A Perturbation Theory for Nonintegrable Equations with Small Dispersion

Abstract

We describe an approach called the “weak asymptotics method” to construct multisoliton asymptotic solutions for essentially nonintegrable equations with small dispersion. This paper contains a detailed review of the method and a perturbation theory to describe the interaction of distorted solitons for equations with small perturbations. All constructions have been realized for the gKdV equation with the nonlinearity uμ, μ ∈ (1, 5).

1. Introduction

We consider the problem of propagation and interaction of soliton-type solutions of nonlinear equations. Our basic example is the nonhomogeneous version of the generalized KdV equation

$\frac{\partial u}{\partial t}+\frac{\partial u^{\mu }}{\partial x}+{\epsilon }^2\frac{{\partial }^{3}u}{\partial x^3}=f\left(u,\epsilon \frac{\partial u}{\partial x}\right),x\in R^1,t>0,$E1

where μ ∈ (1, 5), ε ≪ 1, f(u, z) is a known smooth function such that f(0, 0) = 0. Note that the restriction on μimplies both the soliton-type solution and the stability of the equation with respect to initial data (see, for example [1, 2]).

In the special case f ≡ 0 and μ = 2 (μ = 3), Eq. (1) is the famous KdV (modified KdV) equation. It is well known that KdV (mKdV) solitons are stable and interact in the elastic manner: after the collision, they preserve the original amplitudes and velocities shifting the trajectories only (see [3] and other bibliographies devoted to the inverse scattering transform (IST) method). In the case of μ = 2 (μ = 3) but with f ≠ 0, Eq. (1) is a nonintegrable one. However, using the smallness of ε(or of ffor other scaling), it is possible to create a perturbation theory that describes the evolution of distorted solitons (see the approaches by Karpman and E. Maslov [4] and Kaup and Newell [5] on the basis of the IST method, and the “direct” method by V. Maslov and Omel’yanov [6]). Moreover, the approach by V. Maslov and Omel’yanov [6] can be easily extended to essentially nonintegrable equations (μ ≠ 2, 3), but for a single soliton only. In fact, it is impossible to use any direct method in the classical sense for the general problem of the wave interaction. To explain this proposition, let us consider the homogeneous gKdV equation

$\frac{\partial u}{\partial t}+\frac{\partial u^{\mu }}{\partial x}+{\epsilon }^2\frac{{\partial }^{3}u}{\partial x^3}=0,x\in R^1,t>0.$E2

It is easy to find the explicit soliton solution of (2),

$u\left(x,t,\epsilon \right)=\mathit{A\omega }\left(\beta \left(x-\mathit{Vt}\right)/\epsilon \right),\omega \left(\eta \right)={\text{cosh}}^{-\gamma }\left(\eta /\gamma \right),$E3

$\gamma =2/\left(\mu -1\right),V={\beta }^2,A^{\mu -1}=V\left(\mu +1\right)/2.$E4

Next let us consider two-soliton initial data

${\left.u\right|}_{t=0}=\sum _{i=1}^2{A}_i\omega \left({\beta }_i\left(x-x_{\left(i,0\right)}\right)/\epsilon \right),$E5

where x_(1, 0) > x_(2, 0) and A₂ > A₁. Obviously, since (x_(2, 0) − x_(1, 0))/ε → ∞ as ε → 0, the sum of the waves (3)

$u=\sum _{i=1}^2{A}_i\omega \left({\beta }_i\left(x-V_{i}t-x_{\left(i,0\right)}\right)/\epsilon \right)$E6

approximates the problem (2), (5) solution with the precision O(ε^∞) but for t ≪ 1 only. Conversely, the sum (6) does not satisfy the gKdV equation for t ∼ O(1) in view of the trajectories x = V_it + x_(i, 0) intersection at a point (x^*, t^*).

Let us consider shortly how it is possible to analyze the problem (2), (5). There are some different cases:

1. Let A₁ ≪ A₂. Then, one can construct an asymptotic solution

$u=W\left(\left(x-{\varphi }_2\left(t\right)\right)/\epsilon ,t,x,\epsilon ,\nu \right),$E7

where ν = A₁/A₂ ≪ 1 and W((x − ϕ₂(t))/ε, t, x, ε, ν) = A₂ω(β₂(x − V₂t − x_(2, 0))/ε) + O(ν + ε). Thus, to find the leading term of the asymptotics, we obtain an equation with nonlinear ordinary differential operator; whereas to construct the corrections, it is enough to analyze the linearization of this operator. This construction (with a little bit of other viewpoints) has been realized by Ostrovsky et al. [7].

1. Let A₂ − A₁ ≪ 1. We write again the ansatz in the form (7), where ν = A₂ − A₁ ≪ 1 now, and we assume ν/ε ≪ 1. In fact, this case coincides with the problem considered in [7].

2. The amplitudes A₂ > A₁ are arbitrary numbers. Then, we should write a two-phase ansatz

$u\left(x,t,\epsilon \right)=W\left(\left(x-{\varphi }_1\left(t\right)\right)/\epsilon ,\left(x-{\varphi }_2\left(t\right)\right)/\epsilon ,t,x,\epsilon \right)$E8

without any additional parameter. Substituting (8) into equation (2), we obtain for the leading term W₀(τ₁, τ₂, t):

$\widehat{A}{W}_0+\widehat{B}{W}_0^{\mu }+{\widehat{B}}^3{W}_0=0,\widehat{A}=-\sum _{i=1}^2{\stackrel{\mathrm{̇}}{\varphi }}_i\frac{\partial }{\partial {\tau }_i},\widehat{B}=\sum _{i=1}^2\frac{\partial }{\partial {\tau }_i}.$E9

Since ${\stackrel{\mathrm{̇}}{\varphi }}_1\ne {\stackrel{\mathrm{̇}}{\varphi }}_2$, we can pass to new variables, $\eta =\left({\tau }_1-{\tau }_2\right)/\left({\stackrel{\mathrm{̇}}{\varphi }}_2-{\stackrel{\mathrm{̇}}{\varphi }}_1\right)$, $\zeta =\left({\stackrel{\mathrm{̇}}{\varphi }}_1{\tau }_2-{\stackrel{\mathrm{̇}}{\varphi }}_2{\tau }_1\right)/\left({\stackrel{\mathrm{̇}}{\varphi }}_1-{\stackrel{\mathrm{̇}}{\varphi }}_2\right)$, and transform equation (9) to the gKdV form (2) again

$\frac{\partial W_0}{\partial \eta }+\frac{\partial W_0^{\mu }}{\partial \zeta }+\frac{{\partial }^3{W}_0}{\partial {\zeta }^3}=0.$E10

Therefore, to construct two-phase asymptotics, we should solve (10) explicitly what is impossible for any essentially nonintegrable case.

This difficulty can be overcome by using the weak asymptotics method. The main point here is that solitons tend to distributions as ε → 0. Thus, it is possible to pass to the weak description of the problem, ignore the actual shape of the multiwave solutions, and find only the main solution characteristics, that is, the time dynamics of wave amplitudes and velocities. The weak asymptotics method has been proposed at first for shock wave type solutions [8] and for soliton-type solutions [9] many years ago. Further generalizations, modifications, and adaptations to other problems can be found in publications by M. Colombeau, Danilov, Mitrovic, Omel’yanov, Shelkovich, and others, see, for example, [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and references therein.

The contents of the paper are the following: in Section 2, we present a detailed survey of the weak asymptotics method application to the problem of multisoliton asymptotics and Section 3 contains new results, namely a perturbation theory to describe the evolution and collision of distorted solitons for equation (1).

2. Weak asymptotics method

2.1. Main definitions

Let us associate equation (2) with first two conservation laws written in the differential form:

$\frac{\partial Q_j}{\partial t}+\frac{\partial P_j}{\partial x}={\epsilon }^2\frac{{\partial }^3{R}_j}{\partial x^3},j=1,2,$E11

$Q_1=u,P_1=u^{\mu },Q_2=u^2,P_2=2\mu u^{\mu +1}/\left(\mu +1\right)-3{\left(\epsilon u_x\right)}^2,$E12

and R₁ = u, R₂ = u². Next, we define smallness in the weak sense:

Definition 1.A function v(t, x, ε) is said to be of the value$O_{\mathcal{D}\text{'}}\left({\epsilon }^{\mathrm{\varkappa }}\right)$if the relation${\int }_{-\infty }^{\infty }v\left(t,x,\epsilon \right)\Psi \left(x\right)\mathit{dx}=O\left({\epsilon }^{\mathrm{\varkappa }}\right)$holds uniformly in t for any test function$\Psi \in \mathrm{}\left({\mathbb{R}}_x^1\right)$. The right-hand side here is a${\mathrm{}}^{\infty }$-function for ε = const > 0 and a piecewise continuous function uniformly in ε ≥ 0.

Following [9, 17, 18], we define two-soliton weak asymptotics:

Definition 2.A sequence u(t, x, ε), belonging to${\mathcal{C}}^{\infty }\left(0,T;{\mathcal{C}}^{\infty }\left({\mathrm{ℝ}}_x^1\right)\right)$for ε = const > 0 and belonging to$\mathcal{C}\left(0,T;{\mathcal{D}}^{\text{'}}\left({\mathrm{ℝ}}_x^1\right)\right)$uniformly in ε, is called a weak asymptotic mod$O_{\mathcal{D}\text{'}}\left({\epsilon }^2\right)$solution of(2)if the relations(11)hold uniformly in t with the accuracy$O_{\mathcal{D}\text{'}}\left({\epsilon }^2\right)$.

Let us consider the interaction of two solitary waves for the model (2) with the initial data (5).

Following [9, 17, 18] again, we write the asymptotic ansatz in the form:

$u=\sum _{i=1}^2{G}_i\left(\tau \right)\omega \left({\beta }_i\left(x-{\varphi }_i\left(t,\tau ,\epsilon \right)\right)/\epsilon \right),G_i\left(\tau \right)=A_i+S_i\left(\tau \right).$E13

Here ϕ_i = ϕ_i0(t) + εϕ_i1(τ), where ϕ_i0 = V_it + x_(i, 0) are the trajectories of noninteracting solitary waves, τ = ψ₀(t)/εdenotes the “fast time”, ψ₀(t) = β₁(ϕ₂₀(t) − ϕ₁₀(t)), and the phase and amplitude corrections ϕ_i1, S_iare smooth functions such that with exponential rates

${\varphi }_{i1}\left(\tau \right)\to 0\mathit{as}\tau \to -\infty ,{\varphi }_{i1}\left(\tau \right)\to {\varphi }_{i1}^{\infty }={\mathit{\text{const}}}_i\mathit{as}\tau \to +\infty ,$E14

$S_i\left(\tau \right)\to 0\mathit{as}\tau \to \pm \infty .$E15

2.2. Two-wave asymptotic construction

To construct the asymptotics, we should calculate the weak expansions of the terms from the left-hand sides of the relations (11). It is easy to check that

$u=\epsilon \sum _{i=1}^2{a}_1\frac{G_i}{{\beta }_i}\delta \left(x-{\varphi }_i\right)+O_{\mathcal{D}\text{'}}\left({\epsilon }^3\right),$E16

where δ(x) is the Dirac delta-function. Here and in what follows, we use the notation

$a_k\stackrel{\mathrm{def}}{=}{\int }_{-\infty }^{\infty }{\left(\omega \left(\eta \right)\right)}^k\mathit{d}\mathit{\eta },k>0,a_2^{\prime }\stackrel{\mathrm{def}}{=}{\int }_{-\infty }^{\infty }{\left({\omega }^{\prime }\left(\eta \right)\right)}^2\mathit{d}\mathit{\eta }.$E17

At the same time for any F(u, ε∂u/∂x) ∈ C¹, we have

$\begin{array}{l}{\int }_{-\infty }^{\infty }F\left(\sum _{i=1}^2{G}_i\omega \left({\beta }_i\frac{x-{\varphi }_i}{\epsilon }\right),\sum _{i=1}^2{\beta }_i{G}_i{\omega }^{\prime }\left({\beta }_i\frac{x-{\varphi }_i}{\epsilon }\right)\right)\psi \left(x\right)\mathit{dx}\\ =\epsilon \sum _{i=1}^2\frac{1}{{\beta }_i}{\int }_{-\infty }^{\infty }F\left(A_i\omega \left(\eta \right),{\beta }_i{A}_i{\omega }^{\prime }\left(\eta \right)\right)\psi \left({\varphi }_i+\epsilon {\frac{\eta }{\beta }}_i\right)\mathit{d}\mathit{\eta }\\ +\frac{\epsilon }{{\beta }_2}{\int }_{-\infty }^{\infty }\{F\left(\sum _{i=1}^2{G}_i\omega \left({\eta }_{i2}\right),\sum _{i=1}^2{\beta }_i{G}_i{\omega }^{\prime }\left({\eta }_{i2}\right)\right)\\ -\sum _{i=1}^{2}F\left(A_i\omega \left({\eta }_{i2}\right),{\beta }_i{A}_i{\omega }^{\prime }\left({\eta }_{i2}\right)\right)\}\psi \left({\varphi }_2+\epsilon {\frac{\eta }{\beta }}_2\right)\mathit{d}\mathit{\eta },\end{array}$E18

where

${\eta }_{12}=\mathit{\theta \eta }-\sigma ,{\eta }_{22}=\eta ,\sigma ={\beta }_1\left({\varphi }_1-{\varphi }_2\right)/\epsilon ,\theta ={\beta }_1/{\beta }_2.$E19

We take into account that the second integrand in the right-hand side of (18) vanishes exponentially fast as ∣ϕ₁ − ϕ₂∣ grows; thus, its main contribution is at the point x^*. We write

${\varphi }_{i0}=x^{*}+V_i\left(t-t^{*}\right)=x^{*}+\epsilon V_i\tau /{\stackrel{\mathrm{̇}}{\psi }}_0\text{and}{\varphi }_i=x^{*}+{\mathit{\epsilon \chi }}_i,$E20

where ${\dot{\psi }}_0={\beta }_1\left(V_2-V_1\right)$, ${\chi }_i=V_i\tau /{\stackrel{\mathrm{̇}}{\psi }}_0+{\varphi }_{i1}$. It remains to apply the formula

$f\left(\tau \right)\delta \left(x-{\varphi }_i\right)=f\left(\tau \right)\delta \left(x-x^{*}\right)-{\mathit{\epsilon \chi }}_{i}f\left(\tau \right){\delta }^{\prime }\left(x-x^{*}\right)+O_{\mathcal{D}\text{'}}\left({\epsilon }^2\right),$E21

which holds for each ϕ_iof the form (20) with slowly increasing χ_iand for f(τ) from the Schwartz space. Moreover, the second term in the right-hand side of (21) is $O_{\mathcal{D}\text{'}}\left(\epsilon \right)$. Thus, under the assumptions (14) and (15), we obtain the weak asymptotic expansion of F(u, ε∂u/∂x) in the final form:

$F\left(u,\epsilon u_x\right)=\epsilon \sum _{i=1}^2\left\{\frac{a_{F,i}^{\left(0\right)}}{{\beta }_i}\delta \left(x-{\varphi }_i\right)-\epsilon \frac{a_{F,i}^{\left(1\right)}}{{\beta }_i}{\delta }^{\prime }\left(x-x^{*}\right)\right\}+\frac{\epsilon }{{\beta }_2}\left\{{\mathfrak{R}}_F^{\left(0\right)}\delta \left(x-x^{*}\right)-\epsilon {\overline{\mathfrak{R}}}_F{\delta }^{\prime }\left(x-x^{*}\right)\right\}\mathrm{}+O_{\mathcal{D}\text{'}}\left({\epsilon }^3\right),{\overline{\mathfrak{R}}}_F={\chi }_2{\mathfrak{R}}_F^{\left(0\right)}+{\mathfrak{R}}_F^{\left(1\right)}/{\beta }_2,$E22

where

$a_{F,i}^{\left(n\right)}={\int }_{-\infty }^{\infty }{\eta }^{n}F\left(A_i\omega \left(\eta \right),{\beta }_i{A}_i{\omega }^{\prime }\left(\eta \right)\right)\mathit{d\eta },$E23

${\mathrm{\Re }}_F^{\left(n\right)}={\int }_{-\infty }^{\infty }{\eta }^n\left\{F\left(\sum _{i=1}^2{G}_i\omega \left({\eta }_{i2}\right),\sum _{i=1}^2{\beta }_i{G}_i{\omega }^{\prime }\left({\eta }_{i2}\right)\right)-\sum _{i=1}^{2}F\left(A_i\omega \left({\eta }_{i2}\right),{\beta }_i{A}_i{\omega }^{\prime }\left({\eta }_{i2}\right)\right)\right\}\mathit{d\eta }.$E24

Here, we take into account that to define $\text{∂}{u}^2/\text{∂}t\text{mod}{O}_{\mathcal{D}\text{'}}\left({\epsilon }^2\right)$, it is necessary to calculate u² with the precision $O_{\mathcal{D}\text{'}}\left({\epsilon }^3\right)$. Thus, using (22) with F(u) = u² and transforming (16) with the help of (21), we obtain modulo $O_{\mathcal{D}\text{'}}\left({\epsilon }^3\right)$:

$u=\epsilon \sum _{i=1}^2{a}_1{K}_{i0}^{\left(1\right)}\delta \left(x-{\varphi }_i\right)+\epsilon \sum _{i=1}^2{a}_1{K}_{i1}^{\left(1\right)}\left\{\delta \left(x-x^{*}\right)-{\mathit{\epsilon \chi }}_i{\delta }^{\prime }\left(x-x^{*}\right)\right\},$E25

$u^2=\epsilon \sum _{i=1}^2{a}_2{K}_{i0}^{\left(2\right)}\delta \left(x-{\varphi }_i\right)+\frac{\epsilon }{{\beta }_2}\left\{{\mathfrak{R}}_{u^2}^{\left(0\right)}\delta \left(x-x^{*}\right)-\epsilon {\overline{\mathfrak{R}}}_{u^2}{\delta }^{\prime }\left(x-x^{*}\right)\right\},$E26

where

$K_i^{\left(n\right)}=G_i^n/{\beta }_i,K_{i0}^{\left(n\right)}=A_i^n/{\beta }_i,K_{i1}^{\left(n\right)}=K_i^{\left(n\right)}-K_{i0}^{\left(n\right)}.$E27

Calculating weak expansions for other terms from Definition 2 and substituting them into (11), we obtain linear combinations of εδ^′(x − ϕ_i), i = 1, 2, δ(x − x^*), and εδ^′(x − x^*). Therefore, we pass to the system:

$a_1{V}_i{K}_{i0}^{\left(1\right)}-a_{P_1,i}^{\left(0\right)}/{\beta }_i=0,a_2{V}_i{K}_{i0}^{\left(2\right)}-a_{P_2,i}^{\left(0\right)}/{\beta }_i=0,i=1,2,$E28

$\sum _{i=1}^2{K}_{i1}^{\left(1\right)}=0,{\mathrm{\Re }}_{u^2}^{\left(0\right)}=0,i=1,2,$E29

${\stackrel{\mathrm{̇}}{\psi }}_0\frac{d}{\mathit{d}\mathit{\tau }}\sum _{i=1}^2\left\{K_{i0}^{\left(1\right)}{\varphi }_{i1}+{\chi }_i{K}_{i1}^{\left(1\right)}\right\}=f,{\stackrel{\mathrm{̇}}{\psi }}_0\frac{d}{\mathit{d}\mathit{\tau }}\left\{\sum _{i=1}^2{a}_2{K}_{i0}^{\left(2\right)}{\varphi }_{i1}+{\overline{\mathfrak{R}}}_{u^2}\right\}=F,$E30

where

$f=\frac{1}{a_1{\beta }_2}{\mathfrak{R}}_{P_1}^{\left(0\right)},F=\frac{1}{{\beta }_2}{\mathfrak{R}}_{P_2}^{\left(0\right)}-a_1{\stackrel{\mathrm{̇}}{\psi }}_0\sum _{i=1}^2{\varphi }_{i1}\frac{{\mathit{dK}}_{i1}^{\left(2\right)}}{\mathit{d}\mathit{\tau }}.$E31

The first four algebraic equations (28) imply again the relation (4) among A_i, β_i, and V_i. Furthermore, there exists a number θ^* ∈ (0, 1) such that equations (29), (30) have the required solution S_i, ϕ_i1 with the properties (14) and (15) under the sufficient condition θ ≤ θ^* (see [9, 17]). It is obvious that the existence of the weak asymptotics (13) with the properties (14) and (15) implies that the solitary waves interact like the KdV solitons at least in the leading term.

Theorem 1.Let θ ≤ θ^*. Then(13)describes$\mathit{mod}{O}_{\mathcal{D}\text{'}}\left({\epsilon }^2\right)$the elastic scenario of the solitary waves interaction for the μ-gKdVequation (2).

Numerical simulations ([14, 15, 17]) confirm the traced analysis, see Figure 1. Note that a small oscillating tail appears after the soliton collision, see [15] for detail. Obviously, this effect is similar to the “radiation” appearance for the perturbed KdV [21].

Figure 1.

Evolution of two solitary waves forμ = 4 andε = 0.1.

2.3. Multisoliton interaction

N-wave solutions of the form similar to waves (13) contain 2Nfree functions S_i, ϕ_i1. Thus, to describe an N-soliton collision, we should consider Nconservation laws. However, nonintegrability implies the existence of a finite number of conservation laws only. For this reason, we need to involve into the consideration balance laws. For the gKdV-4 equation, the first conservation and balance laws have the form

$\frac{\partial Q_j}{\partial t}+\frac{\partial P_j}{\partial x}+{\epsilon }^{-1}{K}_j=O_{\mathcal{D}\text{'}}\left({\epsilon }^2\right),$E32

where Q_j, P_j, j = 1, 2, coincide with (12) for μ = 4, K_i = 0, i = 1, 2, 3,

$Q_3={\left(\epsilon u_x\right)}^2-\frac{2}{5}{u}^5,P_3=16u^3{\left(\epsilon u_x\right)}^2-u^8-3{\left({\epsilon }^2{u}_{\mathit{xx}}\right)}^2,$E33

$Q_4=\frac{1}{2}{\left({\epsilon }^2{u}_{\mathit{xx}}\right)}^2+\frac{5}{21}{u}^8-\frac{10}{3}{u}^3{\left(\epsilon u_x\right)}^2,K_4=-{\left(\epsilon u_x\right)}^5,$E34

$P_4=12u^3{\left({\epsilon }^2{u}_{\mathit{xx}}\right)}^2-19u{\left(\epsilon {\mathrm{ }u}_x\right)}^4-\frac{3}{2}{\left({\epsilon }^3{u}_{\mathit{xxx}}\right)}^2+\frac{160}{231}{u}^{11}-\frac{100}{3}{u}^6{\left(\epsilon u_x\right)}^2.$E35

Note that the nondivergent “production” ε⁻¹K₄ has the same value O(ε⁻¹) (in the C-sense and for rapidly varying functions) as the first ones in (32).

The formal scheme of the asymptotic construction is similar to the one described above: we write the ansatz of the form (13) but with Nsummands, found weak representations for all terms in (32), and pass to a system similar to (28)–(30). The main obstacle here is the proof that this system admits a solution with the properties of (14), (15). This idea has been realized in [18, 19] for the problem of three soliton collisions for the gKdV-4 equation.

Theorem 2.Let us denote A_i the amplitudes of the original solitons and x_(i, 0)their initial positions such that A_i + 1 > A_i, x_(i, 0) > x_{(i + 1, 0)}, and i = 1, 2. Let all trajectories x = ϕ_i0(t) have an intersection point(x^*, t^*). Then, under the assumption

${\beta }_2/{\beta }_3={\nu }^3,{\beta }_1/{\beta }_3={\nu }^{3\left(3+\alpha \right)/2},\alpha \in \left[0,1\right)$E36

with sufficiently smallν < 1, the three-phase asymptotic solution exists and describes$\mathit{mod}{O}_{\mathcal{D}\text{'}}\left({\epsilon }^2\right)$the elastic scenario of the solitary waves interaction.

Figure 2 depicts the evolution of a three-wave solution [14].

Figure 2.

Evolution of the soliton triplet withμ = 4,ε = 0.1.

2.4. Asymptotic equivalence

Let us come back to the case of two-phase asymptotics and transform the ansatz (13) to the following form:

$\tilde{u}=\sum _{i=1}^2\left\{G_i\left(\tau \right)\omega \left({\beta }_i\frac{x-{\varphi }_i\left(t,\tau ,\epsilon \right)}{\epsilon }\right)+{\mathfrak{S}}_i\left(\tau \right)W\left({\beta }_i\frac{x-{\varphi }_i\left(t,\tau ,\epsilon \right)}{\epsilon }\right)\right\},$E37

where ${\mathfrak{S}}_i\left(\tau \right)$, i = 1, 2 are arbitrary functions from the Schwartz space,

$W\left(\eta \right)=d^{2l+1}\omega \left(\eta \right)/d{\eta }^{2l+1},$E38

and l ≥ 1 is an arbitrary integer. Calculating the weak representations for $\tilde{u}$and ${\tilde{u}}^2$, we obtain

$\begin{array}{l}\tilde{u}=u+O_{\mathcal{D}\text{'}}\left({\epsilon }^{2l+2}\right),\\ {\tilde{u}}^2=\epsilon \sum _{i=1}^2{a}_2{K}_{i0}^{\left(2\right)}\delta \left(x-{\varphi }_i\right)+\frac{\epsilon }{{\beta }_2}{\mathfrak{R}}_{{\tilde{u}}^2}^{\left(0\right)}\delta \left(x-x^{*}\right)+O_{\mathcal{D}\text{'}}\left({\epsilon }^2\right),\end{array}$E39

where

${\mathrm{\Re }}_{{\tilde{u}}^2}^{\left(0\right)}={\int }_{-\infty }^{\infty }\left\{{\left(\sum _{i=1}^2\left(G_i\omega \left({\eta }_{i2}\right)+{\mathfrak{S}}_{i}W\left({\eta }_{i2}\right)\right)\right)}^2-\sum _{i=1}^2{\left(A_i\omega \left({\eta }_{i2}\right)\right)}^2\right\}\mathit{d\eta },$E40

and uin the right-hand side in (39) is the representation (25). Thus, the difference between uof the forms (13) and (37) is arbitrarily small in the sense ${\mathcal{D}}^{\text{'}}\left({\mathrm{ℝ}}_x\right)$. At the same time, instead of (29), (30), we obtain

$\sum _{i=1}^2{K}_{i1}^{\left(1\right)}=0,{\mathrm{\Re }}_{{\tilde{u}}^2}^{\left(0\right)}=0,i=1,2,$E41

${\stackrel{\mathrm{̇}}{\psi }}_0\frac{d}{\mathit{d}\mathit{\tau }}\sum _{i=1}^2\left\{K_{i0}^{\left(1\right)}{\varphi }_{i1}+{\chi }_i{K}_{i1}^{\left(1\right)}\right\}=\tilde{f},{\stackrel{\mathrm{̇}}{\psi }}_0\frac{d}{\mathit{d}\mathit{\tau }}\left\{\sum _{i=1}^2{a}_2{K}_{i0}^{\left(2\right)}{\varphi }_{i1}+{\overline{\mathfrak{R}}}_{{\tilde{u}}^2}\right\}=\tilde{F},$E42

where $\tilde{f}$, $\tilde{F}$differ from f, Fin the same manner as ${\mathrm{\Re }}_{{\tilde{u}}^2}^{\left(0\right)}$differs from ${\mathrm{\Re }}_{u^2}^{\left(0\right)}$. The system (41) and (42) have again a solution with the properties (14) and (15) [9, 12]; however, it differs from the solution of Eqs. (29) and (30) with the value O(1) in the C-sense. Moreover, the asymptotic solutions (13) and (37) differ with the precision $O_{\mathcal{D}\text{'}}\left(\epsilon \right)$in the sense of Definition 1. This implies the principal impossibility to describe explicitly neither the real shape of the waves at the time instant of the collision nor the real ε-size displacements of the trajectories after the interaction. However, the nonuniqueness of the value O(ε) is concentrated within O(ε^1 − ν)-neighborhood of the time instant t^* of the interaction, ν > 0. Thus, it is small in the ${\mathcal{D}}^{\text{'}}\left(R_{x,t}^2\right)$sense. We set

Definition 3.Functions u₁(x, t, ε) and u₂(x, t, ε) are said to be asymptotically equivalent if for any test function$\psi \in \mathcal{D}\left(R^2\right)$

${\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left\{u_1\left(x,t,\epsilon \right)-u_2\left(x,t,\epsilon \right)\right\}\psi \left(x,t\right)\mathit{dx}\mathit{dt}=O\left({\epsilon }^2\right).$E43

In this sense, the solutions (13) and (37) are asymptotically equivalent.

We now focus attention on another question: how to choose, from the set of all possible conservation and balance laws, those that allow to construct a multiphase asymptotic solution? It seems that there is not any rule and it is possible to use arbitrary combination of the laws. Thus, there appears the next question: what is the difference between such solutions? This problem has been discussed in [20] for two-phase asymptotic solutions of the gKdV-4 equation. Let us define two-phase asymptotics in the following manner:

Definition 4.Let1 ≤ k₀ < k₁ ≤ 4 and let a sequence u_k0, k1 = u_k0, k1(t, x, ε) belong to the same functional space as u(t, x, ε) inDefinition 2. Then, u_k0, k1is called a weak asymptotic mod$O_{\mathcal{D}\text{'}}\left({\epsilon }^2\right)$solution of(2)if therelations (32)hold for j = k₀and j = k₁uniformly in t.

A detailed analysis implies the assertion [20].

Theorem 3.Let θ be sufficiently small. Then, the weak asymptotic solutions u_1, k1and$u_{1,k_1^{\prime }}$of the problem(2),(5)exist and they are asymptotically equivalent for all$k_1,k_1^{\prime }\in \left\{2,3,4\right\}$.

3. Collision of distorted solitons

We consider now the nonhomogeneous version of the gKdV equation (1). It is easy to verify that, in the case of rapidly varying solutions, the right-hand side fcan be treated as a “small perturbation.”

An approach to construct one-phase self-similar asymptotic solutions for (1) had been created in [6] (see also [17]). Let us generalize this approach to the multiphase case. From the beginning, we state that equation (1) is associated with balance laws, the first two of which are

$\frac{\partial Q_j}{\partial t}+\frac{\partial P_j}{\partial x}+K_j=O_{\mathcal{D}\text{'}}\left({\epsilon }^2\right),j=1,2,$E44

where Q_jand P_jcoincide with ones described in (12),

$K_1=-f\left(u,\epsilon u_x\right),K_2=-\mathit{uf}\left(u,\epsilon u_x\right).$E45

Note that, in contrast to K_jin (32), productions here are regularly degenerating functions with the value O(1) in the C-sense.

Let us first construct a two-phase version of self-similar asymptotics, which assumes a special initial data for (1) and discuss afterward how to treat it for more realistic initial data. By analogy with Definition 2, we write:

Definition 5.Let a sequence u = u(t, x, ε) belong to the same functional space as in Definition 2. Then u is called a weak asymptotic mod$O_{\mathcal{D}\text{'}}\left({\epsilon }^{\overline{q}}\right)$solution of(1)if therelation (44)hold uniformly in t ∈ (0, T), $\overline{q}=\text{min}\left\{\mu ,2\right\}$.

Generalizing one-phase asymptotics, we write the ansatz as

$u=\sum _{i=1}^2\left\{G_i\left(\tau \mathit{,}t\right)\omega \left({\eta }_i\right)+\epsilon \left(z_i\left(x,t\right)\mathfrak{H}\left({\eta }_i\right)+{\mathfrak{G}}_i\left(\tau \right)\omega \left({\eta }_i\right)\right)\right\},$E46

$G_i\left(\tau ,t\right)=A_i\left(t\right)+S_i\left(\tau \right),{\eta }_i={\beta }_i\left(t\right)\left(x-{\varphi }_i\right)/\epsilon ,{\varphi }_i\left(t,\tau ,\epsilon \right)={\varphi }_{i0}+{\mathit{\epsilon \phi }}_{i1}.$E47

Here A_i(t), ϕ_i0 = ϕ_i0(t), ${\beta }_i^2\left(t\right)=\gamma A^{\mu -1}\left(t\right)$, ω(η), S_i(τ), ϕ_i1(τ) are the same as in (13); τ = ψ₀(t)/εwith ψ₀(t) = ϕ₂₀(t) − ϕ₁₀(t) denotes the “fast time” again; $z_i\left(x,t\right)\in {\mathcal{C}}^{\infty }$; and ${\mathfrak{G}}_i$, ℌ are smooth functions such that

${\mathfrak{G}}_i\left(\tau \right)\to 0\mathit{as}\tau \to -\infty ,{\mathfrak{G}}_i\left(\tau \right)\to {\mathfrak{G}}_i^{\infty }={\mathit{\text{const}}}_i\mathit{as}\tau \to +\infty ,$E48

$\mathrm{ℌ}\left(\eta \right)\to 1\mathit{as}\eta \to -\infty ,\mathrm{ℌ}\left(\eta \right)\to 0\mathit{as}\eta \to +\infty$E49

with exponential rates. We assume also the intersection of the trajectories x = ϕ_i0(t), i = 1, 2 at a point x^* = ϕ_i0(t^*) namely,

${\left.\exists t^{*}>0\text{such\hspace{0.17em}that}{\varphi }_{10}\left(t^{*}\right)={\varphi }_{20}\left(t^{*}\right),{\stackrel{\mathrm{̇}}{\psi }}_0\stackrel{\mathrm{def}}{=}\frac{d}{\mathit{dt}}\left({\varphi }_{20}\left(t\right)-{\varphi }_{10}\left(t\right)\right)\right|}_{t=t^{*}}\ne 0.$E50

It is easy to verify the weak representations with the precision $O_{\mathcal{D}\text{'}}\left({\epsilon }^2\right)$:

$u=\epsilon \sum _{i=1}^2\left\{a_1{K}_{i0}^{\left(1\right)}\delta \left(x-{\varphi }_i\right)+a_1{K}_{i1}^{\left(1\right)*}\delta \left(x-x^{*}\right)+z_i\left(x,t\right)H\left({\varphi }_i-x\right)\right\},$E51

$\begin{array}{l}\frac{\partial u}{\partial t}=a_1{\stackrel{\mathrm{̇}}{\psi }}_0\frac{\partial }{\partial \tau }\sum _{i=1}^2{K}_{i1}^{\left(1\right)*}\delta \left(x-x^{*}\right)+\epsilon \sum _{i=1}^2\frac{\partial z_i}{\partial t}H\left({\varphi }_i-x\right)\\ +\epsilon a_1\sum _{i=1}^2\left\{\frac{{\mathit{dK}}_{i1}^{\left(1\right)}}{\mathit{dt}}+z_i\left({\varphi }_i,t\right)\frac{d{\varphi }_{i0}}{\mathit{dt}}\right\}\delta \left(x-{\varphi }_i\right)-\epsilon a_1\sum _{i=1}^2{K}_{i0}^{\left(1\right)}\frac{d{\varphi }_{i0}}{\mathit{dt}}{\delta }^{\prime }\left(x-{\varphi }_i\right)\\ {\left.+\epsilon \sum _{i=1}^2\left\{{\stackrel{\mathrm{̇}}{\psi }}_0\frac{\partial }{\partial t}\left(a_1\frac{{\mathfrak{S}}_i}{{\beta }_i}+z_i\left(x^{*},t\right){\varphi }_{i1}\right)-\frac{a_1}{{\beta }_i^2}{S}_i\frac{d{\beta }_i}{\mathit{dt}}\right\}\right|}_{t=t^{*}}\delta \left(x-x^{*}\right)\\ {\left.-\epsilon a_1\sum _{i=1}^2\left\{K_{i1}^{\left(1\right)}\frac{d{\varphi }_{i0}}{\mathit{dt}}+{\stackrel{\mathrm{̇}}{\psi }}_0\left(K_i^{\left(1\right)}\frac{\partial {\varphi }_{i1}}{\partial \tau }+\frac{{\chi }_i}{{\beta }_i}\frac{\partial K_i^{\left(1\right)}}{\partial \tau }\right)\right\}\right|}_{t=t^{*}}{\delta }^{\prime }\left(x-x^{*}\right),\end{array}$E52

where H(x) is the Heaviside function, H(x) = 0 for x < 0 and H(x) = 1 for x > 0; ${\left.g^{*}\stackrel{\mathrm{def}}{=}g\left(\tau ,t\right)\right|}_{t=t^{*}}$, ${\chi }_i\stackrel{\mathrm{def}}{=}{\varphi }_i\left(\Phi \left(\mathit{\epsilon \tau },t^{*}\right),\tau ,\epsilon \right)-x^{*}$, and Φ(ετ, t^*) is the solution of the equation ϕ₂₀(t^* + Φ) − ϕ₁₀(t^* + Φ) = ετ, which exists in accordance with (50).

Next, the existence of nonsoliton summands in (51) implying a correction of formula (22), namely

$F\left(u,\epsilon u_x\right)=\epsilon \sum _{i=1}^2{a}_{F,i}^{\left(0\right)}{\beta }_i^{-1}\delta \left(x-{\varphi }_i\right)+{\mathit{\epsilon \beta }}_2^{-1}{\mathfrak{R}}_F^{\left(0\right)}\delta \left(x-x^{*}\right)\mathrm{}+\epsilon F_u^{\prime }\left(0,0\right)\sum _{i=1}^2{z}_i\left(x,t\right)H\left({\varphi }_i-x\right)+O_{\mathcal{D}\text{'}}\left({\epsilon }^{\overline{q}}\right),$E53

where $a_{F,i}^{\left(0\right)}$and ${\mathrm{\Re }}_F^{\left(0\right)}$are defined in (23), (24), ${\left.F_u^{\prime }\left(0,0\right)=\text{∂}F(u,\mathrm{0)}/\text{∂}u\right|}_{u=0}$.

Repeating the same calculations as above, we obtain linear combinations of εδ(x − ϕ_i), εδ^′(x − ϕ_i), εH(ϕ_i − x), i = 1, 2; δ(x − x^*), εδ(x − x^*), and εδ^′(x − x^*). Equating zero, the coefficients of εδ(x − ϕ_i) and εδ^′(x − ϕ_i) yield

$a_1{A}_i\frac{d{\varphi }_{i0}}{\mathit{dt}}=a_{P_1,i}^{\left(0\right)},a_2\frac{d}{\mathit{dt}}\frac{A_i^2}{{\beta }_i}=\frac{a_{K_2,i}^{\left(0\right)}}{{\beta }_i},$E54

$a_2{A}_i^2\frac{d{\varphi }_{i0}}{\mathit{dt}}=a_{P_2,i}^{\left(0\right)},a_1\frac{d}{\mathit{dt}}\frac{A_i}{{\beta }_i}+z_i\left({\varphi }_i,t\right)\frac{d{\varphi }_{i0}}{\mathit{dt}}=\frac{a_{f,i}^{\left(0\right)}}{{\beta }_i}.$E55

Equation (54) forms the closed system to define A_i(t) and ϕ_i0(t). To simplify it, let us use the equalities (4) and rewrite the model equation for ω(η) as follows:

$\frac{d}{\mathit{d\eta }}\left\{-\omega +\frac{\mu +1}{2}{\omega }^{\mu }+\frac{d^2\omega }{d{\eta }^2}\right\}=0.$E56

Simple manipulations with (56) allow us to find relations between structural constants:

$a_1=\left(\mu +1\right)a_{\mu }/2,a_2=\left(\mu +3\right)a_{\mu +1}/4,a_2^{\prime }=\left(\mu -1\right)a_{\mu +1}/4.$E57

Next, we use (57), the equality ${\beta }_i^2=\gamma A_i^{\mu -1}$, add the initial conditions, and obtain from (54) the Cauchy problem

$\frac{{\mathit{dA}}_i}{\mathit{dt}}=-c_1{a}_{K_2,i}^{\left(0\right)}{A}_i^{-1},\frac{d{\varphi }_{i0}}{\mathit{dt}}=\frac{a_{\mu }}{a_1}{A}_i^{\mu -1},t>0,$E58

$A_i{\left|{}_{t=0}=A_i^0,{\varphi }_{i0}\right|}_{t=0}=x_{\left(i,0\right)},$E59

where c₁ = 2/(a₂(5 − μ)); $A_i^0>0$and x_(i, 0) are arbitrary numbers; and i = 1, 2. Note also that the first equalities in equations (54) and (55) are equivalent.

Next, equating zero the coefficients of the Heaviside functions, we obtain the equations

$\frac{\partial z_i}{\partial t}=f_u^{\prime }\left(0,0\right)z_i,x<{\varphi }_{i0}\left(t\right),t>0,i=1,2.$E60

In view of (58)dϕ_i0/dt > 0, so we use the second equality in (55) to state the correct initial condition for (60)

${\left.z_i\left(x,t\right)\right|}_{x={\varphi }_{i0}\left(t\right)}=\sqrt{\gamma }{a}_{f,i}^{\left(0\right)}{A}_i^{\left(3-4\mu \right)/2}\left(t\right)+c_2{a}_{K_2,i}^{\left(0\right)}{A}_i^{\left(1-3\mu \right)/2}\left(t\right),t>0,$E61

${\left.z_i\left(x,t\right)\right|}_{t=0}=z_i^0\left(x\right),x\le x_{\left(i,0\right)},$E62

where c₂ = a₁(3 − μ)(1 + μ)/(2a₂(5 − μ)), $z_i^0\left(x\right)$is an arbitrary smooth function, which satisfies the consistency condition

${\left.z_i^0\left(x_{\left(i,0\right)}\right)=\left\{\sqrt{\gamma }{a}_{f,i}^{\left(0\right)}{A}_i^{\left(3-4\mu \right)/2}+c_2{a}_{K_2,i}^{\left(0\right)}{A}_i^{\left(1-3\mu \right)/2}\right\}\right|}_{t=0}.$E63

We should note that the nonlinearity u^μin (1) can require the inequality u ≥ 0. To this end, we will assume

${\left.A_i\left(t\right)>0,z_i(x,t)\right|}_{x={\varphi }_{i0}\left(t\right)}\ge 0\text{for}t\ge 0.$E64

Furthermore, equating zero the coefficients of δ(x − x^*) and εδ^′(x − x^*) yield (29), (30) again. Consequently, the condition θ ≤ θ^* guaranties the existence of S_i, ϕ_i1 with the properties of (14), (15). In particular

$S_1=-\theta S_2{S}_1={\mathit{\theta \beta }}_1^{\gamma }\lambda \left(\sigma \right)\left(1+O\left({\theta }^q\right)\right),$E65

where

$q=\mathit{min}\left\{1,\gamma \right\},\lambda \left(\sigma \right)=a_2^{-1}{\int }_{-\infty }^{\infty }\omega \left(\eta \right)\omega \left({\eta }_{12}\right)\mathit{d\eta }.$E66

The last step of the construction is the determination of ${\mathfrak{G}}_i\left(\tau \right)$, i = 1, 2. By setting the coefficients of εδ(x − x^*) zero, we obtain

${\left.\frac{\partial }{\partial \tau }\sum _{i=1}^2\frac{{\mathfrak{G}}_i}{{\beta }_i^{*}}={\mathfrak{F}}_1,\frac{\partial }{\partial \tau }\sum _{i=1}^2\left(\frac{G_i}{{\beta }_i}+\frac{G_{\overline{i}}}{{\beta }_2}\lambda \left(\sigma \right)\right)\right|}_{t=t^{*}}{\mathfrak{G}}_i={\mathfrak{F}}_2,$E67

where $\overline{i}=2$for i = 1 and $\overline{i}=1$for i = 2,

${\mathfrak{F}}_1=-\sum _{i=1}^2\left\{\frac{z_i\left(x^{*},t\right)}{a_1}\frac{d{\varphi }_{i1}}{\mathit{d}\mathit{\tau }}+{\stackrel{\mathrm{̇}}{\psi }}_0^{-1}\frac{\partial }{\partial t}\frac{S_i}{{\beta }_i}\right\}{\left|{}_{t=t^{*}}+\frac{1}{{\beta }_2}{\mathfrak{R}}_f^{\left(0\right)}\right|}_{t=t^{*}},$E68

${\mathfrak{F}}_2=-\frac{\partial }{\partial \tau }\sum _{i=1}^2\left\{a_H{z}_i\left(x^{*},t\right)\frac{S_i}{{\beta }_i}+z_{\overline{i}}\left(x^{*},t\right)\frac{G_i}{{\beta }_2}{\lambda }_{H,i}\left(\sigma \right)\right\}{\left|{}_{t=t^{*}}+\frac{{\mathfrak{R}}_{K_2}^{\left(0\right)}}{c_H}\right|}_{t=t^{*}},$E69

$a_H=a_1/\left(2a_2\right),c_H=2a_2{\dot{\psi }}_0{\beta }_2,{\lambda }_{H,i}\left(\sigma \right)=a_2^{-1}{\int }_{-\infty }^{\infty }\omega \left({\eta }_{i2}\right)\mathrm{ℌ}\left({\eta }_{\overline{i}2}\right)\mathit{d\eta }.$

Calculating the determinant Δof the matrix in the left-hand part of (67) and using (65), we conclude

$\mathrm{\Delta }=\left(G_2-G_1+\lambda \left(G_1-\theta G_2\right)\right){\left|{}_{t=t^{*}}={\beta }_2^{\gamma }\left(1-{\theta }^{\gamma }-\lambda \left(\theta -{\theta }^{\gamma }\right)-{\lambda }^2{\theta }^{\gamma }\left(1+O\left({\theta }^q\right)\right)\right)\right|}_{t=t^{*}}.$E70

Obviously, Δ ≠ 0 for sufficiently small θ. Since the right-hand sides ${\mathfrak{F}}_i$belong to the Schwartz space, the functions ${\mathfrak{G}}_i$exist and satisfy the assumption (48).

Henceforth, we pass to the final result:

Theorem 4.Let θ be sufficiently small and let the assumptions(50),(63), and(64), if it is necessary, be fulfilled. Then, the self-similar two-wave weak asymptotic mod$O\left({\epsilon }^{\overline{q}}\right)$solution of theequation (1)exists and has the form(46).

Let us finally stress that the self-similarity implies a special choice of the initial data: for the classical asymptotics in the C-sense, there appears a very restrictive condition for small correction of the soliton A(0)ω((x − x₀)/ε) (see [6, 17]), and for weak asymptotics, there appears the restriction (63). If it is violated, then the perturbed soliton generates a rapidly oscillating tail of the amplitude o(1) (“radiation”) instead of the smooth tails εu⁻(x, t) (see [21] and numerical results [14, 15, 17]). Nowadays, this radiation phenomenon can be described analytically only for integrable equations, so that we should use self-similar approximation for essentially nonintegrable equations. However, the smooth tail εu⁻(x, t), which can be treated as an average of the radiation, describes sufficiently well the tendency of the radiation amplitude behavior, see graphics depicted in Figures 3 and 4, and other numerical results in [15, 17]).

Figure 3.

Example of noninteracting solitary waves,μ = 4,f = u(1 − u).

Figure 4.

Example of interacting solitary waves,μ = 4,f = u(1 − u),ε = 0.1.

Acknowledgments

The research was supported by SEP-CONACYT under grant no. 178690 (Mexico).