Asymptotic theory of not completely integrable soliton equations

If a nonlinear wave system supports propagation of solitons, then an intensive enough initial pulse evolves eventually into a certain number $N$ of solitons and some amount of linear radiation. In view of universality of this phenomenon, it is very important to be able to predict parameters of emerging solitons (e.g., their velocities and amplitudes) from given initial distributions of the wave variables. This problem admits a straightforward solution in case of completely integrable nonlinear wave equations; see, e.g., Refs. nmpz-80 ; as-81 ; newell-85 and references therein. Such equations are associated with some linear spectral problems where the wave distributions play the role of ‘potentials’. Though these ‘potentials’ evolve with time according to the wave equation under consideration, the spectrum of the associated linear problem does not change, i.e. the evolution is isospectral, and each discrete eigenvalue corresponds to a certain soliton at the asymptotic stage of evolution at $t\to\infty$. Hence, if we find the spectrum of the linear problem for given initial wave distributions, then the discrete eigenvalues provide all necessary information about solitons’ parameters at asymptotically large time $t$.

This theory considerably simplifies if the final number of solitons is large, $N\gg 1$, and the initial distributions are smooth enough, so we can apply the quasiclassical WKB method to the linear spectral problem. For example, in case of Korteweg-de Vries (KdV) equation associated with the stationary Schrödinger equation ggkm-67 , the well-known Bohr-Sommerfeld quantization rule readily gives approximate values of the discrete eigenvalues very simply related with solitons’ velocities and amplitudes karpman-67 ; karpman-73 . As usual in the WKB method, the asymptotic formulas give quite accurate results even for small quantum numbers. Similar Bohr-Sommerfeld quantization rule was derived in Refs. JLML-99 ; kku-02 for the Zakharov-Shabat spectral problem associated with nonlinear Schrödinger (NLS) equation zs-73 . However, when the nonlinear wave equation under consideration is not completely integrable, such a direct approach becomes impossible.

To find a more general method for calculation of parameters of asymptotic solitons, we need to consider in some more detail the process of transformation of an initially smooth pulse to a sequence of solitons. One can distinguish in this process three typical stages. At first, the pulse’s profile gradually steepens due to nonlinear effects, but it keeps a smooth enough form without any oscillations, so its evolution obeys with high accuracy the dispersionless (hydrodynamic) limit of nonlinear wave equations under consideration. When the profile’s slope becomes very large, the dispersion effects start to influence on the evolution of the wave, so oscillations are generated after the so-called wave breaking moment of time. Such regions of nonlinear oscillations are called dispersive shock waves (DSWs) and, according to Gurevich and Pitaevskii gp-73 , they can be approximated by modulated periodic solutions of the nonlinear wave equations. Evolution of the modulation parameters obeys the Whitham modulation equations whitham ; Whitham-74 . In Gurevich-Pitaevskii approximation, a DSW occupies a finite expanding region of space. At one edge of the DSW solitons are gradually formed and the opposite small-amplitude edge propagates with some group velocity along a smooth part of the evolving pulse. Thus, at the second stage of evolution the wave structure consists of one or two DSWs adjoined with smooth parts of the pulse and the DSWs expand gradually over the whole pulse. Numbers of oscillations in DSWs increase with time due to difference between phase and group velocities of linear waves at the small-amplitude edges where oscillations enter into the DSWs regions gp-87 . At the last third stage of evolution of an initially localized pulse with a finite length its smooth parts disappear and the number of oscillations in DSWs is stabilized, so the solitons acquire their asymptotic values of parameters as $t\to\infty$, when distances between solitons become much greater than their widths. Thus, asymptotic solitons form gradually from small-amplitude oscillations which enter into the DSW region at its small-amplitude edge, so we have to consider propagation of this edge in some more detail.

Let the nonlinear wave dynamics under consideration be described by two variables which for definiteness we shall call “density” $\rho$ and “flow velocity” $u$. The periodic solutions $\rho=\rho(x,t),u=u(x,t)$ are periodic functions of the phase $\theta$ which in non-modulated case has the form $\theta=kx-\omega t=k(x-Vt)$, where $V$ is the phase velocity of the traveling wave. In a slightly modulated wave, the wave number $k$ and the frequency $\omega$ become slow functions of $x$ and $t$. Since locally the wave can still be considered in the leading approximation as uniform, they can be defined as

for some phase $\theta(x,t)$, and then one of the modulation equations can be written in the form of ‘the number of waves’ conservation law whitham ; Whitham-74

where $k$ plays the role of density of waves and $\omega$ is their flux.

At the small-amplitude edge we can distinguish the short wavelength oscillation at the length scale $\sim 2\pi/k$ and the slowly changing “background distributions” $\rho_{b}(x,t),u_{b}(x,t)$ which change at the length scale of the initial pulse length $\sim l$. Hence we can write $\rho(x,t)=\rho_{b}(x,t)+\rho^{\prime}(x,t),u(x,t)=u_{b}(x,t)+u^{\prime}(x,t)$ where the smooth functions $\rho_{b}(x,t),u_{b}(x,t)$ obey the motion equation in the dispersionless (hydrodynamic) limit obtained by neglecting term with higher order space and/or time derivatives, and the small-amplitude oscillations $\rho^{\prime}(x,t),u^{\prime}(x,t)$ obey the equations linearized with respect to deviations from the background flow. Since at the length scale $\sim 2\pi/k$ the background flow can be considered as constant, we obtain in the main approximation from the linearized equations with constant coefficients the harmonic wave solutions $\rho^{\prime},u^{\prime}\propto\exp[i(kx-\omega t)]$ with the dispersion relation $\omega=\omega(k,\rho_{b},u_{b})$. A slightly modulated linear wave packet propagating along a slowly changing background still consists of harmonics with the same dispersion law which satisfies the number of waves conservation law (2), but $\rho_{b},u_{b}$ become in this case the slowly changing solutions $\rho=\rho(x,t),u=u(x,t)$ of the dispersionless equations (see Ref. Whitham-74 ). In Gurevich-Pitaevskii theory gp-73 of DSWs, the small-amplitude edge is represented by such a linear wave packet and according to the conservation law (2) oscillations enter into the DSW region where their amplitudes grow from zero at the small-amplitude edge to soliton-like propagation at the opposite soliton edge. Since the small-amplitude edge propagates with the group velocity $v_{g}=\partial\omega(k,\rho,u)/\partial k$, along its path the flux is Doppler-shifted, so the number of oscillations inside the DSW region changes with time as (see Refs. gp-87 ; kamch-21a )

The expression in the right-hand side coincides, up to a constant factor, with Lagrangian of a point particle associated with the small-amplitude edge of the DSW, if in accordance with the well-known optics-mechanical analogy (see, e.g., Ref. lanczos ) we identify $k$ and $\omega$ with the particle’s momentum and Hamiltonian, correspondingly. Then the number $N$ of solitons formed from an intensive initial pulse is equal to the mechanical action produced by an associated with the packet point-like particle,

where $dx=v_{g}dt$ and integration is taken over the packet’s path. If the initial distributions $\rho_{0}(x),u_{0}(x)$ correspond to unidirectional simple wave propagation of the background pulse, that is there exists a functional dependence between $\rho$ and $u$ corresponding to a constant value of one of the Riemann invariants of the dispersionless equations LL-6 , then the wave number $k$ can be expressed as a function of the value of $\rho$ at the point where the small-amplitude edge is located at this moment of time, $k=k(\rho)$. Then we have also $\omega=\omega(k,\rho)$, where $\rho=\rho(x,t)$ is a solution of the Hopf equation

to which the dispersionless equations can be reduced, $k=k(\rho)$ is the solution of the equation

with the initial condition

where it is assumed that wave breaking occurs at the point with $\rho=0$. Equation (6) was first obtained in Ref. el-05 by means of analysis of Whitham equations for DSWs at their small-amplitude edge and then it was derived from the Hamilton equations for propagation of linear wave packets along simple wave pulses in Refs. kamch-20a ; ks-21 . When the function $k=k(\rho)$ is found, then, as was shown in Ref. kamch-19 , one can find the path $x=x(t)$ of the small-amplitude edge of a DSW and, consequently, calculate the integral in Eq. (4) for some particular nonlinear wave equations (see Refs. kamch-20a ; kamch-21 ; cbk-21 ). In this case the final result can be written in the form

where $\rho_{0}(x)$ is the initial distribution of $\rho$. Such a formula was suggested earlier in Refs. egkkk-07 ; egs-08 under supposition that one can extend solution of the Whitham equations to the unmodulated yet part of the pulse in such a way, that the total number of oscillations in DSW and unmodulated part remains constant during evolution of the pulse. Generalization of this rule to more general solutions $k=k(\rho,\overline{q})$, $\overline{q}$ is an integration constant in a solution of Eq. (6), yields also parameters of solitons at the asymptotic stage of evolution egs-08 . This theory agrees with numerical solutions of nonlinear wave equations and with experimental results obtained in Ref. mfweh-20 for a viscous fluid conduit.

So far this theory was limited to an important but particular case of an initial simple-wave pulse evolving eventually to a number of solitons and the aim of this paper is to generalize this approach to the general case of nonlinear wave evolutions describes by two variables $\rho(x,t),u(x,t)$ and admitting solitonic propagation of waves. To this end, we notice that equality of two expressions (4) and (8) can be interpreted as a special case of preservation of a constant value of the Poincaré-Cartan integral invariants poincare ; cartan for a mechanical system with the phase space $(x,k)$ and Hamiltonian $\omega(k,x,t)$. However, in this special case the tube of trajectories preserving the invariant is defined by solutions of Eq. (5) rather than by solutions of the Hamilton equations under consideration. In this paper, we extend this definition of Poincaré-Cartan integral invariants to general hydrodynamic flows for two variables $\rho,u$ not restricted by the condition that one of the Riemann invariants is constant and arrive at generalization of formula (8)

where $\rho_{0}(x),u_{0}(x)$ are the initial distributions of these two variables and $q_{N}$ is an integration constant in the solution of equations which define the function $k=k(\rho,u)$ of two variables. One can say that $q_{N}$ corresponds to the $N$-th soliton in the asymptotic distribution of solitons. Then if we decrease $N$ by one, then we get the value $q_{N-1}$ corresponding to the $(N-1)$-th soliton, and in this way we relate each $n$-th soliton with the corresponding value $q_{n}$ of the generalized ‘Bohr-Sommerfeld quantization’ rule (9), which extends the above mentioned method karpman-67 ; karpman-73 ; JLML-99 ; kku-02 of finding solitons parameters for completely integrable wave equations to not completely integrable equations. Relationship of the parameters $q_{n}$ with such physical parameters of solitons as, for example, their velocities is established by means of the Stokes remark lamb ; stokes (see also ai-77 ; dkn-03 ; kamch-20a ) that linear waves and exponentially small soliton tails obey the same linearized equations, so the expressions for their phase or soliton velocities are transformed to each other by the replacement $k\leftrightarrow-i\widetilde{k}$, where $\widetilde{k}$ is the inverse half-width of a soliton.

We illustrate the formulated here approach by its application to the generalized NLS (gNLS) equation and confirm its accuracy by comparison with numerical solutions.

Let the hydrodynamic (dispersionless) equations for the background flow evolution can be written in the form of equations of the compressible fluid dynamics,

where $c=c(\rho)$ is the ‘sound velocity’ related with the density $\rho$ by means of equation of state $p=p(\rho)$, $c^{2}=dp/d\rho$, $p$ is the pressure. The characteristic velocities of this system,

correspond to the sound waves propagating upstream and downstream the background flow with velocity $u$. Equations (10) can be transformed to the Riemann diagonal form

for the variables

called Riemann invariants. If we substitute here the function $\rho=\rho(c)$, then we get the Riemann invariants in the form

so that $c$ can be used as a wave variable instead of $\rho$. If Eqs. (12) are solved, then the physical variables $u,\rho,c$ can be expressed in terms of the Riemann invariants,

In case of unidirectional propagation of a background pulse we get a simple wave solution with one of the Riemann invariants constant. Let such a pulse propagate to the right through the uniform quiescent medium with constant values of $\rho=\rho_{R}$, $c=c_{R}$, $u=u_{R}=0$. Then along this pulse the variables $u$ and $c$ are related by the formula $u/2-\sigma(c)=-\sigma(c_{R})$, so that $c=c(u)$ and $r_{+}=u+\sigma(c_{R})$, $v_{+}=u+c(u)\equiv V_{0}(u)$. Consequently, the second equation (12) is fulfilled identically and the first one reduces to the Hopf equation

We are interested in propagation of the small-amplitude edge of a DSW and we identify this propagation with motion of a point particle with coordinate $x(t)$ and momentum $k(t)$ which satisfy the Hamilton equations

In usual situations the Hamiltonian (dispersion relation) $\omega$ is a function of $k,x,t$, $\omega=\omega(k,x,t)$. Following the excellent textbook gant-66 , let us define in the three-dimensional extended phase space $(x,k,t)$ an arbitrary closed curve

where the points corresponding to $\eta=0$ and $\eta=1$ coincide with each other. Then we can calculate along this curve the integral

where $\delta x$ and $\delta t$ denote differentials of the functions $x=x_{0}(\eta)$, $t=t_{0}(\eta)$ at the points of the contour $C_{0}$. Now, we define the flow in the phase space by the equations

and, when solutions of these equations cross the contour $C_{0}$, we obtain a tube of trajectories

where $x(t_{0}(\eta),\eta)=x_{0}(\eta)$, $k(t_{0}(\eta),\eta)=k_{0}(\eta)$. If we take now another function $t=t_{1}(\eta)$, then we get a new contour

and a new value of the integral similar to Eq. (19),

which is taken over the contour $C_{1}$ around the tube of trajectories generated by the flow (21). Now one can ask, for which Eqs. (20) and corresponding flows (21) the integrals (19) and (23) are equal to each other independently of the choice of the contour $C_{1}$ (that is the choice of the function $t=t_{1}(\mu)$). As was shown in Refs. poincare ; cartan ; gant-66 , this is true for

that is when the flow obeys the Hamilton equations (17). Thus, the invariance of the Poincaré-Cartan integral

means that the flow (20) is Hamiltonian.

Now we change the perspective and look at invariance of the integral (25) from a different point of view. As was indicated in Introduction, we are interested in invariance of this integral with respect to flows generated by the hydrodynamic equations (10) or (16). In this case the Hamiltonian $\omega$ depends on $x$ and $t$ only via solutions $\rho=\rho(x,t)$, $u=u(x,t)$ of these equations, i.e.,

and the flow is generated by the equation

Then invariance of the integral (25) imposes certain conditions on the dependence of the momentum (wave number) $k$ on the background variables,

If the condition of invariance is fulfilled, then the action (4) can be reduced to the generalized Bohr-Sommerfeld quantization rule (9) which provides important information about parameters of asymptotic solitons. We shall consider the condition of invariance of the integral (25) separately for the simple wave flow (16) and for the general solutions of Eqs. (10),

Now, when each soliton is labeled by a specific value of the parameter $\overline{q}_{n}$ or $q_{n}$, we need to relate it with soliton’s physical parameters, as, for example, its velocity $V_{n}$. This can be done with the use of Stokes remark lamb ; stokes (see also ai-77 ; dkn-03 ) that the small amplitude tails $\propto\exp[\pm\widetilde{k}(x-Vt)]$ of a soliton obey the same linearized equations as a small-amplitude harmonic wave $\propto\exp[i(kx-\omega t)]$. Consequently, the soliton’s velocity $V_{n}$ can be expressed in the form

and $\widetilde{k}$ denotes the soliton’s inverse half-width (see Refs. el-05 ; kamch-20a ; kamch-21 ). This means analytical continuation of the function $\omega=F(k^{2})$ from the interval of positive values of its argument $k^{2}$ to the interval of its negative values of the argument $k^{2}=-\widetilde{k}^{2}<0$. In practice it means that we consider the same function $F$ at different intervals of its argument $k^{2}$.

We generalize this Stokes observation to other functions of $k^{2}$ that can also be continued to the region $k^{2}<0$ where they express some assertions about the inverse half-widths $\widetilde{k}$ of solitons. For example, suppose we have found the exact solution (31) of Eq. (30) and it can be expressed in the form

This formula can be continued to the solitonic region to give

Suppose we have found from the Bohr-Sommerfeld rule the value $\overline{q}_{n}$ for the $n$-th soliton and at asymptotically large time this soliton propagates along zero background $u=0$. Then its inverse half-width is given by the formula

and, consequently, its velocity is readily obtained from Eq. (46). This approach reproduces in a different form the theory developed for simple wave pulses in Refs. egs-08 ; mfweh-20 .

We can extend the above approach to the case of exact solutions of Eqs. (38), so we imply that the condition (39) is fulfilled. Again we express the exact solution in the form

and analytically continue it to the soliton region to obtain

If we found for the $n$-th soliton the eigenvalue $q_{n}$ from the Bohr-Sommerfeld rule (45) and if the asymptotic soliton propagates along the uniform background with $\rho=\rho_{R},u=0$, then its inverse half-width is given by the formula

and its velocity is to be found from the Stokes rule (46).

However such an analytical continuation becomes impossible in case of asymptotic solutions (41) which are only correct for $k\gg c\sim u$. Nevertheless, we can overcome this difficulty in the following way. Suppose we have found the asymptotic solution in the form (50). Naturally, this general solution remains correct in case of simple-wave pulses when there is the functional relationship between $\rho$ and $u$, say, $\rho=\rho_{\mathrm{sw}}(u)$, so we get the asymptotic formula

Now, the asymptotic soliton trains propagate along simple-wave solutions, so their inverse half-widths are asymptotic specifications of the formula (48) obtained by analytic continuation of Eq. (47). If we take its series expansion with respect to small parameter $u$, then the resulting formula

must coincide with Eq. (53). This matching condition allows one to find the relationship between the parameters $q$ and $\overline{q}$:

Consequently, the inverse half-widths of asymptotic solitons are given by Eq. (49),

where the parameters $q_{n}$ are to be obtained from the Bohr-Sommerfeld rule (45) and we assumed that the asymptotic solitons propagate along the background with $u=0$. At last, the soliton’s velocities are to be found according to the Stokes rule (46).

Let us illustrate this theory by examples.

Here we shall consider evolution of a pulse according to the defocusing generalized NLS equation (generalized Gross-Pitaevskii equation for a Bose-Einstein condensate of atoms with repulsive interaction)

where the nonlinearity function is positive: $f(\rho)>0$, $f(0)=0$. It is well known that by means of the substitution

this equation can be transformed to the hydrodynamic-like system

where

Linearization of this system with respect to small deviations from a uniform flow yields the Bogoliubov dispersion relation

that is $c$ has the meaning of the sound velocity in the dispersionless limit $k\to 0$. Equations (59) reduce in the same limit to the standard Euler equations (10).

Now we can check whether the derivatives (38) commute. In this case it is convenient to replace $\rho$ by the variable $c$ where the function $\rho=\rho(c)$ is obtained by inversion of the function $c=c(\rho)$ (see Eq. (60)) so Eqs. (38) are cast to the form

The straightforward calculation yields (see also sk-23 )

and we see that the compatibility condition is only fulfilled for the case of the Kerr-like nonlinearity with $f(\rho)=\rho$ and $\rho=c^{2}$. Otherwise, it is fulfilled only in the limit $k\to\infty$, so it is natural to consider these two situations separately.