Interaction of Solitons for Sine-Gordon-Type Equations
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 with some smooth nonlinearities and the parameter . It is well known  that the unique completely integrable representative of the family (1) is the sine-Gordon equation, that is, (1) with (see, e.g., [2, 3]).
At the same time, there are many nonlinearities such that (1) admits exact traveling wave solutions of the soliton type: Traditionally, solution (2) with the sign “plus” is called “kink,” whereas (2) with the sign “minus” is called “antikink.” It is easy to check that the conditions (A) for , (B) , where and , and or , are sufficient for the existence of kink/antikink solutions such that Moreover, under the periodicity condition(C)
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 (2).
The first results of this topic have been obtained in [4–6], where the two-wave asymptotics have been constructed and then simulated numerically. In the present paper, we continue the cited investigations considering the interaction of three or more waves. The point is that there is a hypothesis (Danilov and Subochev , Boris Dubrovin’s, private communication) that there are sufficiently many equations with sine-Gordon scenario of two solitary waves interaction, but three waves can interact in the same manner for the completely integrable equations only. In fact, the situation is more complicated. Our main result consists in the conclusion that, in the leading term with respect to , two solitary waves (2) interact like sine-Gordon solitons but the stability of three or more waves depends on their parameters. Namely, structures with one kink and two antikinks or with one antikink and two kinks remain stable for special choice of the velocities, whereas interaction of four or more waves does not change the structure only for the entities of the same type (all of them are kinks or antikinks). Our main tool here is the numerical simulation.
The contents of the paper is the following. In Section 2 we present the asymptotics for the interaction of two solitary waves; the description of the finite difference scheme is contained in Section 3; in Section 4 we consider the numerical results.
2. Asymptotic Solution
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., [4, 5, 8–11] and references therein). The main advantage of this approach is the possibility of reducing the problem of describing nonlinear waves interaction to a qualitative analysis of some ordinary differential equations (instead of partial differential equations). This method takes into account the fact that kinks (as well as solitons [9, 10]) which are smooth for become nonsmooth in the limit as . So it is possible to treat such solutions as a mapping for and only as uniformly in . Accordingly, the remainder should be small in the weak sense. This rather trivial observation allowed to reach a progress for some old problems about nonlinear wave interaction for nonintegrable equations. As for the equations of the form (1), it should be noted that there is an obstacle to apply the standard construction. Indeed, in the sense, the differential terms of (1) are subordinated to the nonlinear term. Moreover, the left-hand side of (1) is of the value in the weak sense for any of the form (4) and . Obviously, this prevents the construction of the correct asymptotics for the Cauchy problem. To overcome this obstacle, in  has been constructed a new definition of asymptotic solutions, which involves in the leading term the derivatives of with arguments and .
Definition 1. A sequence , belonging to for and belonging to uniformly in , is called a weak asymptotic mod solution of (1) if the relation holds uniformly in for any test function .
Here the right-hand side is a -function for and a piecewise continuous function uniformly in . The estimate is understood in the sense: The left-hand side of (5) is the result of multiplication of (1) by and integration by parts in the case of smooth . Therefore, the relation (5) is satisfied automatically for any exact solution. On the other hand, the relation (5) is just the orthogonality condition that appears for single-phase asymptotics [12, 13]. This condition both guarantees the first correction existence and allows to find an equation for the distorted kink’s front motion.
Definition 2. A function is said to be of the value if the relation holds uniformly in for any test function .
Let us consider the interaction of two kinks, where , , and the initial front positions are such that . Obviously, it is assumed that the trajectories have a joint point at a time instant .
The asymptotic ansatz for the problem (1), (8) has the following form: Here are the trajectories of the noninteracting kinks, denotes the “fast time,” . The phase corrections are smooth functions such that with a rate not less than . Furthermore, are exponentially vanishing as functions, is a sufficiently small parameter, , and where is an arbitrary number and is a sufficiently fast vanishing function as .
Theorem 3 (see ). Let the assumptions (A)–(C) hold. Set the additional assumptions (D), (E) let the function be such that the inequality
holds uniformly in . Then the interaction of kinks in the problem (1), (8) preserves the sine-Gordon scenario with accuracy in the sense of Definition 1. The weak asymptotic solution of (1), (8) has the form (9) with a special choice of the amplitudes and of the parameter .
Remark 4. The symmetry (D) has been assumed to simplify the asymptotic analysis and it is not very important.
Remark 5. The sense of assumption (E) is the following. The phase corrections are solutions of a -dynamical system with a singularity whose support divides the phase plane into two parts with the possible exception of the point . Assumptions (10) are satisfied (consequently, the sine-Gordon scenario takes place) if and only if there exists a specific trajectory which goes from one half-plane to the other one through the point . When in (9) are equal to zero, the existence of the trajectory implies the appearance of an additional very complicated assumption. This condition can be made more coarse and transformed into the simplest form (12). Such version can be treated as an admissible one since it is satisfied for the sine-Gordon equation for any velocities . The same is true for the nonlinearity Taking into account a freedom in the choice of the amplitudes , assumption (12) can be made weaker. However, the dynamical system with , is very complicated and its complete analysis remains undone.
Obviously, all stated above remains true for the antikink-antikink interaction.
Technically, the construction of (15) is similar to the kink-kink case. However, the resulting dynamical system for the phase corrections becomes much more complicated. Moreover, it is impossible to simplify the additional assumption, which appears here also, without loss of the adequacy. For this reason we do not present the explicit form of the additional condition but state only the existence of the weak asymptotics (15) under some restrictions for and . We refer the readers to  for the explicit statement.
Finally we note that there is a correspondence between weak asymptotic solutions and energy relations for (1).
3. Finite Differences Scheme
The actual numerical simulation for (1) is realized for a finite -interval, . For this reason we simulate the Cauchy problem by the following mixed problem: where is a combination of kinks and antikinks of the form (4) and denotes its time derivative calculated at , . To simulate by (18) the interaction phenomena, we assume that , , and the initial front positions , are such that the intersection point of the solitary wave fronts belongs to . Furthermore, let , , and be such that uniformly in for some sufficiently small . Since it is impossible to create any finite difference scheme for the problem (18), which remains stable uniformly in and , we will treat as a small but fixed constant. However, we will fix any relation between and finite differences scheme parameters.
To create a finite differences scheme for (18) we should choose appropriate approximations for the differential terms and for the nonlinear term. Let us do it separately.
3.1. Preliminary Nonlinear “Scheme”
As usually, we define a mesh over and denote Let us consider the following system of nonlinear equations: where is such that last equality in (18) is approximated with accuracy . Obviously, the local approximation accuracy of (21) is .
To simplify the notation, we will write
So the short form of (21) is the following:
Our first result consists in obtaining the boundedness condition for the problem (21) solution.
Lemma 7 (see ). Let be a sufficiently small constant and let Suppose that system (21) is solvable for any . Then uniformly in where and are the norms, namely, Here and in what follows denotes a const > 0 which does not depend on , , and .
As a consequence of this lemma and the identity we obtain the inequality where denotes the right-hand side in (25). Obviously, this estimate is very rough. However, it can be improved a little for the specific initial data (8) and (14).
Now we should verify the solvability of (21) for any fixed , that is, of the equation as well as select a way to linearize the nonlinearity. To this aim let us construct the sequence of functions , , such that and for satisfies the equation
The solvability of the algebraic system (31) is obvious for sufficiently small and . To simplify the notation we write . Let also define
Thus we immediately conclude that the terms of the -sequence vanish very rapidly, By virtue of (33), the terms of the -sequence are bounded uniformly in Furthermore, for any This implies the main statement of this subsection.
3.3. Algorithm for the Numerical Simulation
For any fixed , we(i)define ,(ii)calculate , accordingly with (31),(iii)define , redefine , and go back to (i).
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.
Theorem 12 (see ). Let the assumption (24) be satisfied and = const. Then the solution of the above described finite differences scheme converges to the solution of the problem (18) as in the sense.
Finally, in view of the boundedness of the sequence , , it is easy to establish our last statement.
4. Results of Numerical Simulation
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 (13) and the following: Obviously, these functions satisfy the assumptions (A)–(D). Note also that functions (13), (42), and (43) for have one, three, and five critical points respectively. At the same time, for (13) if and . Therefore, the explicit kink type solution for this case, that is, tends to zero as when . Conversely, for functions (42), and (43) if and . Therefore, the kink type solutions for these cases tend to zero with exponential rates when . However, their explicit form remains unknown; that is, why we simulate them solving numerically the Cauchy problem and, by virtue of the condition (C), define with negative argument as . To calculate the solution of (45) we use the Runge-Kutta method of the forth order with the mesh step .
4.1. Two-Wave Interaction
In accordance with the asymptotic analysis, two solitons interact preserving the shape; see Figures 1 and 2 for the nonlinearity (13) and (here and in what follows we numerate the waves from the left to the right). Moreover, this remains true independently of the wave parameters , for all nonlinearities under consideration. Thus we can conclude that the additional conditions, which appear for the asymptotics, are restrictions of the asymptotic method only.
4.2. Three-Wave Interaction
There are 32 combinations of three solitons (kinks and antikinks) with trajectories which intersect at one instant of time. In view of the symmetry for (1), we can reduce this number to 16, considering only admissible combinations of three kinks and of two kinks and one antikink. Next we reduce the number of combinations to 12 using the symmetry . All these combinations have been analyzed numerically with the same result for each of the nonlinearities (13), (42), (43): three kinks preserve the shape after the interaction for all admissible velocities, whereas there are only two stable combinations for two kinks and one antikink. The corresponding numerical results are depicted in Figures 3–5 for (43) with . In Figure 6 we present the stable structure of one kink and two antikinks which is dual to Figure 4 in the sense the symmetry . So that, there are 12 stable structures of kinks (K) and antikinks (A):(1)KKK: (, , ), (, , ), (, , ), (, , );(2)KKA: (, , );(3)KAA: (, , );(4)AAK: (, , );(5)AKK: (, , );(6)AAA: (, , ), (, , ), (, , ), (, , ).
4.3. Interaction of Four or More Waves
We obtain again that the interaction of waves of the same type does not destroy the structure; see, for example, Figure 7 with 6 kinks, , and , . As for waves of different types, we did not find any stable combination. In fact, even for four waves the number of possible combinations is too large to simulate each of them, so we checked a part of the combinations only. However, we guess that there are no stable structures for the following reason: each combination of four waves can be considered either as a union of a triplet with an additional wave on its left, or as a union of a triplet with an additional wave on its right. It turns out that one of these triplets should be unstable since the list of stable triplets is too scanty. Thus the total combination is unstable. We illustrate this in Figure 8 for the interaction of two kinks and two antikinks. Indeed, such combination can be treated as the union of the stable triplet depicted in the Figure 4 and the antikink on his right. On the contrary, the same combination is the union of the kink and the kink-antikink-antikink triplet with , which is unstable.
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 . Let us note also that our algorithm (see Section 3) allows to eliminate “radiation due to discreteness effects” which appears for the trivial linearization; see .
The research was supported by SEP-CONACYT under grant 178690 (Mexico).
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