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).
- generalized Korteweg-de Vries equation
- weak asymptotics method
- 2010 Mathematics Subject Classification: 35D30
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
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 μ implies 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  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 f for 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  and Kaup and Newell  on the basis of the IST method, and the “direct” method by V. Maslov and Omel’yanov ). Moreover, the approach by V. Maslov and Omel’yanov  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
It is easy to find the explicit soliton solution of (2),
Next let us consider two-soliton initial data
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)
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 = Vit + x(i, 0) intersection at a point (x*, t*).
Let A1 ≪ A2. Then, one can construct an asymptotic solution
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. .
The amplitudes A2 > A1 are arbitrary numbers. Then, we should write a two-phase ansatz
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  and for soliton-type solutions  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:
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 valueif the relationholds uniformly in t for any test function. The right-hand side here is a-function for ε = const > 0 and a piecewise continuous function uniformly in ε ≥ 0.
Definition 2.A sequence u(t, x, ε), belonging tofor ε = const > 0 and belonging touniformly in ε, is called a weak asymptotic modsolution of (2) if the relations (11) hold uniformly in t with the accuracy.
Here ϕi = ϕi0(t) + εϕi1(τ), where ϕi0 = Vit + x(i, 0) are the trajectories of noninteracting solitary waves, τ = ψ0(t)/ε denotes the “fast time”, ψ0(t) = β1(ϕ20(t) − ϕ10(t)), and the phase and amplitude corrections ϕi1, Si are smooth functions such that with exponential rates
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
where δ(x) is the Dirac delta-function. Here and in what follows, we use the notation
At the same time for any F(u, ε∂u/∂x) ∈ C1, we have
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
where , . It remains to apply the formula
which holds for each ϕi of the form (20) with slowly increasing χi and for f(τ) from the Schwartz space. Moreover, the second term in the right-hand side of (21) is . Thus, under the assumptions (14) and (15), we obtain the weak asymptotic expansion of F(u, ε∂u/∂x) in the final form:
Calculating weak expansions for other terms from Definition 2 and substituting them into (11), we obtain linear combinations of εδ′(x − ϕi), i = 1, 2, δ(x − x*), and εδ′(x − x*). Therefore, we pass to the system:
The first four algebraic equations (28) imply again the relation (4) among Ai, βi, and Vi. Furthermore, there exists a number θ* ∈ (0, 1) such that equations (29), (30) have the required solution Si, ϕi1 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) describesthe elastic scenario of the solitary waves interaction for the μ-gKdV equation (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  for detail. Obviously, this effect is similar to the “radiation” appearance for the perturbed KdV .
2.3. Multisoliton interaction
N-wave solutions of the form similar to waves (13) contain 2N free functions Si, ϕi1. Thus, to describe an N-soliton collision, we should consider N conservation 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
where Qj, Pj, j = 1, 2, coincide with (12) for μ = 4, Ki = 0, i = 1, 2, 3,
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 N summands, 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 Ai the amplitudes of the original solitons and x(i, 0)their initial positions such that Ai + 1 > Ai, x(i, 0) > x(i + 1, 0), and i = 1, 2. Let all trajectories x = ϕi0(t) have an intersection point (x*, t*). Then, under the assumption
with sufficiently smallν < 1, the three-phase asymptotic solution exists and describesthe elastic scenario of the solitary waves interaction.
2.4. Asymptotic equivalence
Let us come back to the case of two-phase asymptotics and transform the ansatz (13) to the following form:
where , i = 1, 2 are arbitrary functions from the Schwartz space,
and l ≥ 1 is an arbitrary integer. Calculating the weak representations for and , we obtain
and u in the right-hand side in (39) is the representation (25). Thus, the difference between u of the forms (13) and (37) is arbitrarily small in the sense . At the same time, instead of (29), (30), we obtain
where f ˜ , F ˜ differ from f, F in the same manner as differs from . 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 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(ε1 − ν)-neighborhood of the time instant t* of the interaction, ν > 0. Thus, it is small in the 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
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  for two-phase asymptotic solutions of the gKdV-4 equation. Let us define two-phase asymptotics in the following manner:
Definition 4.Let 1 ≤ k0 < k1 ≤ 4 and let a sequence uk0, k1 = uk0, k1(t, x, ε) belong to the same functional space as u(t, x, ε) inDefinition 2. Then, uk0, k1is called a weak asymptotic modsolution of (2) if the relations (32) hold for j = k0and j = k1uniformly in t.
A detailed analysis implies the assertion .
Theorem 3.Let θ be sufficiently small. Then, the weak asymptotic solutions u1, k1andof the problem (2), (5) exist and they are asymptotically equivalent for all.
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 f can be treated as a “small perturbation.”
An approach to construct one-phase self-similar asymptotic solutions for (1) had been created in  (see also ). 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
where Qj and Pj coincide with ones described in (12),
Note that, in contrast to Kj in (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 modsolution of (1) if the relation (44) hold uniformly in t ∈ (0, T), .
Generalizing one-phase asymptotics, we write the ansatz as
Here Ai(t), ϕi0 = ϕi0(t), , ω(η), Si(τ), ϕi1(τ) are the same as in (13); τ = ψ0(t)/ε with ψ0(t) = ϕ20(t) − ϕ10(t) denotes the “fast time” again; ; and , ℌ are smooth functions such that
with exponential rates. We assume also the intersection of the trajectories x = ϕi0(t), i = 1, 2 at a point x* = ϕi0(t*) namely,
It is easy to verify the weak representations with the precision :
where H(x) is the Heaviside function, H(x) = 0 for x < 0 and H(x) = 1 for x > 0; , , and Φ(ετ, t*) is the solution of the equation ϕ20(t* + Φ) − ϕ10(t* + Φ) = ετ, which exists in accordance with (50).
Repeating the same calculations as above, we obtain linear combinations of εδ(x − ϕi), εδ′(x − ϕi), εH(ϕi − x), i = 1, 2; δ(x − x*), εδ(x − x*), and εδ′(x − x*). Equating zero, the coefficients of εδ(x − ϕi) and εδ′(x − ϕi) yield
Simple manipulations with (56) allow us to find relations between structural constants:
Next, equating zero the coefficients of the Heaviside functions, we obtain the equations
where c2 = a1(3 − μ)(1 + μ)/(2a2(5 − μ)), is an arbitrary smooth function, which satisfies the consistency condition
We should note that the nonlinearity uμ in (1) can require the inequality u ≥ 0. To this end, we will assume
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 Si, ϕi1 with the properties of (14), (15). In particular
The last step of the construction is the determination of , i = 1, 2. By setting the coefficients of εδ(x − x*) zero, we obtain
where for i = 1 and for i = 2,
Obviously, Δ ≠ 0 for sufficiently small θ. Since the right-hand sides belong to the Schwartz space, the functions 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 modsolution of the equation (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  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]).
The research was supported by SEP-CONACYT under grant no. 178690 (Mexico).