Theory of quasi-simple dispersive shock waves and number of solitons evolved from a nonlinear pulse
TTheory of quasi-simple dispersive shock waves and number of solitonsevolved from a nonlinear pulse
A. M. Kamchatnov Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia a) (Dated: 25 August 2020) The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed.It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of thebackground flow that interacts with edge wave packets or edge solitons. A conjecture about existence of a certainsymmetry between equations for the small-amplitude and soliton edges is formulated. In case of localized simple wavepulses propagating through a quiescent medium this theory provided a new approach to derivation of an asymptoticformula for the number of solitons produced eventually from such a pulse.PACS numbers: 47.35.Jk, 47.35.Fg, 02.30.Ik
In quite general situations, a localized intensive nonlin-ear wave pulse splits during its evolution into two pulsespropagating in opposite directions. Such individual pulseswith unidirectional propagation are called simple waves and they can be described by evolution of a single vari-able. Simple waves break with formation of dispersiveshock waves (DSWs) that can be represented as modulatednonlinear periodic waves whose evolution is governed inGurevich-Pitaevskii approach by the Whitham modula-tion equations. We call such type of DSWs quasi-simpleshocks and show that in this case the motion of the small-amplitude DSW edge is governed by the Hamilton equa-tions with the dispersion law for linear waves playing therole of the Hamiltonian. The Hamilton equations have anintegral which plays the role of the limiting modulationparameter in the Whitham system at this edge. On the ba-sis of old Stokes’ observation about expression of soliton’sspeed in terms of the dispersion law of linear waves andother similar findings, we formulate a conjecture aboutrelationship between limiting equations for the two edges.This theory leads to derivation of an asymptotic formulafor the number of solitons produced from an initially local-ized simple wave pulse. The developed theory is applicableto a quite wide class of nonlinear wave equations which isnot limited to completely integrable equations.
I. INTRODUCTION
As is known, if a nonlinear wave system supports soliton-like propagation, then an intensive enough initial pulseevolves eventually into a certain number N of solitons andsome amount of linear radiation which is negligibly small forlarge N . Therefore, possibility of prediction of this number N for a given initial pulse is very important for the theoretical de- a) Also at Moscow Institute of Physics and Technology, Institutsky lane9, Dolgoprudny, Moscow region, 141700, Russia.; Electronic mail:[email protected]. scription of behavior of nonlinear pulses in many experimen-tal situations. If the nonlinear wave equation is completelyintegrable, then this problem can be solved in principle byconsidering the associated with this equation linear spectralproblem: N is equal to the number of discrete eigenvalues forgiven initial data, whereas the eigenvalues λ i , i = , , . . . , N ,determine the parameters of solitons emerging from the pulseat asymptotically large time t → ∞ (see Ref. 1). For large N (cid:29) λ i andsimple asymptotic expression for N (see Ref. 2). However,such a general method does not exist for non completely in-tegrable equations. Nevertheless, some particular results canbe obtained if we confine ourselves to the initial pulses of asimple-wave type and trace in some detail a gradual process ofsolitons formation from an initially smooth pulse. In fact, thisrestriction is not very strong since in hydrodynamic approxi-mation with neglected dispersion effects any typical localizedpulse splits during its evolution into two pulses propagatingin opposite directions. If this splitting takes place at the stageof evolution before the wave breaking moment then the abovecondition is fulfilled due to the natural wave dynamics. Wewill consider in what follows the simple-wave initial pulsesonly.In dispersive nonlinear systems, wave breaking leads to for-mation of a dispersive shock wave (DSW), that is a region ofstrong nonlinear oscillations. As was shown in Ref. 3, sucha region can be presented as a modulated periodic solution ofthe wave equation under consideration and then the Whithammodulation equations, Ref. 4, can be applied to descriptionof its evolution. In Gurevich-Pitaevskii approach, a DSWdegenerates at one its edge to a sequence of solitons and atanother edge to a linear wave packet with vanishing ampli-tude. Each edge propagates along the corresponding parts ofthe hydrodynamic simple-wave solution of dispersionless ap-proximation. Generally speaking, even evolution of initiallysimple-wave pulses can lead to formation of quite compli-cated wave structures with several DSWs, rarefaction wavesand plateau regions, if the system is not genuinely nonlinear,that is if its characteristic velocities can vanish at some valuesof wave amplitude, or the initial pulse profile has several localextrema or inflection points. However, if we confine ourselves a r X i v : . [ n li n . PS ] A ug to a simple-wave type of initial conditions with a single localextremum of the amplitude for genuinely nonlinear systems,then a single DSW evolves after wave breaking moment. Sit-uation simplifies even more, if the pulse propagates into a qui-escent medium. As was noticed in Ref. 5 for the completelyintegrable Korteweg-de Vries (KdV) equation, in this case theDSW is described by only two varying parameters and it wascalled quasi-simple by analogy with hydrodynamical simplewaves with a single varying parameter. The shall generalizethe notion of quasi-simple DSWs to all situations with wave-breaking of initially simple wave smooth pulses. If such aDSW propagates through a quiescent medium, then this sub-class of quasi-simple DSWs admits more complete investiga-tions and even in this restricted formulation, the problem ofdescription of DSW formation is applicable to a huge numberof realistic experimental situations. In this paper we shall con-sider genuinely nonlinear physical systems and simple-wavetype of initial conditions for pulses propagating into a quies-cent medium. II. FORMULATION OF THE PROBLEM
Here we define in more explicit terms the class of physi-cal wave systems to which our approach can be applied. Weconsider some nonlinear dispersive system and assume thatin the so-called dispersionless limit, when the higher orderderivatives of physical variables are neglected, the resultingequations can be written in a hydrodynamics-like form ∂ ρ∂ t + ∂ ( ρ u ) ∂ x = , ∂ v ∂ t + v ∂ v ∂ x + c ρ ∂ ρ∂ x = , (1)where ρ plays the role of “density”, v is the “flow velocity”and c ( ρ ) has the meaning of the “local sound velocity” whichis related with ρ according to the “equation of state” p = p ( ρ ) according to the relationship c = d p / d ρ . It is known thatin many physical situations nonlinear wave equations can bewritten in this form (see, e.g., Refs. 6 and 7). The system (1)has a standard for compressible fluid dynamics form and canbe cast to a diagonal Riemann form (see, e.g., Ref. 8) ∂ r + ∂ t + v + ( r + , r − ) ∂ r + ∂ x = , ∂ r − ∂ t + v − ( r + , r − ) ∂ r − ∂ x = , (2)where r ± = v ± (cid:90) ρρ c ( ρ ) ρ d ρ (3)the Riemann velocities v ± = v ± c (4)are expressed in terms of r ± by means of solving Eqs. (3)with respect to v and c = c ( ρ ) and substitution of the resultinto Eqs. (4).In simple waves, one of the Riemann invariants r ± is con-stant and for definiteness we assume that this is r − . Besides that, we consider here pulses propagating into a uniform qui-escent medium with constant density ρ = ρ and zero flowvelocity v =
0, that is r − = r + = v + = v + ( r + , ) , v + ( , ) = c ( ρ ) . Insteadof r + , we can choose for our convenience as a physical vari-able some other function u = u ( r + ) and change the referenceframe by the replacement x → x + c ( ρ ) t . Then the function u ( x , t ) obeys the equation ∂ u ∂ t + V ( u ) ∂ u ∂ x = , (5)where V = v + ( r + ( u ) , ) and V → u →
0. As was con-jectured by Gurevich and Meshcherkin in Ref. 9, the constantRiemann invariant r − preserves the same value at both edgesof the DSW in spite of its fast oscillations within the DSWregion. In systems described by completely integrable equa-tions, this condition is fulfilled by the Gurevich-Pitaevskiiconstruction of the solution of Whitham’s equations, andGurevich and Meshcherkin generalized this property to non-completely integrable situations.The full system of wave equations, which includes higherorder derivatives of ρ and v , can be linearized with respectto small deviations ρ (cid:48) , v (cid:48) from their “background” values ρ , v which can be considered locally as constant. Then the linearwave solutions ρ (cid:48) , v (cid:48) ∝ exp [ i ( kx − ω t )] yield two branches ω = ω ± ( ρ , v , k ) = ω ± ( r + , r − , k ) (6)of the dispersion law. Again we put here r − = , r + = r + ( u ) and take the branch for which the phase velocity V ± = ω ± / k converges to V ( u ) in the limit k →
0. This means that weconsider linear waves for which their long wavelength limit isconsistent with linearization of Eq. (5): dispersionless evolu-tion coincides locally with unidirectional propagation of longwavelength linear waves. As a result, we arrive at the disper-sion law ω = ω ( u , k ) , V ( u , k ) = ω ( u , k ) / k (7)with its dispersionless limit ω = V ( u ) k , V ( u ) = V ( u , ) , V ( ) = . (8)For definiteness, we will consider physical systems with V ( u ) > d ω / dk <
0) which sup-port “bright” soliton solutions in the form of humps of thevariable u propagating along the background with u =
0. Weassume that the initial distribution u ( x ) = u ( x , ) is a smoothfunction and belongs to the simple-wave type of unidirectionalpropagation. Its dispersionless evolution according to Eq. (5)leads to steepening of the front so that the wave breaks at somemoment of time. To simplify the notation, we take t = u ( x ) > − l < x < u ( x ) = u m atsome point x m (see Fig. 1(a)).After wave breaking moment a DSW appears and in theGurevich-Pitaevskii approach the wave number k ( x , t ) of the xu ( x ) u m x m − l (a) u x ( u ) u m x m − l x x (b) FIG. 1. (a) The initial profile u ( x ) . (b) Two branches x ( u ) and x ( u ) of the inverse function. locally periodic modulated wave is considered in Whitham ap-proximation as one of the modulation variables, so that k / ( π ) presents “a density of waves” within the DSW. Hence thenumber of waves spanned by DSW is equal to N DSW ( t ) = π (cid:90) x R ( t ) x L ( t ) k ( x , t ) dx , (9)where we assume that x L ( t ) and x R ( t ) denote the coordinatesof the small-amplitude and soliton edges, correspondingly.Evolution of k ( x , t ) obeys the number of waves conservationlaw (Ref. 4) ∂ k ∂ t + ∂ ω∂ x = , (10)where ω = kV is the frequency of the periodic travelling wavesolution, V is its phase velocity. The coordinate x R ( t ) cor-responds to the position of the leading soliton whose motionalong a smooth background does not change N DSW ( t ) . On thecontrary, at the small-amplitude edge the wave number k ( x , t ) is not equal to zero and here we have a flux ω of waves intothe DSW region.Our starting point is an important remark made by Gurevichand Pitaevskii in Ref. 10 that since the small-amplitude edgeof the DSW propagates with the group velocity v g = ∂ ω∂ k (11)of the wave at this edge, where ω = ω ( u , k ) is the wave fre-quency of a linear wave propagating along the backgroundwith the amplitude u , differs from the phase velocity V ( u , k ) = ω ( u , k ) k (12)of a linear wave, then the length of DSW increases at thisedge by ( v g − V ) dt in the time interval dt , so that the numberof waves inside DSW increases with time as dN DSW dt = π k ( v g − V ) = π (cid:18) k ∂ ω∂ k − ω (cid:19) . (13)Up to the sign, this expression can be regarded as a Doppler-shifted frequency representing the flux of waves into the DSW region. If we integrate the above formula upon time from thewave breaking moment to t = + ∞ , then we get the followingformula for the number of solitons (see Ref. 11) N = π (cid:90) ∞ k ( v g − V ) dt = π (cid:90) ∞ (cid:18) k ∂ ω∂ k − ω (cid:19) dt . (14)All the parameters in the integrand are to be calculated at thesmall-amplitude edge x L ( t ) of the DSW at the moment t of itsevolution.In Whitham’s approximation, a typical wavelength inside aDSW is much smaller than the size of the whole DSW andthis corresponds to the quasi-classical approximation of wavepropagation. At the small-amplitude edge the wave is lin-ear and the well-known Hamilton’s optico-mechanical anal-ogy (see, e.g., Ref. 12) can be applied to propagation of thewave packet moving along the path of the small-amplitudeedge. According to this analogy, the motion of this edge canbe interpreted as a motion of a classical particle with momen-tum k and Hamiltonian ω ( u , k ) . Then the integrand in (14) isinterpreted as a Lagrangian of this classical particle and theintegral is equal to the action S produced by such a particleduring its motion: N = S π . (15)Thus, our task is to develop the Hamilton theory of prop-agation of the small-amplitude edge, extend it to the solitonedge, and to calculate asymptotic number of solitons with theuse of Eq. (14). III. GENERAL THEORY
Now we take into account that the dependence of theHamiltonian ω ( u , k ) on the coordinate x of the particle is car-ried on via the dependence of the background simple wave u ( x , t ) along which the small-amplitude short wavelength per-turbation of DSW propagates at this edge. Evolution of u ( x , t ) ,on the contrary to the short wavelength propagation of thewave packet perturbation, is determined by the dispersion-less hydrodynamic approximation of simple-wave type, sothat u ( x , t ) obeys the equation (5). This Hopf equation for thesimple-wave evolution can be easily solved for a given initialdistribution u ( x ) (see, e.g., Ref. 13), x − V ( u ) t = x ( u ) , (16)where x ( u ) is the function inverse to the initial distribution u = u ( x ) . If we consider initial pulses in the form of a lo-calized hump (see Fig. 1(a)) then the inverse function consistsof two branches x ( u ) and x ( u ) (see Fig. 1(b)) and Eq. (16)determines in an implicit form the dependence u = u ( x , t ) foreach branch.The specific dependence of the Hamiltonian ω ( u ( x , t ) , k ) on x and t via the solution (16) of Eq. (5) leads to importantconsequences. In particular, the Hamilton equations dxdt = ∂ ω∂ k , dkdt = − ∂ ω∂ x (17)together with Eq. (5) give at once dkdt = − ∂ ω∂ u ∂ u ∂ x , dudt = ∂ u ∂ x dxdt + ∂ u ∂ t = − (cid:18) V − ∂ ω∂ k (cid:19) ∂ u ∂ x , and their ratio yields the equation dkdu = ∂ ω / ∂ uV − ∂ ω / ∂ k (18)obtained by El in Ref. 14. Here the right-hand side dependsonly on u and k , so its solution gives k = k ( u , q ) , (19)where q is the integration constant. The value u = k →
0. This determines the boundary condition k = u = h along the small-amplitude edge path. After such a specifica-tion, the wave number k = k ( u ) depends solely on u . As a re-sult, if we consider the evolution of the pulse with a step-likeinitial condition u = u = const, then the solution k = k ( u ) gives us the value k ( u ) of the wave number at the small-amplitude edge propagating along the constant background u = u and, hence, the velocity v g ( k ( u )) of its propagation.This approach suggested by El, Ref. 14, permitted one to solvea number of interesting problems with step-like initial condi-tions, Refs. 14–22.The solution (19) satisfying the initial condition (20) de-scribes the motion of the wave packet (its ray) at the small-amplitude DSW edge. Apparently, the Hamilton equations(17) have more general character and describe the motion ofwave packets along the background u = u ( x , t ) with arbitraryinitial conditions. Since along each ray found in this way wehave q = const, then the variable q = q ( u , k ) (21)defined implicitly by Eq. (19) must satisfy the equation q t + v g q x =
0. Combining this equation with Eq. (5), we arrive atthe system ∂ u ∂ t + V ( u ) ∂ u ∂ x = , ∂ q ∂ t + v g ( u ) ∂ q ∂ x = q can be regarded as a Riemann invariant of theWhitham equations in this limit. Obviously, the system (22)has a very general nature and it often arises in description ofthe problem of interaction of linear wave packets with meanflow (see, e.g. Ref. 23 and references within). It is worthnoticing that in our approach the expression (21) is obtained by means of solving Eq. (18) rather then by diagonalizationof Eqs. (5) and (10) although both methods are equivalent, ofcourse.Evidently, a similar reduction of the Whitham equationsmust exist at the soliton edge of DSW and the question is howto find the Riemann invariant which corresponds to the char-acteristic velocity V s equal to the speed of the leading solitonin DSW. A hint to answering this question can be found inan old remark of Stokes fist published in §252 of the bookRef. 24 and later reproduced in the form of the letter to Lambin Ref. 25. Stokes noticed that propagation of the small ampli-tude soliton’s tails (“outskirts” according to his terminology)is governed by the same linearized equations that are used fordescription of propagation of linear travelling waves, so thatthe expression exp [ i ( kx − ω t )] for the linear wave is replacedby the expression exp [ − (cid:101) k ( x − V s t )] for the tail at x → + ∞ . Thismeans that if we make the replacement k → i (cid:101) k in the disper-sion law ω ( u , k ) for linear waves and define (cid:101) ω ( u , (cid:101) k ) = − i ω ( u , ik ) , (23)then the soliton velocity is given by V s = (cid:101) ω ( u , (cid:101) k ) (cid:101) k , (24)where (cid:101) k has the physical meaning of the inverse half-width ofsoliton. This remark turned out to be very useful both in con-crete studies of nonlinear wave propagation (see, e.g., Refs. 26and 27) and in the theory of DSWs for non-completely-integrable equations (see Refs. 14–22).We assume that the same replacement transforms Eq. (21)into the Riemann invariant (cid:101) q = (cid:101) q ( u , (cid:101) k ) (25)for the reduction ∂ u ∂ t + V ( u ) ∂ u ∂ x = , ∂ (cid:101) q ∂ t + V s ∂ (cid:101) q ∂ x = dxdt = ∂ (cid:101) ω∂ (cid:101) k , d (cid:101) kdt = − ∂ (cid:101) ω∂ x , (27)and again these equations together with the first Eq. (26) yield d (cid:101) kdu = ∂ (cid:101) ω / ∂ uV − ∂ (cid:101) ω / ∂ (cid:101) k . (28)Under certain assumptions, this equation was derived by El,Ref. 14, from the number of waves conservation law (10).By construction, the invariant (cid:101) q ( u , (cid:101) k ) is constant along tra-jectories defined as solutions of Eqs. (27), so these trajectoriescan be regarded as paths of solitons with fixed values of (cid:101) q .However, (cid:101) q changes along the path x R ( t ) of the soliton edgedetermined by the solution of the equation dx R dt = V s = (cid:101) ω ( u , (cid:101) k ) (cid:101) k . (29)Apparently, this leading soliton path should be an envelope ofpaths of solitons with fixed values of (cid:101) q . In a sense, at each mo-ment of time the leading soliton is represented by an instantlocation of some soliton having invariant (cid:101) q when it touchesthe curve representing the path of the soliton edge of DSW.In fact, this mechanism of edge formation as envelopes func-tions applies to the general form of DSW appearing after wavebreaking including its small-amplitude edge. In such generalsituations the Whitham system does not reduce to one (forstep-like initial conditions) or two (for quasi-simple DSWspropagating into the quiescent medium) equations and we donot know beforehand the value of the corresponding edge Rie-mann invariant q or (cid:101) q . This qualitative picture of DSW evolu-tion agrees with known particular solutions of Whitham equa-tions for the KdV case which describe evolution of shocksafter wave breaking of quadratic and cubic initial profiles (seeAppendix B).So far an initial distribution was assumed to be quite ar-bitrary. Now we turn to the case of localized initial pulseshown in Fig. 1(a), so that it evolves into N solitons. Toestimate the integral in Eq. (14) and to find N , we need totrace the variation of u with time t at the small-amplitudeedge for the general form of the initial simple-wave pulse.This can be achieved by means of the following reasoning (seeRefs. 28 and 29). The small-amplitude edge propagates withthe group velocity (11), that is during the time interval dt itmoves to the distance dx = v g dt . Since this path lies on thesurface u = u ( x , t ) of the dispersionless solution, the relation dx / dt = v g must be compatible with Eq. (16) representing thissurface. For a parametric representation t = t ( u ) and x = x ( u ) of the small-amplitude path, the differentiation of (16) withrespect to u and elimination of dx / dt = v g yields the equation [ v g ( u ) − V ( u )] dtdu − V (cid:48) ( u ) t = x (cid:48) ( u ) . (30)This linear differential equation t ( u ) can be easily solved withthe initial condition t = u = t ( u ) of time t on u for the period of evolution when thesmall-amplitude edge propagates along the first branch of thedispersionless solution corresponding to x . After the momentwhen t reaches the time t ( u m ) , we have to solve Eq. (30) withthe initial condition t = t ( u m ) at u = u m and this gives us thedependence t ( u ) corresponding to propagation of the small-amplitude edge along the second branch of the dispersionlesssolution. As a result, we obtain the function t = t ( u ) for thetotal process of the pulse evolution and this function togetherwith the already known functions k ( u ) and ω ( u , k ( u )) permitus to calculate the number of solitons with the use of Eq. (14).In concrete situations such a calculation can often be donewithout much difficulty and we shall illustrate the method bya simple example in the next section. IV. EXAMPLE
We consider here formation of solitons from a pulse u ( x ) whose evolution is governed by the generalized KdV equation u t + V ( u ) u x + u xxx = , (31)which under certain conditions for V ( u ) , V ( ) =
0, has peri-odic and soliton solutions (see Ref. 14). Linearization of thisequation yields the dispersion law of linear waves ω ( u , k ) = V ( u ) k − k , (32)so that Eq. (18) reduces to3 k dkdu = V (cid:48) ( u ) , and its solution with the boundary condition Eq. (20) has theform (see Ref. 14) k ( u ) = (cid:114) V ( u ) . (33)Consequently, the group velocity at the small-amplitude edgepropagating along background with the amplitude u is equalto v g ( u ) = − V ( u ) . Then Eq. (30) becomes − V ( u ) dtdu − V (cid:48) ( u ) t = x (cid:48) ( u ) (34)and for two branches shown in Fig. 1(b) its solution reads (seeRef. 28) t ( u ) = t ( u ) = − (cid:112) V ( u ) (cid:90) u x (cid:48) ( u ) (cid:112) V ( u ) du , t ( u ) = t ( u ) = − (cid:112) V ( u ) (cid:40) (cid:90) u m x (cid:48) ( u ) (cid:112) V ( u ) du + (cid:90) uu m x (cid:48) ( u ) (cid:112) V ( u ) du (cid:41) . (35)Substitution of these expressions into Eq. (14) leads after sim-ple transformations to the formula N = ( / ) / π (cid:40) (cid:90) u m duV (cid:48) ( u ) (cid:90) u m u ( x (cid:48) − x (cid:48) ) du (cid:112) V ( u )+ (cid:90) u m (cid:112) V ( u )( x (cid:48) − x (cid:48) ) du (cid:41) . (36)Here the double integral reduces to the ordinary one by meansof evident integration by parts with account of V ( ) =
0, sothat we get the final expression N = π (cid:90) u m (cid:114) V ( u ) ( x (cid:48) − x (cid:48) ) du = π (cid:90) − l (cid:114) V ( u ( x )) dx . (37) xt x L ( t ) x R ( t ) DSW dispersionlesssolution u = 0 − l FIG. 2. Three regions distinguished in the waves structure evolvesfrom the initial profile u ( x ) : dispersionless solution for x < x L ( t ) ,DSW for x L ( t ) ≤ x ≤ x R ( t ) , quiescent medium for x > x R ( t ) . Dashedlines describe paths of wave packets propagating along the disper-sionless solution and forming the distribution of k ( x , t ) . Remembering the formula (33) for the wave number, we ob-tain N = π (cid:90) k [ u ( x )] dx . (38)The expression Eq. (38) agrees with asymptotic formu-las for the number of solitons known for completely inte-grable equations and similar calculations for some other non-completely integrable equations lead to the final expressionswhich can also written in the form (38) what indicates its gen-erality. The general proof of this expression was suggested inRefs. 16 and 30 on the basis of extension of the notion of theDSW wave number k beyond the DSW region. In the nextSection we present modification of this proof which clarifiessome its important points. V. FORMULA FOR THE NUMBER OF SOLITONS
The pulse evolved from the initial distribution depicted inFig. 1 consists in the Gurevich-Pitaevskii approximation fromthree parts: on the right of the soliton edge x R ( t ) we havethe quiescent medium with u =
0, the DSW is located be-tween the two edges x L ( t ) ≤ x ≤ x R ( t ) , and on the left of thesmall-amplitude edge x L ( t ) we have the dispersionless solu-tion (see Fig. 2). The smooth evolution of the pulse outsideDSW obeys Eq. (5) whose solution u ( x , t ) is given in implicitform by Eqs. (16) for two branches x ( u ) and x ( u ) . Lin-ear wave packets can propagate along this smooth backgroundand paths of these packets are given by solutions of the Hamil-ton equations (17) for certain choice of initial conditions. Weknow that at the small-amplitude edge the limiting Riemanninvariant q of the Whitham equations can be defined and itsvalue here is equal to q =
0. This equality can be regarded asan expression of the Gurevich-Meshcherkin assumption (seeRef. 9) that in quasi-simple DSWs the value of preserved dis-persionless Riemann invariant is transferred through a DSW. This means that we can define an additional Riemann invari-ant q in the smooth region as an extension of one of the lim-iting Riemann invariants u , q of Whitham equations to thewhole dispersionless region. Thus, in the dispersionless re-gion we have q = k ( u ) = k ( u , ) to the smooth region which yields distribution k = k ( x , t ) = k [ u ( x , t )] of wave numbers as an extension ofDSW’s wave number at the small-amplitude edge to the wholesmooth region (see Fig. 2). One can say that this is a specificproperty of the Whitham approximation: although the ampli-tude of oscillations is equal here zero in this approximation,the notion of the wavelength of waves, entering into the DSWregion, still has physical meaning. Obviously, k ( x , t ) = x < − l since here u = h =
0. Solutions of the equation dxdt = ∂ ω∂ k (cid:12)(cid:12)(cid:12)(cid:12) k = k ( x , t ) (39)with the initial condition x ( ) = x , − l ≤ x ≤
0, give usa family of rays along which wave packets propagate whenthey are radiated from points x = x with the carrying wavenumbers k [ u ( x )] ; they are shown by dashed lines in Fig. 2.In particular, the ray radiated from the wave breaking point x = x = x L ( t ) .Now we define the number of waves N smooth ( t ) correspond-ing to the defined above distribution k ( x , t ) : N smooth = π (cid:90) x L ( t ) − l k ( x , t ) dx = π (cid:90) x L ( t ) − l k [ u ( x , t )] dx . (40)It complements the number of waves N DSW entered into theDSW region up to the moment of time t (see Eq. (9)). Thenumber N DSW changes with time according to Eq. (13), so letus calculate the derivative of N smooth with respect to t : dN smooth dt = π (cid:26) dx L dt k ( x a ( t ) , t ) + (cid:90) x L ( t ) − l ∂ k ( x , t ) ∂ t dx (cid:27) . We can substitute Eq. (39) with x = x a ( t ) into the first term.Then, by definition the function k ( x , t ) satisfies the numberof waves conservation law (10) and this statement can eas-ily be checked with the help of Eqs. (5) and (18), so in thesecond term the integrand ∂ k ( x , t ) / ∂ t can be replaced by − ∂ ω ( x , t ) ∂ x , and after integration we get dN smooth dt = π (cid:18) k ∂ ω∂ k − ω (cid:19) x = x L ( t ) . (41)This is equal to Eq. (13) with opposite sign, that is N smooth + N DSW = const. At last, since in the limit t → ∞ we have N DSW → N , N smooth → t → N DSW → N smooth → ( / ( π )) (cid:82) ∞ − ∞ k [ u ( x , )] dx and u ( x , ) = u ( x ) , wearrive at the final formula for the number of solitons N = π (cid:90) ∞ − ∞ k [ u ( x )] dx , (42)where the function k ( u ) is the solution of Eq. (18) with theboundary condition Eq. (20). The presented here proof ofEq. (42) provides an explicit construction of the function k ( x , t ) for wave numbers in the smooth region of the pulseintroduced earlier in Refs. 16 and 30. The formula (42) wasconfirmed by numerical solutions of nonlinear wave equationsand it agrees very well with the results of recent experimentspresented in Ref. 31.Under some additional assumptions, the asymptotic distri-bution of solitons parameters was obtained in Ref. 30. VI. CONCLUSION
The notion of quasi-simple DSWs was first introduced inRef. 5 for the KdV equation case as the shocks with only twoRiemann invariants changing along them. In this paper, wegeneralized this notion to nonlinear wave situations with wavebreaking of simple waves, so that, according to Gurevich-Meshcherkin conjecture, one dispersionless Riemann invari-ant has the same value at both edges of the DSW under con-sideration. This definition is not limited to the class of com-pletely integrable equations and is applicable to any nonlin-ear wave equations admitting propagation of solitons. As fol-lows from the Gurevich-Pitaevskii remark made in Ref. 10on the number of waves entering into the DSW region in aunit of time, propagation of the high-frequency wave packetat the small-amplitude edge of DSW satisfies the Hamiltonequations with the linear dispersion law playing the role ofthe Hamiltonian, Ref. 11. This system of Hamilton equationsis coupled with the Hopf equation for evolution of the back-ground field what results in the diagonal form of the Whithamequations at the DSW edges.Long ago G. G. Stokes remarked in Refs. 24 and 25 that theexpression for soliton’s velocity can be obtained from the dis-persion law for linear waves because the tails of a soliton obeythe same linearized equations as the small-amplitude travel-ling waves. We generalize here this observation to the sym-metry relationships between equations at the small-amplitudeand soliton edges: k ⇔ (cid:101) k , (43) ω ( u , k ) ⇔ (cid:101) ω ( u , (cid:101) k ) = − i ω ( u , i (cid:101) k ) , (44) dkdu = ∂ ω / ∂ uV − ∂ ω / ∂ k ⇔ d (cid:101) kdu = ∂ (cid:101) ω / ∂ uV − ∂ (cid:101) ω / ∂ (cid:101) k , (45) v g = ∂ ω∂ k ⇔ V s = (cid:101) ω ( (cid:101) k ) (cid:101) k , (46) q ( u , k ) ⇔ (cid:101) q ( u , (cid:101) k ) = q ( u , i (cid:101) k ) , (47) ∂ q ∂ t + v g ∂ q ∂ x = ⇔ ∂ (cid:101) q ∂ t + V s ∂ (cid:101) q ∂ x = , (48) k is the wave number at the small-amplitude edge, (cid:101) k is in thesoliton’s inverse half-width at the soliton edge, and at bothedges the background field obeys the same equation ∂ u ∂ t + V ∂ u ∂ x = . (49) Equations (48) and (49) comprise the limiting Whitham equa-tions at the DSW edges for the Riemann invariants ( u , q ) or ( u , (cid:101) q ) , respectively.Generally speaking, paths of small-amplitude and solitonedges of DSW are represented by envelopes of characteristicsof Whitham equations and their finding is not an easy task innon-completely-integrable case. Important exceptions are thesituations with initial step-like distributions when velocitiesof both edges can be found (see Ref. 14) and a quasi-simpleDSW propagating into a quiescent medium when the path ofone its edge can be calculated and the asymptotic velocity ofthe other edge can be found in the case of localized pulses (seeRefs. 28 and 29).In case of localized quasi-simple DSW propagating into aquiescent medium and evolving into a train of solitons, theasymptotic formula for their number can be derived with theuse of Gurevich-Pitaevskii theorem on the number of oscilla-tions entering into the DSW region. The resulting formulasagree with the expression derived in Refs. 16 and 30.At last, although we considered here a concrete problem ofevolution of quasi-simple DSWs, some results can be appliedto other problems of interaction of linear modulated waveswith mean flow; see, e.g., Ref. 23.To sum up, the presented in this paper theory unifies thepreviously obtained result into a consistent approach applica-ble to a wide class of quasi-simple DSWs. ACKNOWLEDGMENTS
I am grateful to G. A. El, N. Pavloff and L. P. Pitaevskii foruseful discussions. This study was funded by RFBR, projectnumber 20-01-00063.
Appendix A: Limiting Whitham equations: KdV equation case
We shall consider the KdV equation u t + uu x + u xxx = r , r , r .Near the small-amplitude edge a DSW is approximated by thesmall-amplitude solution (see, e.g., Ref. 11) u ( x , t ) = r + ( r − r ) cos [ √ r − r ( x − V t )] , V = ( r + r ) , r − r (cid:28) r − r , (A2)where the Riemann invariants r , r obey the limitingWhitham equations ∂ r ∂ t + ( r − r ) ∂ r ∂ x = , ∂ r ∂ t + r ∂ r ∂ x = . (A3)As follows from Eq. (A2), the background field and the wavenumber are expressed in terms of r , r by the formulas u = r , k = √ r − r . (A4)Hence, we get r = u − k / = q and Eqs. (A5) take the form ∂ q ∂ t + ( u − k ) ∂ q ∂ x = , ∂ u ∂ t + u ∂ u ∂ x = , (A5)which coincides with the system Eqs. (22) with account of thedispersion law ω ( u , k ) = uk − k , v g = ∂ ω∂ k = u − k , V ( u ) = u (A6)of linear waves for a linearized Eq. (A1).Now, at the opposite edge of DSW the leading soliton hasthe form u ( x , t ) = r + ( r − r ) cosh [ √ r − r ( x − V s t )] , V s = ( r + r ) , (A7)and the Whitham equations reduce to ∂ r ∂ t + r ∂ r ∂ x = , ∂ r ∂ t + ( r + r ) ∂ r ∂ x = . (A8)We get expressions for the background field and the inversehalf-width of soliton from Eq. (A7), u = r , (cid:101) k = √ r − r , (A9)so r = r + (cid:101) k / = (cid:101) q and Eqs, (A8) take the form ∂ u ∂ t + u ∂ u ∂ x = , ∂ (cid:101) q ∂ t + ( u + (cid:101) k ) ∂ (cid:101) q ∂ x = , (A10)coinciding with Eqs. (26) with account of (cid:101) ω ( u , k ) = u (cid:101) k + (cid:101) k , V s = (cid:101) ω (cid:101) k = u + (cid:101) k , V ( u ) = u . (A11)Similar symmetry between equations for the DSW’s edgescan be proved for other completely integrable equations. Appendix B: Paths of DSW edges as envelopes
First we consider situation with wave breaking of aparabolic pulse with u ( x ) = √− x , x ≤ , which belongs to thequasi-simple type. In the KdV equation theory, at the solitonedge with r = u = , r = (cid:101) q the Whitham system Eqs. (A8)reduces to ∂ (cid:101) q ∂ t + (cid:101) q ∂ (cid:101) q ∂ x = x − (cid:101) qt = − (cid:101) q . (B2)These are the characteristic curves near the soliton edge andfor their envelope the differentiation of Eq. (B2) with respect to (cid:101) q gives the relation (cid:101) q = ( / ) t . Then V s = (cid:101) k = (cid:101) q = t and the path of the soliton edge is given by x R = t (B3)in agreement with the known result (see Ref. 5 and 11).Now we turn to a more complicated situation with thegeneric Gurevich-Pitaevskii problem on wave breaking of acubic initial profile with u ( x ) = ( − x ) / . In this case allthree Riemann invariants are changing within the DSW andthe global solution of Whitham equations was found in Ref. 32(see also Ref. 11). At the small-amplitude edge it reduces to x − ( r − r ) t = ( − r + r r + r r + r ) . It was shown that at this edge we have r = u , r = − u /
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