Interaction of Solitons for Sine-Gordon-Type Equations

Abstract

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.

1. Introduction

We consider the semilinear wave equation 𝜀²𝑢_𝑡𝑡−𝑢_𝑥𝑥+𝐹^′(𝑢)=0,𝑥∈ℝ¹,𝑡>0, (1) with some smooth nonlinearities 𝐹(𝑢) and the parameter 𝜀→0. It is well known [1] that the unique completely integrable representative of the family (1) is the sine-Gordon equation, that is, (1) with 𝐹^′(𝑢)=sin(𝑢) (see, e.g., [2, 3]).

At the same time, there are many nonlinearities 𝐹(𝑢) such that (1) admits exact traveling wave solutions of the soliton type: 𝑢(𝑥,𝑡,𝜀)=𝜔±𝛽𝑥−𝑉𝑡𝜀,𝛽=1−𝑉^2−1/2,𝜔𝜂∈𝒞^∞(ℝ),𝜔𝜂⟶0for𝜂⟶−∞,𝜔𝜂⟶1for𝜂⟶+∞. (2) 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)  𝐹(𝑧)∈𝒞^∞(ℝ),𝐹(𝑧)>0 for 𝑧∈(0,1), (B)  𝐹^(𝑖)(𝑧₀)=0,𝑖=0,1,…,𝑘,𝐹^(𝑘+1)(𝑧₀)>0, where 𝑧₀=0 and 𝑧₀=1, and 𝑘=1 or 𝑘=3, are sufficient for the existence of kink/antikink solutions such that 𝜔^′𝜂≤𝑐𝜂⁻²as𝜂⟶±∞. (3)Moreover, under the periodicity condition(C)𝐹(𝑧+1)=𝐹(𝑧)

any combination of kink-antikink waves, that is, 𝑢_Σ=^𝑁_𝑖=1𝜔±𝛽_𝑖𝑥−𝑉_𝑖𝑡−𝑥⁰_𝑖𝜀,𝑥⁰_𝑖+1−𝑥⁰_𝑖>1,0<𝑡≪1, (4)

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 [7], 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 𝜀, 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 𝜀>0 become nonsmooth in the limit as 𝜀→0. So it is possible to treat such solutions as a mapping 𝒞^∞((0,𝑇);𝒞^∞(ℝ¹_𝑥)) for 𝜀=const>0 and only as 𝒞((0,𝑇);𝒟^′(ℝ¹_𝑥)) uniformly in 𝜀≥0. 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 𝒟^′ construction. Indeed, in the 𝒟^′ sense, the differential terms of (1) are subordinated to the nonlinear term. Moreover, the left-hand side of (1) is of the value 𝑂(𝜀²) in the weak sense for any 𝑢 of the form (4) and 𝑡≪1. Obviously, this prevents the construction of the correct asymptotics for the Cauchy problem. To overcome this obstacle, in [4] has been constructed a new definition of asymptotic solutions, which involves in the leading term the derivatives of 𝑢 with arguments 𝑥/𝜀 and 𝑡/𝜀.

Definition 1. A sequence 𝑢(𝑡,𝑥,𝜀), belonging to 𝒞^∞((0,𝑇);𝒞^∞(ℝ¹_𝑥)) for 𝜀=const>0 and belonging to 𝒞((0,𝑇);𝒟^′(ℝ¹_𝑥)) uniformly in 𝜀, is called a weak asymptotic mod 𝑂_𝒟′(𝜀²) solution of (1) if the relation 2𝑑𝑑𝑡^∞_−∞𝜀²𝑢_𝑡𝑢_𝑥+𝜓𝑑𝑥_∞−∞𝜀𝑢_𝑡²+𝜀𝑢_𝑥²−2𝐹(𝑢)𝜓_𝑥𝑑𝑥=𝑂𝜀² (5) holds uniformly in 𝑡 for any test function 𝜓=𝜓(𝑥)∈𝒟(ℝ¹).

Here the right-hand side is a 𝒞^∞-function for 𝜀=const>0 and a piecewise continuous function uniformly in 𝜀≥0. The estimate is understood in the 𝒞(0,𝑇) sense: 𝑔(𝑡,𝜀)=𝑂𝜀^𝑘⟷max_{[]𝑡∈0,𝑇}𝑔(𝑡,𝜀)≤𝑐𝜀^𝑘. (6) The left-hand side of (5) is the result of multiplication of (1) by 𝜓(𝑥)𝑢_𝑥 and integration by parts in the case of smooth 𝑢. 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 𝑣(𝑡,𝑥,𝜀) is said to be of the value 𝑂_𝒟′(𝜀^𝑘) if the relation ^∞_−∞𝑣(𝑡,𝑥,𝜀)𝜓(𝑥)𝑑𝑥=𝑂𝜀^𝑘 (7) holds uniformly in 𝑡 for any test function 𝜓∈𝒟(ℝ¹_𝑥).

Let us consider the interaction of two kinks, 𝑢|_𝑡=0=²_𝑖=1𝜔𝛽_𝑖𝑥−𝑥⁰_𝑖𝜀,𝜀𝜕𝑢𝜕𝑡_𝑡=0=−²_𝑖=1𝛽_𝑖𝑉_𝑖𝜔^′𝛽_𝑖𝑥−𝑥⁰_𝑖𝜀, (8) where 𝛽_𝑖=1/1−𝑉²_𝑖, |𝑉_𝑖|∈(0,1), and the initial front positions 𝑥⁰_𝑖 are such that 𝑥⁰₂−𝑥⁰₁>1. Obviously, it is assumed that the trajectories 𝑥=𝑉_𝑖𝑡+𝑥⁰_𝑖 have a joint point 𝑥=𝑥^∗ at a time instant 𝑡=𝑡^∗.

The asymptotic ansatz for the problem (1), (8) has the following form: 𝑢=²_𝑖=1𝜔𝛽_𝑖𝑥−Φ_𝑖(𝑡,𝜏,𝜀)𝜀+𝐴_𝑖(𝜏)𝑈𝛽_𝑖𝑥−Φ_𝑖(𝑡,𝜏,𝜀)𝜇²𝜀. (9) Here Φ_𝑖=𝜙_𝑖0(𝑡)+𝜀𝜙_𝑖1(𝜏),𝜙_𝑖0=𝑉_𝑖𝑡+𝑥⁰_𝑖 are the trajectories of the noninteracting kinks, 𝜏=𝜓₀(𝑡)/𝜀 denotes the “fast time,” 𝜓₀(𝑡)=𝜙₂₀(𝑡)−𝜙₁₀(𝑡). The phase corrections 𝜙_𝑖1 are smooth functions such that 𝜙_𝑖1𝜙⟶0as𝜏⟶−∞,_𝑖1⟶𝜙^∞_𝑖1=const_𝑖as𝜏⟶+∞, (10) with a rate not less than 1/|𝜏|. Furthermore, 𝐴_𝑖(𝜏)∈𝒞^∞ are exponentially vanishing as |𝜏|→∞ functions, 𝜇 is a sufficiently small parameter, 𝜀<𝜇≪1, and 𝑈𝜂=𝑑^𝑚𝑈₀𝜂𝑑𝜂^𝑚, (11) where 𝑚≥1 is an arbitrary number and 𝑈₀(𝜂)∈𝒞^∞ is a sufficiently fast vanishing function as |𝜂|→∞.

The main result, which is known for the problem (1), (8), is the following.

Theorem 3 (see [4]). Let the assumptions (A)–(C) hold. Set the additional assumptions (D)𝐹(1/2+𝑧)=𝐹(1/2−𝑧), (E) let the function 𝐹(𝑧) be such that the inequality ^∞_−∞𝐹𝜔𝜂≤+𝜔𝜃𝜂𝑑𝜂_∞−∞𝐹𝜔𝜂+𝐹𝜔𝜃𝜂²𝑑𝜂 (12)
holds uniformly in 𝜃∈(0,∞). Then the interaction of kinks in the problem (1), (8) preserves the sine-Gordon scenario with accuracy 𝑂_𝒟′(𝜀²) in the sense of Definition 1. The weak asymptotic solution of (1), (8) has the form (9) with a special choice of the amplitudes 𝐴_𝑖 and of the parameter 𝜇.

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 𝜙_𝑖1 are solutions of a 2×2-dynamical system with a singularity whose support divides the phase plane into two parts with the possible exception of the point (0,0). 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 (0,0). When 𝐴_𝑖 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 𝑉_𝑖,𝑖=1,2. The same is true for the nonlinearity 𝐹(𝑢)=sin⁴(𝜋𝑢). (13) Taking into account a freedom in the choice of the amplitudes 𝐴_𝑖,𝑖=1,2, assumption (12) can be made weaker. However, the dynamical system with 𝐴_𝑖≠0,𝑖=1,2, is very complicated and its complete analysis remains undone.

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

Let us focus our attention in the kink-antikink interaction, that is, in (1) with initial data 𝑢|_𝑡=0=²_𝑖=1𝜔𝑆_𝑖𝛽_𝑖𝑥−𝑥⁰_𝑖𝜀,𝜀𝜕𝑢𝜕𝑡_𝑡=0=−²_𝑖=1𝑆_𝑖𝛽_𝑖𝑉_𝑖𝜔^′𝑆_𝑖𝛽_𝑖𝑥−𝑥⁰_𝑖𝜀, (14) where 𝑆₁=1,𝑆₂=−1, and the notation 𝛽_𝑖, 𝑉_𝑖, 𝑥⁰_𝑖 is the same as in (8).

The asymptotic ansatz for the solution of the problem (1), (14) differs a little bit from (9), namely, 𝑢=²_𝑖=1𝜔𝑆_𝑖𝛽_𝑖𝑥−Φ_𝑖(𝑡,𝜏,𝜀)𝜀+𝐴_𝑖(𝜏)𝑈𝑆_𝑖𝜇𝛽_𝑖𝑥−Φ_𝑖(𝑡,𝜏,𝜀)𝜀, (15) with the same notation and assumption (10).

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 𝑉₁ and 𝑉₂. We refer the readers to [5] for the explicit statement.

Finally we note that there is a correspondence between weak asymptotic solutions and energy relations for (1).

Theorem 6. Let the assumptions of Theorem 3 hold. Then two kinks (9) preserve mod 𝑂_𝒟′(𝜀²) their forms after the interaction if and only if they satisfy the conservation law 𝑑𝑑𝑡^∞_−∞𝑢_𝑡𝑢_𝑥𝑑𝑥=0 (16) and the energy relation 2𝑑𝑑𝑡^∞_−∞𝑥𝜀²𝑢_𝑡𝑢_𝑥+𝑑𝑥_∞−∞𝜀𝑢_𝑡²+𝜀𝑢_𝑥²−2𝐹(𝑢)𝑑𝑥=0. (17)

Similar conclusion is true for the kink-antikink pair (15) [5].

3. Finite Differences Scheme

The actual numerical simulation for (1) is realized for a finite 𝑥-interval, 𝑥∈[0,𝐿]. For this reason we simulate the Cauchy problem by the following mixed problem: 𝜀²𝑢_𝑡𝑡−𝑢_𝑥𝑥+𝐹^′(𝑢)()(),𝑢|=0,𝑥∈0,𝐿,𝑡∈0,𝑇_𝑥=0=𝜈_ℓ,𝑢|_𝑥=𝐿=𝜈_𝑟,𝑢|_𝑡=0=𝑢⁰𝑥𝜀,𝜀𝜕𝑢𝜕𝑡_𝑡=0=𝑢¹𝑥𝜀, (18) where 𝑢⁰ is a combination of kinks and antikinks of the form (4) and 𝑢¹ denotes its time derivative calculated at 𝑡=0, 𝜈_ℓ=𝑢⁰|_𝑥=0,𝜈_𝑟=𝑢⁰|_𝑥=𝐿. To simulate by (18) the interaction phenomena, we assume that 𝐿, 𝑇, and the initial front positions 𝑥⁰_𝑖,𝑖=1,2, are such that the intersection point of the solitary wave fronts belongs to 𝑄_𝑇=(0,𝑇)×(0,𝐿). Furthermore, let 𝐿, 𝑇, and 𝑥⁰_𝑖 be such that uniformly in 𝑡≤𝑇𝑢_{Σ𝑥∈[0,𝛿]}−𝜈_ℓ≤𝑐𝜀²,𝑢_{Σ𝑥∈[𝐿−𝛿,𝐿]}−𝜈_𝑟≤𝑐𝜀² (19) for some sufficiently small 𝛿>0. Since it is impossible to create any finite difference scheme for the problem (18), which remains stable uniformly in 𝜀→0 and 𝑡∈(0,𝑇),𝑇=const, we will treat 𝜀 as a small but fixed constant. However, we will fix any relation between 𝜀 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 𝑄_{𝑇,𝜏,ℎ}={(𝑥_𝑖,𝑡_𝑗)=(𝑖ℎ,𝑗𝜏),𝑖=0,…,𝐼,𝑗=0,…,𝐽} over 𝑄_𝑇 and denote 𝑦^𝑗_𝑖=𝑢𝑥_𝑖,𝑡_𝑗,𝑦^𝑗_𝑖𝑡=𝑦_𝑖^𝑗+1−𝑦^𝑗_𝑖𝜏,𝑦^𝑗_𝑖𝑡=𝑦^𝑗_𝑖−𝑦_𝑖^𝑗−1𝜏,𝑦^𝑗_𝑖𝑥=𝑦^𝑗_𝑖+1−𝑦^𝑗_𝑖ℎ,𝑦^𝑗_𝑖𝑥=𝑦^𝑗_𝑖−𝑦^𝑗_𝑖−1ℎ,𝑦^𝑗_𝑖𝑡𝑡=𝑦^𝑗_𝑖𝑡𝑡,𝑦^𝑗_𝑖𝑥𝑥=𝑦^𝑗_𝑖𝑥𝑥. (20) Let us consider the following system of nonlinear equations: 𝜀²𝑦^𝑗_𝑖𝑡𝑡−𝑦^𝑗+1_𝑖𝑥𝑥+𝐹^′𝑦_𝑖^𝑗+1𝑦=0,𝑖=1,…,𝐼−1,𝑗=2,3,…,_𝑗0=𝜈_ℓ,𝑦^𝑗_𝐼=𝜈_𝑟𝑦,𝑗=0,1,…,_0𝑖=𝑢⁰𝑥_𝑖𝜀,𝜀𝑦⁰_𝑖𝑡=𝑢¹𝑥_𝑖𝜀,𝜏,𝑖=0,…,𝐼, (21) where 𝑢¹(𝑥_𝑖/𝜀,𝜏) is such that last equality in (18) is approximated with accuracy 𝑂(𝜏²). Obviously, the local approximation accuracy of (21) is 𝑂(𝜏²+ℎ²).

To simplify the notation, we will write 𝑦∶=𝑦^𝑗_𝑖,𝑦∶=𝑦_𝑖^𝑗+1,𝑦̌∶=𝑦_𝑖^𝑗−1. (22)

So the short form of (21) is the following: 𝜀²𝑦_𝑡𝑡−𝑦_𝑥𝑥+𝐹^′𝑦=0. (23)

Our first result consists in obtaining the boundedness condition for the problem (21) solution.

Lemma 7 (see [6]). Let 𝜀 be a sufficiently small constant and let 𝜏𝜀²≤𝑐𝑜𝑛𝑠𝑡. (24) Suppose that system (21) is solvable for any 𝑗=2,…,𝐽. Then uniformly in 𝑗𝜀𝑦_𝑡²+𝜀𝑦_𝑥²+2𝐹(𝑦)²+𝜏𝜀²𝜀²𝑦_𝑡𝑡²𝑗+𝜀²𝑦_𝑥𝑥²𝑗≤𝜀𝑦⁰_𝑡²+𝜀𝑦¹_𝑥²+2𝐹𝑦¹²𝑒^{𝑐𝑡𝑗𝜏/𝜀2}(1+𝑂(𝜏)), (25) where ‖⋅‖ and |‖⋅‖|(𝑗) are the ℒ² norms, namely, 𝑓²=ℎ^𝐼−1_𝑖=1𝑓_𝑖²,𝑓²𝑗=𝜏^𝑗_𝑘=1𝑓^𝑘2. (26) Here and in what follows 𝑐 denotes a const > 0 which does not depend on ℎ, 𝜏, and 𝜀.

As a consequence of this lemma and the identity 𝑦^𝑗_𝑖=𝑦⁰_𝑖+𝜏^𝑗−1_𝑘=0𝑦^𝑘_𝑖𝑡, (27) we obtain the inequality 𝑦^𝑗2≤2𝑦⁰²+2𝑡_𝑗𝜏^𝑗−1_𝑘=0𝑦^𝑘_𝑡²≤2𝑦⁰²𝑡+2^2𝑗𝜀²𝑐₀, (28) where 𝑐₀>0 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).

Lemma 8 (see [6]). Let the assumptions of Lemma 7 be satisfied. Then for the initial data 𝑢⁰(𝑥_𝑖/𝜀), 𝑢¹(𝑥_𝑖/𝜀,𝜏), which approximate the Cauchy data (8) or (14), the following estimate holds uniformly in 𝑗: √𝜀𝑦^𝑗_𝑡+𝑦^𝑗_𝑥+𝑦^𝑗≤𝑐. (29)

3.2. Linearization

Now we should verify the solvability of (21) for any fixed 𝑗≥1, that is, of the equation 𝑦^𝑗+1−𝜏²𝑦_𝑥^𝑗+1_𝑥+𝜏²𝜀²𝐹^′𝑦^𝑗+1=𝐺^𝑗,𝐺^𝑗=𝑦^𝑗+𝜏𝑦^𝑗_𝑡, (30) as well as select a way to linearize the nonlinearity. To this aim let us construct the sequence of functions 𝜑(𝑠)∶={𝜑₀(𝑠),…,𝜑_𝐼(𝑠)}, 𝑠≥0, such that 𝜑(0)=𝑦^𝑗 and 𝜑(𝑠) for 𝑠≥1 satisfies the equation 𝜑(𝑠)−𝜏²𝜑_𝑥𝑥(𝑠)+𝜏²𝜀²𝐹^′𝜑(𝑠−1)+𝐹^′′𝜑()×𝜑(𝑠)()𝑠−1−𝜑𝑠−1=𝐺^𝑗,𝜑₀(𝑠)=𝜈_ℓ,𝜑_𝐼(𝑠)=𝜈_𝑟. (31)

The solvability of the algebraic system (31) is obvious for sufficiently small 𝜏 and 𝜏/𝜀²≤const. To simplify the notation we write 𝜑∶=𝜑(𝑠),𝜑∶=𝜑(𝑠−1),𝜑∶=𝜑(𝑠−2). Let also define 𝑤∶=𝜑−𝜑,𝑤∶=𝜑−𝜑. (32)

Lemma 9 (see [6]). Let assumption (24) be satisfied and let 𝜏 be sufficiently small. Then 𝜑²+𝜏²𝜀²𝜀𝜑_𝑥²≤(1+𝑐𝜏)𝑦^𝑗2+𝑐√𝜏𝑦^𝑗2+𝜀𝑦^𝑗_𝑡²+𝑐‖𝑤‖²+𝜏²𝑤_𝑥²²,𝑔(1)≤𝑐𝜏^3/2,𝑔(𝑠)≤𝑐𝜏𝑔²(𝑠−1)𝑓𝑜𝑟𝑠>1, (33)(34) where 𝑔(𝑠)∶=‖𝑤(𝑠)‖²+𝜏²𝑤_𝑥(𝑠)². (35)

Thus we immediately conclude that the terms of the 𝑤-sequence vanish very rapidly, ‖𝑤(1)‖²≤𝑐𝜏^3/2,‖𝑤(2)‖²≤𝑐𝜏⁴,‖𝑤(3)‖²≤𝑐𝜏⁹,…. (36) By virtue of (33), the terms of the 𝜑-sequence are bounded uniformly in 𝑠𝜑(𝑠)²≤𝑦^𝑗21+𝑂√𝜏. (37) Furthermore, for any 𝑛>0𝜑()(𝑠≤𝑠+𝑛−𝜑)_𝑛𝑖=1𝑤_𝑠+𝑖≤𝑤_𝑠+1^∞_𝑖=1𝑤_𝑠+𝑖𝑤_𝑠+1≤𝑐𝑤_𝑠+1. (38) This implies the main statement of this subsection.

Theorem 10 (see [6]). Let assumption (24) be satisfied and 𝜀 = const. Then for sufficiently small 𝜏 the sequence 𝜑 converges in the ℒ²_ℎ sense to the solution of (30). Moreover, 𝑦^𝑗+1−𝜑(2)≤𝑐𝜏^9/2. (39)

3.3. Algorithm for the Numerical Simulation

Since the accuracy 𝑂(𝜏^9/2) is much less than the accuracy of the finite differences scheme (21), we obtain the following algorithm for the numerical simulation of the problem (18) solution:

For any fixed 𝑗=1,2,…,[𝑇/𝜏],𝑇=const, we(i)define 𝜑(0)∶=𝑦^𝑗,(ii)calculate 𝜑(𝑠),𝑠=1,2, accordingly with (31),(iii)define 𝑦^𝑗+1∶=𝜑(2), redefine 𝑗∶=𝑗+1, and go back to (i).

By virtue of the estimates (29) and (39), this algorithm allows to calculate bounded in ℒ²(𝑄_{𝑇,ℎ,𝜏}) numerical solution of the problem (18).

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.

Lemma 11 (see [6]). Let assumption (24) be satisfied and 𝜀 = const. Then uniformly in 𝑠𝜀𝜑_𝑡(𝑠)≤𝑐𝑜𝑛𝑠𝑡,𝜀𝜑_𝑥(𝑠)≤𝑐𝑜𝑛𝑠𝑡. (40) Moreover, uniformly in 𝑗𝜀²𝑦^𝑗_𝑥𝑡+𝜀²𝑦_𝑥^𝑗+1_𝑥≤𝑐√𝜀,𝜀𝑦_𝑡^𝑗+1−𝜀𝜑_𝑡(𝑠)≤𝑐𝜀^𝑝(𝑠),𝜀𝑦_𝑥^𝑗+1−𝜀𝜑_𝑥(𝑠)≤𝑐𝜀^𝑝(𝑠), (41) with some 𝑝(𝑠) that tend to infinity as 𝑠→∞.

An immediate consequence of the Lemmas 7–11 is the following.

Theorem 12 (see [6]). Let the assumption (24) be satisfied and 𝜀 = const. Then the solution of the above described finite differences scheme converges to the solution of the problem (18) as 𝜏,ℎ→0 in the 𝑊¹₂(𝑄_𝑇) sense.

Finally, in view of the boundedness of the sequence 𝜑^𝑗(𝑠), 𝑠=1,2,𝑗=2,3,…, it is easy to establish our last statement.

Theorem 13 (see [6]). Under the assumptions of Theorem 12 the above described finite differences scheme is stable in the 𝑊¹₂(Ω_{𝑇,𝜏,ℎ}) sense.

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: 𝐹(𝑢)()(),𝐹(𝑢)5=2−cos2𝜋𝑢−cos4𝜋𝑢=1+sin2𝜋𝑢+45cos4𝜋𝑢+4. (42)(43) Obviously, these functions satisfy the assumptions (A)–(D). Note also that functions (13), (42), and (43) for 𝑢∈(0,1) have one, three, and five critical points respectively. At the same time, for (13) 𝐹^(𝑖)(0)=0 if 𝑖=0,…,3 and 𝐹⁽⁴⁾(0)>0. Therefore, the explicit kink type solution for this case, that is, 𝜔𝜂=1𝜋arccot−√2𝜋𝜂, (44) tends to zero as 𝜂⁻¹ when 𝜂→−∞. Conversely, for functions (42), and (43) 𝐹^(𝑖)(0)=0 if 𝑖=0,1 and 𝐹⁽²⁾(0)>0. Therefore, the kink type solutions for these cases tend to zero with exponential rates when 𝜂→−∞. However, their explicit form remains unknown; that is, why we simulate them solving numerically the Cauchy problem 𝑑𝜔=𝑑𝜂2𝐹(𝜔),𝜂>0,𝜔|_𝜂=0=12 (45) and, by virtue of the condition (C), define 𝜔 with negative argument as 𝜔(𝜂)=1−𝜔(−𝜂). To calculate the solution of (45) we use the Runge-Kutta method of the forth order with the mesh step ℎ_𝜂=0.01.

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 𝜀=0.1 (here and in what follows we numerate the waves from the left to the right). Moreover, this remains true independently of the wave parameters 𝑉₁, 𝑉₂ 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.

Figure 1 

Evolution of the kink-kink pair, 𝑉1>0, 𝑉2<0.

Figure 2 

Evolution of the kink-antikink pair, 𝑉1>0, 𝑉2<0.

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 𝑥→−𝑥,𝑡→−𝑡 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 𝑡→−𝑡. 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 𝜀=0.1. 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 𝑥→−𝑥,𝑡→−𝑡. So that, there are 12 stable structures of kinks (K) and antikinks (A):(1)KKK: (𝑉₁>0, 𝑉₂>0, 𝑉₃>0), (𝑉₁>0, 𝑉₂>0, 𝑉₃<0), (𝑉₁>0, 𝑉₂<0, 𝑉₃<0), (𝑉₁<0, 𝑉₂<0, 𝑉₃<0);(2)KKA: (𝑉₁>0, 𝑉₂<0, 𝑉₃<0);(3)KAA: (𝑉₁>0, 𝑉₂>0, 𝑉₃<0);(4)AAK: (𝑉₁>0, 𝑉₂<0, 𝑉₃<0);(5)AKK: (𝑉₁>0, 𝑉₂>0, 𝑉₃<0);(6)AAA: (𝑉₁>0, 𝑉₂>0, 𝑉₃>0), (𝑉₁>0, 𝑉₂>0, 𝑉₃<0), (𝑉₁>0, 𝑉₂<0, 𝑉₃<0), (𝑉₁<0, 𝑉₂<0, 𝑉₃<0).

Figure 3 

Evolution of the kink-triplet, 𝑉1>0, 𝑉2<0, 𝑉3<0.

Figure 4 

Evolution of the kink-kink-antikink triplet, 𝑉1>0, 𝑉2<0, 𝑉3<0.

Figure 5 

Evolution of the antikink-kink-kink triplet, 𝑉1>0, 𝑉2>0, 𝑉3<0.

Figure 6 

Evolution of the antikink-antikink-kink triplet, 𝑉1>0, 𝑉2<0, 𝑉3<0.

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, 𝑉_𝑖>0,𝑖=1,2,3, and 𝑉_𝑗<0, 𝑗=4,5,6. 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 𝑉_𝑖<0,𝑖=1,2,3, which is unstable.

Figure 7 

Interaction of 6 kinks for the nonlinearity (43).

Figure 8 

Evolution of four waves for the nonlinearity (43), 𝑉1>0, 𝑉2<0, 𝑉3<0, 𝑉4<0.

5. Conclusion

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 [11]. 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 [14].

Acknowledgment

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