall not dwell on 2 ! pg H ), and corresponds bettel' to the findings of this matter here. J. Scott Russell. But then (as already mentioned) In retrospect one can concede the right to Airy's the mathematically correct equation is RLWE. So, conclusion that when no dispersion exists, the per­ one can see that, in fact, RLWE is the mathemati­ manent localized solutions with long wavelengths cal object that fulfills the programme of Boussinesq are not possible. We have a case opposite to Boussi­ for explaining Russell's experiments, rather than nesq's: a mathematically correct problem, albeit the aBE which was derived by Boussinesq himself. of no physical relevance! Experiments did show Boussinesq ruled out negative waves (depressions) solitary-like behaviour and highly crested wave and only considered one-side moving waves (x - xt). trains, whose study finally led to the development Yet, {3 of order of unity or larger should not be of a new chapter in mathematical physics thanks in considered for RLWE in the context of long surface particular to the works of Gardner et al. [1967], waves because such values violate the main assump­ Zabusky & Kruskal [1965], among others. How­ tion of long wavelength (see, e.g., Hunter & Vanden ever, solitons are, ideal entities, very much like hard Broeck [1983]). As already noted, the long-wave as­ spheres used to define a perfect gas, and like the sumption enters the model through the requirement hard spheres, which have only purely repulsive {3 « 1 while the dimensionless length of the waves force and no attraction, they cannot allow for is supposed to be of the order of unity. Then the bound states and condensation. Besides, the soliton­ sech solitons of arbitrary wave speed are not long bearing equations are structurally unstable and the waves in this sense. They are rather short, their integrability property is lost when the slightest Inelastic Interaction of Boussinesq Solitons 1099 dissipation is added which is always the case in real shift is negative and it re-emerges after collision in life. For specific discussion of this problem and the a place ahead in the direction of its motion. existence of "dissipative" solitons, see Christov & The reader should note here that Toda & Wa­ Velarde [1994a, 1994b]. dati called "phase shift" the quantity kioi. We call An analytical two-soliton solution is derived in "phase shift" the quantity Oi which is in fact the Toda & Wadati [1973] only for the improper aBE displacement of the center of a soliton with respect by means of Hirota's bilinear technique. It has the to its projected position, i.e., the displacement of form the center of the "phase pattern" called soliton. It is clear that the latter is the phase shift which can be observed in experiments. Although in Toda & Wadati [1973] only the im­ (7) proper equation (13 > 0) was considered, one finds 1/J = Alekl(x-qt) A e k2 (x-c2t) 1 + + 2 that the two-soliton solution is valid also for the

+ A e(k1 +k2)X-(kl q +k2C2)t , proper equation, provided that when the wave num­ 3 ber is calculated the opposite sign (13 < 0) is taken, where c; == 1 + 13k; and i.e., ki = VI - CI- The only difference is that is now not always positive. We tabulated the func­ A3 = tion for various values C = Cl = C2 and discov­ ered that the sign of changes approximately at (clkl - C2k2)2 - (k l - k2)2 - j3(k l - k2)4 - A I A2. C = Cl = C2 = 0.867 while being always bigger (clkl + C2k2)2 - (k l + k2)2 - j3(k l + k2)4 than unity (the solitons are lagging in comparison For t ----; ±oo the two-soliton solution degener­ with their projected positions). In fact ----; +00 ates into the respective one-soliton solutions, while C approaches 0.867 from above, and ----; -00 namely, while c approaches the same value from bellow. It goes beyond the frame of the present work to inves­ tigate in detail the zeros of the denominator of (9) t ----; ±oo. for the case when CI =I C2. (8) But the important difference between the two­ 2.3. Boussinesq paradigm soliton solution and the sheer sum of two one-soliton Boussinesq's was certainly a great mind and he cor­ solutions is the occurrence of phase shift which is de­ rectly understood the basic theoretical aspect un­ fined for each soliton as Oi = 0; - 0;. The "faster" derlying Scott Russell's discovery. Yet Boussinesq's (or the "steeper") soliton (the one with the larger theory can only be considered in a purely heuris­ wave number) experiences the smaller phase shift tic manner. We believe the latter is enough for according to the formula our purpose here. I Our stand is that the intrin­ sic physical relevance of a certain class of models klo -k 0 log , l = 2 2 = should be sound and hopefully with robust prop­ = _ (clkl - C2 k2)2 - (k 1 - k2)2 - ,8(kl - k2)4 erty, beyond their dubious foundation, and not cru­ - (clkl + CZk2)2 - (kl + k2)2 - j3(k l + k2)4' cially depending on the existence or nonexistence (9) of analytical solutions. This conviction, concerning in particular Boussinesq's theory, is the motivation As shown in Toda & Wadati [1973], 0 < < of the present work in which by direct numerical 1, which means that the logarithm always exists simulation we show that, for instance, the main and has negative value. Then the phase shift (the "phase lag") for the right-going soliton in aBE is always negative, i.e., it is being accelerated tem­ 1Another great mind, Lord Rayleigh, also appreciated Scott porarily during the collision and appears at a posi­ Russell's discovery. In a different problem, Benard convec­ tion farther ahead in the direction of motion, rather tion, he also clearly appreciated the relevant finding and provided a theory that later on was found not applicable than the expected position of arrival if there were to Benard's experiment. Yet Benard convection and Lord no interaction with another soliton. The same is Rayleigh's paradigmatic work on buoyancy-driven convection valid to the left-going soliton for which the phase are today outstanding model problems for nonlinear science. 1100 C. 1. Christov & M. G. Velarde properties of the two-soliton interaction, e.g. the 1988] are concerned with infinite intervals we con­ sign and size of phase shift, are "universal" within sider a finite spatial interval which is the case in what we shall from now on call the "Boussinesq numerical implementations. The shape of a local­ paradigm." Thus we shall not concern ourselves with ized solution approaches at each infinity a constant the quantitative correspondence of Boussinesq's the­ and hence all its derivatives decay automatically to ory to water waves, but rather shall deliberately zero, and when treating the problem analytically consider here the Boussinesq equation(s) as a uni­ one may also impose boundary conditions on the versal out-oj-context model problem (a paradigm) second, third; etc. derivatives that follow from the containing nonlinearity and dispersion in appropri­ condition for decay. These are called asymptotic ate balance which allows propagation of both boundary conditions. However, in numerics and in left- and right-going waves of permanent form. The physics as well, one never has a true infinity and the latter allows for investigation of the head-on colli­ exact form of the boundary conditions is of crucial sion of solitons. For extension of the Boussinesq importance. Indeed, any conservation or balance Paradigm to "dissipative" solitons, see Christov & law is a property of the boundary value problem as Velarde [1994a, 1994b], Nepomnyashchy & Velarde a whole, not only of the equation itself. [1994]. Upon introducing an auxiliary function q one It is clear now that for (3 > 0 the seches exist shows that the BE follows from the system only if Icl > 1, i.e. they are "supersonic" or "su­ perluminous" traveling signals (OBE and RLWE). Ut = qxx, (10) Respectively for (3 < 0 they exist if lei < 1, be­ ing "subsonic" (PBE). Then the OBE supersonic dU(u) qt = u - ~ - (3u xx , (11) solitons cannot be simulated directly because of the linear instability of the equation (as it is incorrect in The last system admits conservation laws if one the sense of Hadamard). A similar situation occurs of the following sets of boundary conditions are im­ with the RLWE subsonic solitons when the corre­ posed: sponding sign of (3 leads to incorrect problems in the sense of Hadamard. This sets up the limitation u = 0, q = 0 for x = -L1 , L 2 , (12) of the numerical studies of collisions of Boussinesq solitons, namely subsonic case for PBE and super­ or sonic, for RLWE. U = 0, qx = 0 for x = -L1 , L 2 , (13)

3. Hamiltonian Representation where - L 1, L2 are the values of the spatial co­ ordinate at which we truncate the infinite interval Before turning to numerical implementation, ("actual infinities"). let us show the Hamiltonian form of the problem Defining the mass M, the pseudomomentum P, under consideration. For the BEs it was shown in and energy E of the system as Manoranjan et at. [1984, 1988]. It is interesting to note that the original Boussinesq system (first £2 order in (3 truncation of the equations governing M ~f udx, the free-surface ideal flow in a shallow layer) could 1-£1 not be put into a formally Hamiltonian form (en­ clef 1£2 ergy might not be positive definite). As mentioned P = uqxdx, (14) -£1 in Kaup [1975] it has "... too much nonlinearity." As a result, the representation of the one-equation clef 1£2 1 2 2 2 E = -2 [u + qx + U(u) + {3u x]dx, Boussinesq model as a Hamiltonian system does -£1 not coincide with the original 0((3) system of two equations. one can show that the following conservation/ Here we extend the results of Manoranjan et al. balance laws hold: [1984, 1988J to the case of RLWE but in terms of dP a somewhat different auxiliary function first intro­ dM =0 dE = 0 duced in Christov [1994]. Another generalization dt ' dt ' dt is that while the works [Manoranjan et al., 1984, (15) Inelastic Interaction of B01J.ssinesq Solitons 1101

The first can be called a conservation law for the The first boundary condition (18) provides for mass of the wave, and the second, conservation of the conservation of pseudoenergy but not of mass. the energy. Following Maugin [1992], Maugin & The second boundary condition secures the con­ Trimarco [1992] we call the third equality a balance servation of both mass and pseudoenergy. On the law for the pseudomomentum P, with F being the other hand, the first one yields also u = 0 at x = pseudoforce. In the elasticity applications envisaged - L 1, £2 provided that the velocity has trivial values in Maugin [1992], Maugin & Trimarco [1992], the at the boundaries at the initial moment of time. In energy is the kinetic energy plus stored elastic en­ this instance it corresponds better to the real phys­ ergy. In fluid dynamics applications the interpreta­ ical situation. Numerically speaking, the second tion is not so unambiguous and depends on whether boundary condition yields on each iteration to a lin­ the function u is thought of as velocity or as surface ear problem of Neumann type that has ill­ elevation. In order to be on the safe side we call E conditioned matrix unless some special care is taken. pseudoenergy. For this reason we consider here the first boundary The conservation laws ensure that the mass and condition. In fact the lack of conservation of the the pseudoenergy remain constant during evolution mass of wave is felt only if the seches draw very near of the solution. For the case of asymptotic bound­ to the boundaries. This is not a real limitation here ary conditions the balance law suffices to Claim that since for RLWE we are only concerned with colli­ the pseudomomentum remains constant as well. Nu­ sions among the solitons rather than with reflection merically speaking, however, the boundary condi­ from the boundaries. Anyway, it is a shortcoming tions are not asymptotic and it may happen that of the RLWE model. U x i= 0 at the actual infinities. Only for strictly symmetric cases, when the collision of two identical solitary waves is considered, is pseudomomentum 4. Difference Scheme conserved after a full cycle of reflections from the For the Proper Boussinesq Equation (PBE) a con­ boundaries x = - L , L . 1 2 servative fully implicit scheme was created in Chris­ Let us now turn to the Hamiltonian representa­ tov [1994]. Here we extend it to RLWE. tion of RLWE. Introducing the same auxiliary func­ ~f tion q we show that Eq. (3) follows from the system Consider the set function Ui u(xd on the regular mesh in the interval [-L 1, L2] with spacing (16) h, i.e.,

dU(u) h = L1 + L2 qt - (3qtxx = u - ~ . (17) Xi -L + (i - l)h, (21) = 1 N -1 ' The last system is of a lower order with re­ where N is the total number of grid points in the spect to x and only requires one condition at each said interval. end. One has the choice between the following two We leave the problem of linearization to a later boundary conditions: moment and show first the strictly conservative scheme which is inevitably nonlinear, namely q = 0 or qx = 0 for x = -L1, L 2 . (18) n+! n-! Respectively the pseudomomentum and pseu­ qi 2 - qi 2 doenergy read T 2 n 1 def ( U~+1)2 + u + un + (u n )2 P = 1£2 u(qx - (3qxxx)dx, -a t t t t -£1 3 (19) . def 1 2 2 2 E= 1£2 -2[u +qx+ U(u)+(3utldx. -£1

1 1 The conservation laws for mass and pseudoen­ n 1 n+!2 n+ -2 n+ -2 U + - U~ q 2q + q ergy have the same form as for OBE while the bal­ t t _ i+1 - i i-I 2 (23) ance law for pseudomomentum now adopts the form T h with boundary condition (20) (24) 1102 C. I. Christou & M. G. Velarde

We can prove that the difference approximation however, that the dependent and independent vari­ £ of the pseudoenergy, ables of the equations that belong to the Boussinesq paradigm can be re-scaled so as to reduce each case n 1 1 N-l [(u(n+l))2 + (u )2 to the one considered here. Thus the only indepen­ £n+"2 = _ L t t 2 i=2 2 dent parameters that govern the evolution of a wave system are the phase speeds of the initial seches. We begin with the results for PBE. Its most important trait is that the analytical solution for particle-like waves is only known for subsonic phase velocities. Respectively, the amplitudes are nega­ tive (i.e., the solitons are "depressions" on the wa­ ter surface). Unlike the KdV case the faster solitons is conserved by the difference scheme in the sense of PBE are of smaller amplitudes. This defines the that £n+l = £n. The balance law for pseudomo­ essential feature of the Boussinesq solitons in this mentum is also strictly satisfied on the difference case. level within the round-off error of the calculations When the phase velocities are close to the char­ with double precision. acteristic velocity (i.e., the weakly nonlinear limit), In order to iteratively implement the nonlinear the phase shift is insignificant and the interaction scheme we introduce an "inner" iteration for the of the two "particles" is completely elastic. The sought functions results for c 2: 0.99 are very instructive for under­ standing the advantage of a conservative scheme. The amplitudes of seches in this case are so small and approximate the nonlinear term as that if the energy was not strictly conserved by the scheme, even the slightest "leakage" of energy dur­ u·n+l,k u·n+l,k-l + u·n+l,kUn + (u·n)2 ing the calculations would have led to eventual lin­ -0: t t t t t ear dispersion of the solution and disappearance of 3 the permanent shapes (seches). We can also add The "inner" iterations start from the initial here that the calculations for PBE were conducted conditions for very large times and after multiple reflection from the boundaries, and the solitons perfectly pre­ and (26) served their shapes and individuality. In Fig. 1 is presented a typical case of head­ and terminate at k = K when on collision of two equal sech Boussinesq solitons n+l,K n+l,K-11 governed by PBE. The interaction is perfectly elas­ max Iui - ui ~ 10-11 . (27) tic save the significant positive phase shift. The max lu~+I,K I solitons are being retarded because of the collision The value 10-11 is selected well above the and re-emerge in positions that are behind the posi­ round-off error 10-15 for computations with dou­ tions they would have reached if there were no col­ ble precisions. It is important to note that in all lision. Qualitatively the same is the case in Fig. 2, cases described below the number of iterations was but the phase lag is more than twice larger. In K = 4-6. After the inner iterations converge, the Fig. 3 and Fig. 4 are depicted two cases with un­ solution for the "new" time stage n + 1 of the non­ equal solitons. It is seen that the smaller soliton linear conservative difference scheme is obtained, (here it is the faster one!) experiences larger phase n+l def n+l K lag. Table. 1 presents the systematic investigation namely u i = ui '. of the phase shift as compared to the analytical for­ The conservative scheme is stable in time pro­ mula (9). The quantitative agreement is very good. vided that the "inner" iterations converge. In this instance, we may state that the interac­ tions presented in Figs. 1-4 are just the correspond­ ing two-soliton solutions as obtained by the direct 5. Results and Discussion simulation. For the sake of definiteness we select in our experi­ What is more important is that our calculations ments 0: = -3 and 1,61 = 1. It is important to note, allow one to establish the threshold of nonlinear Inelastic Interaction of B01/.ssinesq Solitons 1103

time time ____-----l 90. 90. ------184. 84. ~------l ------178. -----178.

------1 n. ~------1n. ------166. ~------I 66.

f~::_------160. ~------j60. ~;====~54. ~~54. m'l;;------1 48. /~ 48. 0.1 0.1 m'l;;------j 1l\'I.------142. ro:..------j 42. -u 't------j 36. -u ~~~~~~~~~~36. 00 -im- 0.0 -trrrmTTllrrrm'iiTrrrrmmrrmnmij'TTrmrTTJTTT1TITTTfT'lTIrrmrmmrrn-pi'ilT1TTT)TTTTTTTTTj -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 X X

-24 -16 -8 a 8 16 24 -24 -16 -8 0 8 16 24

70 70 70 70 (l) shift = 2.934-- (l) shift = 1.538 - .hirt- = 2.146 E .-E 50 50 ._ 50 50 4-J 4-J

30 - 30 30 -24 -16 -8 0 8 16 -24 -16 -8 0 8 16 24 X X

Fig. l. Collision of two equal negative seches (depressions) Fig. 3. Collision of two unequal seches in PBE for Cl = 0.95, in the PBE for Cl = -Cl = 0.9: (a) shape of wave function; C2 = -0.9: (a) shape of wave function; (b) trajectories of (b) trajectories of centers of solitons. centers of solitons.

time time 100. 90. 92. 84. 84. 78. 76. n. 68. 66. 60. 60. 52. 54. 44. 48. 0.1 36. 42. 01 28. -u -u 36. o 0 -1-nTITTTl"'r'i'rrTTT.-l'hTTTTTTlTTTTTTTTTTTTTl-rl'i-TTTlmlTnTITm o 0 -trrrnTITTTl'rn-rrnTTl'TTTITt1TTl'TTTrrTITlTTTTrm-rnTTTTTTrri -60 -40 -20 o 20 40 60 -60 -40 -20 a 20 40 60 X X

-40 -30 -20 -10 0 10 20 30 40 -21 -14 -7 0 7 14 21 28 100 ' 100

80 80 70 70 (l) (l) shift = 1231 - shift = 2.-4·97 60 60 E .-E 50 50 4-J 4-J 40 40

20 20 30 30 -40 -30 -20 -10 a 10 20 30 40 -21 -14 -7 a 7 14 21 28 X X

Fig. 2. Collision of two equal seches in PBE near the thresh- Fig. 4. Collision of two seches of significantly different arn- old of nonlinear blow-up for Cl = -C2 = 0.87: (a) shape of plitudes in the PBE for Cl = 0.97, C2 = -0.87: (a) shape of wave function; (b) trajectories of centers of solitons. wave function; (b) trajectories of centers of solitons. 1104 C. I. Christov & M. G. Velarde

Table 1. Comparison of the Phase Shifts for PBE ({3 < 0). time:

C1 81 Eq. (9) 81 (num.) C2 82 Eq. (9) 82 (num.) 0.000 0.97 0.859 0.911 -0.97 0.859 0.911 0.015 0.90 2.791 2.934 -0.90 2.791 2.934 0.87 6.716 6.777 -0.87 6.716 6.777 0.030 0.867 9.456 9.290 -0.867 9.456 9.290 0.045 0.95 2.117 2.146 -0.90 1.517 1.538 0.97 2.457 2.497 -0.87 1.212 1.231 0.060

0.225 0.240 blow-up [Turitzyn, 1993a, 1993b]. We discovered that the smallest phase velocity for which the cal­ 0.255 culations were possible with two equal solitons was 0.270 c = 0.867. All the three digits of this result are in perfect agreement with the threshold of validity of 0.285 the two-soliton solution as established here on the basis of the formula (9) (see Sec. 2.2). This means 0.000 that the analytical two-soliton solution (9) of the PBE ceases to exist at the threshold of the nonlin­ 0.015 ear blow-up of solution. 0.030 It is interesting to check for the existence of permanent waves of positive amplitudes in PBE. 0.045 This case was treated in Christov & Maugin [1991, 0.060 1994], Christov [1994] starting from different initial shapes for the positive bump. The numerical results 0.075 show, for instance, that if a seck is taken with a pos­ itive amplitude, it cannot preserve its shape. The 0.090 respective evolution is depicted in Fig. 5 and Fig. 6, for short time and long time respectively. One sees 0.105 that a signal of smaller amplitude is emitted from 0.120 the main hump and the reminder gradually trans­ forms into an oscillatory pulse. Due to the recoil 0.135 from the small signal the pulse is accelerated to the characteristic speed and then it continues its prop­ agation with unit phase velocity. The evolution of 10 15 the pulse is peculiar in the sense that although it re­ o 5 tains its individuality, it is not a stationary creature. x Its support increases with time while its amplitude decreases. This kind of behavior was behind the Fig. 5. Short-time evolution of a positive hump of sech­ shape moving with phase speed c = 0.9. coinage "Big-Bang" pulses introduced by Christov & Maugin [1991, 1994]. Note that the total energy is conserved in our calculations up to 10-12 . We did actually perform numerical experiments with any. For the mass center defined on the basis of the collisions of "Big-Bang" pulses and they did behave modulus of the wave shape, we discovered no ap­ as solitons in the sense that they retained their in­ preciable phase shift. All this suffices to claim that dividuality during the collision, and so did the to­ the pulses are solitons in the usual sense, although tal energy of the system. Since the definition of they are in fact "aging" solitons. During the cross­ a "mass-center" of a pulse is somewhat ambigu­ section of the collision they do behave as particles ous one cannot strictly assess the phase shift, if while the aging effect has much longer time scales Inelastic Interaction of Boussinesq Solitons 1105 u u

4 4

o o

- 4-+r-rrrTT1rTTln-T""n-rTTTTTTTTTTrlTTln-T""n-rTTTTTTl -6 0 6 12 18 24 30 36 - 4 --!-T"""""""1"'"T""l"TT"1 -5 0 5 -5 a 5 5 -:~:",:~~;"", o -6 0 6 12 18 24 30 36

-5 -5 0 5 10 15 20 4 -:tl~=>~~:"", -6 0 6 12 18 24 30 36

0

-4 -5 0 5 10 15 20

4

0

-4 -:~ -5 0 5 10 15 20 -6 0 6 12 18 24 30 36 x-ct x-ct Fig, 6, Long-time evolution of the right-going pulse from Fig, 5,

and is felt on very long times, say one or two orders The RLWE solitons exist for supersonic phase of magnitude larger than the time scale of the in­ speeds. Once again we start with an investigation teraction. On the other hand, the aging (with pos­ of the weakly nonlinear case, e.g., when c :::::; 1.05. sible extinction) is clearly a characteristic of "dissi­ Figure 7 gives an impression about the dynamics in pative" solitons [Christov & Velarde, 1994a, 1994b]. this case. One sees that the interaction is perfectly Here, due to conservation of pseudoenergy, the ex­ elastic. tinction is strictly impossible and the aging means The signs of residual signal (inelastic collision) transformation of shape in time. appeared to be first noticeable for c :::::; 1.2 (see The "reversed" behavior Of solitons of PBE is Fig. 8). In order to clarify the issue, we investi­ a clear warning that the latter represents a lim­ gated systematically the inelastic collision and dis­ ited model. As argued above, within the Boussi­ covered that the threshold of nonlinear blow-up of nesq Paradigm the particular equation is a matter the solutions is c ::; 1.64, i.e. for larger c the long­ of choice, so that we investigate also the solitonic time evolution was impossible. Indeed, we encoun­ properties of sech solutions of RLWE. tered it for all cases for which c > 1.64, i.e. the 1106 C. J. Christav & M. G. Velarde

time time

80. 32. 72. 28.8 64, 25.6 56. 22.4 48. 19.2 40. 16. 32, 12.6 0.10 24. 9.6 16. U 6.4 8. u 3.2 0.00 -100 -80 -60 -40 -20 0 20 40 60 80 100 -30 -10 10 30 50 X X

-20 -10 0 10 20 -30 -20 -10 0 10 20 30 50 50 30 30

Q) 40 40 Q) 20 E ~ 20 shift = 1.16 :;::; 30 30 E :;::; 10 10 20 20 -20 -10 0 10 20 a X -20 -10 0 10 20 30 X Fig. 7. Perfectly elastic interaction in RLWE for slightly supersonic phase velocities C1 = -C2 = 1.05: (a) shape of Fig. 9. The inelastic interaction in RLWE near to the thresh- wave function; (b) trajectories of centers of solitons. old of nonlinear blow-up, C1 = -C2 = 1.5: (a) shape of wave function; (b) trajectories of centers of solitons.

0.01 time 40, 36, 32.

28. u 0,00 24, 20. 16. 0.2i==~~ 12, 6. -0,01 U 0.1 -500-400-300-200-100 0 100 200 300 400 500 4. X -30 -10 10 30 50 (a) X

-10 -5 o 5 10 0.003 ]1

Q) 20 20 ~:::: t.\I\!\,~!\I,\I.J E u -50 -40 -30 -20 -10 0 10 20 30 40 50 X 10 +''T'T-M'T-rTTTT""rn'T'T'T'TTTTTTTTT"rri'''rrrr+- 10 -10 -5 0 5 10 (b) X Fig. 10. The shape of residual signal in RLWE for the case Fig. 8. The first signs of inelastic interaction in RLWE for of Fig. 9 for extremely large dimensionless time, t = 500 slightly supersonic phase velocities Cl = -C2 = 1.2: (a) shape (100000 time steps): (a) the full residual signal; (b) shape of of wave function; (b) trajectories of centers of solitons. signal in the vicinity of point of collision. Inelastic Interaction of Boussinesq Solitons 1107 additional signal that develops in the collision We discovered that the residual signal is com­ site adopts a shape which is dangerous enough in prised of a small breather in the geometric center a sense that it can grow according to the provision of the collision and two "Big Bang" pulses being of the theorem of Turitzyn [1993a, 1993b]. Tech­ constantly "fed" by the breather (see the profiles nically speaking there develops a portion of the for larger times in Fig. 9). After the collision was solution which if considered separately has negative over and the seches separated sufficiently, we re­ pseudoenergy. moved them and solved numerically with the rest For this reason, when tracking the long-time of the profile taken as initial condition. We went behavior of the residual signal the phase velocity to dimensionless times as large as 500 (106 time c = 1.5 was specially selected to be nonlinear enough steps for the case in Fig. 9) and used spatial res­ within the limits where long-time evolution is still olution of 104 nodes. The shape of the residual is possible. Figure 9 exhibits a residual signal of a shown in Fig. 10(a), where the most striking fea­ larger amplitude than the case for Fig. 8 and the ture is the nonuniform wavelength of the signal (the phase shift is also larger. However, the residual "Big-Bang" feature). The shape with shorter wave­ signal created at the site of bygone collision must length near the origin of the coordinate system is be of very small pseudoenergy content, because the shown in Fig. 10(b) which is a zoom of Fig. 10(a). total pseudoenergy of the system is conserved in Note that the total pseudoenergy of the signal in our scheme, at the time when the seches re-emerge Fig. 10 is exactly equal to the pseudoenergy of the from the collision virtually unchanged in shape (and residual in Fig. 9, although the signal j Fig. 10 hence almost at their original pseudoenergy levels). looks much more complicated and the first intu­ To be specific, the pseudoenergy of the residual in itive guess is to say that it has much larger energy Fig. 6 appears to be 0.4% of the total pseudoenergy content. This is one of the apparently paradoxical of the initial system of waves.

time ~\-"V ~. t ime 70. A" 10. 67. 9. 64. 8. 61. 7. 58. .\ ~ 6. 55. 5. 52. \~ ~. 4. 49. J. u 46. U 2. 43. 1. f ~ o / 1 '- -30 -10 10 30 50 -40 -20 o 20 40 X X

-4 o 4 8 ~===~===:±:::=~==:::;~tt24

shift = 1.24 20

-4 o 4 8 X

Fig. 11. Asymmetric RLWE collision for a case without Fig. 12. Symmetric RLWE collision for moderate supersonic blow-up Cl = 1.6, Cz = -1.2: (a) shape of wave function; phase velocities Cl = -cz = 2: (a) shape of wave function; (b) trajectories of centers of solitons. (b) trajectories of centers of solitons. H08 C. I. Christov & M. G. Velarde

time time ./I" 10.6 9.2 /" -of' 10.4 9.66 10.0 6.56 :/ "/ 9.6 6.24 9.2 7.92 ./ f\f'-..... 6.6 7.60 40 /""'- 6.4 7.26 ../ 20 6.0 6.96 U 20 ../ 7.6 6.64 U /'-.. 6.32 o -30 -10 10 30 0 -20 -10 0 10 20 X

-6 -3 -0 3 6 shift - 0.31 10.2 8.80 8.80 9.2 OJ shift = 1.273 8.2 .~ 7.80 7.80 +-'

6.80 6.80 -6 -3 -0 3 6 X Fig. 15. The inelastic collision for two significantly differ­ Fig. 13. Symmetric RLWE collision for very large super­ ent large phase speeds Cl = 5, C2 = -2: (a) shape of wave sonic phase velocities Cl = -C2 = 5: (a) shape of wave func­ function; (b) trajectories of centers of solitons. tion; (b) trajectories of centers of solitons.

features of systems whose energy functional is not positive definite. The first numerical results revealing the inelas­ time tic properties of RLWE solitons were reported by 11.90 Bogolubsky [1977] who tackled a case roughly cor­ 11.46 responding to our Fig. 9 by means of an explicit 11.02 scheme. The finding of Bogolubsky [1977] was 10.56 10.14 verified by Iskander & Jain [1980], whose scheme 9.70 was already implicit but still not conserving. The 9.26 lack of conservativeness of the schemes of these cited 10~ 6.62 works left the door open for suspicions that the in­ 6.36 elasticity of collision could be a numerical artifact. u 7.94 Because of conservation of energy we can claim that

-20 -10 o 10 20 30 our results prove the fact that the RLWE solitons X behave inelastically during their collisions. An asymmetric evolution is presented in Fig. 11 -10 -8 -6 -4 -2 -0 10 2 4 6 8 Cl C2 11.5 11.5 for two different phase velocities = 1.6 and = 1.2, which are bellow the threshold of nonlinear 1·0.5 shift = 1.66 shift. 0.87 10.5 OJ blow-up. E 9.5 9.5 +-' The stability domain of our scheme significantly 8.5 8.5 exceeds the schemes of Bogolubsky [1977], Iskan­ 7.5 7.5 -10 -8 -6 -4 -2 -0 2 4 6 8 10 der & Jain [1980] and we were able to proceed in X a region of rather strongly pronounced inelasticity. Fig. 14. Asymmetric inelastic collision for large phase speeds The inelastic portion of the signal that appears af­ Cl = 4, C2 = -3: (a) shape of wave function; (b) trajectories ter the interaction increases rapidly with the in­ of centers of solitons. crease of the supersonic phase speeds of the solitons. Inelastic Interaction of Bo'Ussinesq Solitons 1109

Table 2. Comparison for the Phase Shift for the RLWE (f3 > 0).

Cl 81 (9) 81 C2 (9) 81 (num.) C2 82 (9) 82Cl (9) 82 (nUll.)

1.2 0.9808 1.177 1.09 -1.2 0.9808 1.177 1.09 1.5 0.8773 1.316 1.18 -1.5 0.8773 1.316 1.18 2. 0.6805 1.361 1.24 -2. 0.6805 1.361 1.24 5. 0.2768 1.384 1.273 -5. 0.2768 1.384 1.273 1.6 0.6020 0.723 0.6 -1.2 1.133 1.813 1.84 4. 0.3245 0.9735 0.87 -3. 0.4444 1.778 1.66 5. 0.1689 0.3378 0.31 -2. 0.4777 2.389 2.44

Figure 12 provides an illustration of this statement significant sign of the universality for the soliton for C1 = C2 = 2. The phase shift is now conspic­ dynamics within the Boussinesq Paradigm. uous although it is not clear whether it could be Note that the results presented in Figs. 14 and safely interpreted physically, since the calculations 15 are valid only for finite times and were presented are blown-up promptly after the maximum of the here mostly for completeness and for the sake of residual signal exceeds the amplitude of the origi­ examination of the phase shift. nal seches. Anyway, within the time interval of the existence of the solution we calculated the phase shift (see the lower figure in Fig. 12). It is interest­ 6. Concluding Remarks ing to look at the dependence of the phase shift on the phase velocities. For this reason we investigated In the present work we have studied the mathemati­ also the case C1 = C2 = 5 as presented in Fig. 13. cal objects called Boussinesq Equations (Boussinesq As it should have been expected, the value of the Paradigm). Two improved versions of the Boussi­ phase shift increases with the increase of the phase nesq equation [called the Proper Boussinesq Equa­ velocities of the solitons (qualitatively in agreement tion (PBE) and Regularized Long Wave Equation with formula (9). (RLWE)] have been considered which are well-posed First we discuss the result in Fig. 11 where the (correct in the sense of Hadamard) as an initial­ phase shift can be defined for each time stage since value problem. Fully implicit difference schemes the calculations are stable and long-time evolution have been developed representing, strictly on differ­ is possible. ence level, the conservation laws for the mass and In order to examine more closely the phase shift, the pseudoenergy of the wave and the balance law two more cases with different phase speeds (Fig. 14 for the pseudomomentum. The head-on collision and Fig. 15) are considered. Results are presented of seches (Boussinesq solitons) has been thoroughly in Table 2. It is seen that there is no quantita­ investigated. In PBE they are subsonic and of neg­ tive agreement with the formula of Toda & Wa­ ative amplitude, while in the RLWE they are su­ dati. It is hardly surprising since RLWE corre­ personic and positive. sponds to OBE with a dispersion coefficient that The negative seches (the depressions) of is decreased c times. There is a conspicuous rela­ PBE are subject to positive phase shift (phase lag) tion, however, between the phase shifts for OBE after they re-emerge from the collision but they re­ (9) and RLWE. To this testify the third and sev­ tain their shape perfectly and no residual signals enth columns in Table 2. If one scales the result are detected. The numerically obtained phase lag for Sl with C2 and S2 with C1 then there is con­ is in very good quantitative agreement with the sistent, good agreement with the numerical results analytical prediction based on the two-soliton of our study. The above "experimental correla­ solution. tion" fits very well all the cases of different phase Our simulations reveal that the subsonic humps velocities considered here. We deem the qualita­ of positive amplitude are not stable and gradually tive agreement for the phase shifts, as well as the transform into oscillatory pulses whose supports in­ very existence of the mentioned relationship, as a crease with time and whose amplitudes decrease

Inelastic Interaction of Boussinesq Solitons 1111

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