The subject of our consideration is a family of semilinear wave equations with a small parameter and nonlinearities which provide the existence of kink-type solutions (solitons). Using asymptotic analysis and numerical simulation, we demonstrate that solitons of the same type (kinks or antikinks) interact in the same manner as for the sine-Gordon equation. However, solitons of the different type preserve the shape after the interaction only in the case of two or three waves, and, moreover, under some additional conditions.

We consider the semilinear wave equation

At the same time, there are many nonlinearities

where

any combination of kink-antikink waves, that is,

will approximately sufficiently well the exact solution of the corresponding Cauchy problem. This brings up the question about the character of the interaction between the entities (

The first results of this topic have been obtained in [

The contents of the paper is the following. In Section

For essentially nonintegrable interaction problems it is impossible to construct either explicit solutions (classical or weak) or asymptotics in the classical sense. However, it is possible to construct an asymptotic solution in the weak sense (see, e.g., [

A sequence

Here the right-hand side is a

A function

Let us consider the interaction of two kinks,

The asymptotic ansatz for the problem (

The main result, which is known for the problem (

Let the assumptions (A)–(C) hold. Set the additional assumptions

let the function

holds uniformly in

The symmetry (D) has been assumed to simplify the asymptotic analysis and it is not very important.

The sense of assumption (E) is the following. The phase corrections

Obviously, all stated above remains true for the antikink-antikink interaction.

Let us focus our attention in the kink-antikink interaction, that is, in (

The asymptotic ansatz for the solution of the problem (

Technically, the construction of (

Finally we note that there is a correspondence between weak asymptotic solutions and energy relations for (

Let the assumptions of Theorem

Similar conclusion is true for the kink-antikink pair (

The actual numerical simulation for (

To create a finite differences scheme for (

As usually, we define a mesh

To simplify the notation, we will write

So the short form of (

Our first result consists in obtaining the boundedness condition for the problem (

Let

As a consequence of this lemma and the identity

Let the assumptions of Lemma

Now we should verify the solvability of (

The solvability of the algebraic system (

Let assumption (

Thus we immediately conclude that the terms of the

Let assumption (

Since the accuracy

For any fixed

define

calculate

define

By virtue of the estimates (

Note that this result can be improved. Moreover, it turns out that the algorithm is absolutely stable. To prove this we state firstly the proposition.

Let assumption (

An immediate consequence of the Lemmas

Let the assumption (

Finally, in view of the boundedness of the sequence

Under the assumptions of Theorem

The numerical algorithm has been realized as a program and tested using the sine-Gordon equation in the cases of one, two, three, and four solitary waves. Next we used the nonlinearities (

In accordance with the asymptotic analysis, two solitons interact preserving the shape; see Figures

Evolution of the kink-kink pair,

Evolution of the kink-antikink pair,

There are 32 combinations of three solitons (kinks and antikinks) with trajectories which intersect at one instant of time. In view of the symmetry

KKK: (

KKA: (

KAA: (

AAK: (

AKK: (

AAA: (

Evolution of the kink-triplet,

Evolution of the kink-kink-antikink triplet,

Evolution of the antikink-kink-kink triplet,

Evolution of the antikink-antikink-kink triplet,

We obtain again that the interaction of waves of the same type does not destroy the structure; see, for example, Figure

Interaction of 6 kinks for the nonlinearity (

Evolution of four waves for the nonlinearity (

Summarizing all stated above, we can deduce that there exists a family of nonlinearities such that kink-kink and kink–antikink pairs preserve the sine-Gordon scenario of interaction at least in the leading term in the asymptotic sense. Apparently, this family can be specified by assumptions (A)–(D).

As for multiwave interactions, the situation is more complicated. Apparently, a sufficiently large number of solitons of the same type interact preserving the shape, whereas there are only four stable combinations of kinks and antikinks. In fact, this is rather unexpected, since single kink and antikink waves have the same properties. The second strange phenomenon is that there does not appear any perturbation of the radiation type, comparable with the collision of solitons for the KdV-type equations [

The research was supported by SEP-CONACYT under grant 178690 (Mexico).