Advances in Mathematical Physics

Volume 2018, Article ID 3586514, 17 pages

https://doi.org/10.1155/2018/3586514

## A Newton-Type Approach to Approximate Travelling Wave Solutions of a Schrödinger-Benjamin-Ono System

Departamento de Matemáticas, Universidad del Valle, Calle 13 Nro. 100-00, Cali, Colombia

Correspondence should be addressed to Juan Carlos Muñoz Grajales; moc.oohay@zmlracj

Received 1 October 2017; Revised 1 March 2018; Accepted 13 March 2018; Published 19 April 2018

Academic Editor: Remi Léandre

Copyright © 2018 Juan Carlos Muñoz Grajales. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce a Newton’s iterative method to approximate periodic and nonperiodic travelling wave solutions of the Schrödinger-Benjamin-Ono system derived by M. Funakoshi and M. Oikawa. We analyze numerically the influence of the model’s parameters on these solutions and illustrate the collision of two unequal-amplitude solitary waves propagating with different speeds computed by using the proposed numerical scheme.

#### 1. Introduction

This paper is concerned with the nonlinear one-dimensional system (hereafter called the Schrödinger-Benjamin-Ono system (SBO))for , with periodic spatial boundary conditions. This nonlinear dispersive system was derived by Funakoshi and Oikawa [1] to describe the motion of two fluids with different densities under capillary-gravity waves in a deep water regime. The functions and denote the short and long wave terms, respectively. The parameters , are real numbers, and denotes the Hilbert transformwhere stands for the integration in the principal value sense. Other physical scenarios, where system (1) with arises, are the sonic-Langmuir wave interaction in plasma physics (Karpman [2]), the capillary-gravity interaction waves (Djordjevic and Redekopp [3], Grimshaw [4]), and water-wave interaction in a nonlinear medium (see, e.g., [5, 6]).

Due to a balance between dispersive and nonlinear effects, system (1) admits the so-called* travelling-wave solutions* in the formwhere and are periodic real-valued functions or smooth real-valued functions such that, for each , and , as . In this last case, these solutions are called* solitary waves*. In the literature there are some previous analytical results about periodic and nonperiodic travelling wave solutions to the SBO system. For instance, Angulo and Montenegro [7] established the existence of even solitary wave solutions by employing the concentration compactness method (Lions [8, 9]). Existence and stability of a new family of solitary waves was established in [10] for system (1) and a coupled Schrödinger-KdV model. On the other hand, when , the nonperiodic initial value problem corresponding to the SBO system has been considered by Bekiranov et al. [11], who proved well-posedness in the Sobolev space , with . In the case of , Pecher [12] showed local well-posedness for , and Angulo et al. [13] proved global well-posedness for and . However, there are only a few results known for the spatial periodic problem. For instance, assuming that , Angulo et al. [13] showed that system (1) is locally well posed in the Sobolev space for and they further established existence of a smooth branch of periodic travelling wave solutions of the SBO system for and close to 0, and , where is the fundamental period.

In this paper, we introduce a numerical scheme for approximating the periodic and nonperiodic travelling wave solutions of system (1), with whose existence was established in previous works ([7, 10, 13]). This scheme combines a Fourier spectral discretization together with a Newton-type iteration which is initialized by means of well-known exact solutions for the case . By using the spectral Crank–Nicolson numerical scheme introduced in [14] for solving system (1), we validate the approximate travelling waves computed and also illustrate the phenomenon of collision of two right-running unequal-amplitude solitary waves propagating with different speeds. The behavior of the colliding solitary waves is found to be similar to solitary wave solutions of other nonintegrable dispersive-type equations, such as the scalar Benjamin equation [15] and the regularized Benjamin-Ono equation [16]. Such numerical study on the travelling wave solutions of the SBO system has not been considered in previous works to the best knowledge of the author. Furthermore, we obtain numerically new periodic travelling wave solutions which are not included in the analytical theory presented in the work by Angulo et al. [13]. These are the main contributions of the present study. Other contribution is to explain the application of a numerical scheme which belongs to the spectral collocation methods to approximate the solutions of integrodifferential equations, such as system (1). These numerical methods have been intensively used in the solution of dispersive-type equations in the last years. We point out that exact solutions for system (1) are not known when . Therefore, a numerical strategy is very important in order to investigate the properties of the solution space, such as to establish the parameter regime for existence of periodic and nonperiodic travelling waves, orbital stability under small initial disturbances, and interactions among these solutions, for example. We point out that the constant in the SBO system is positive for physical meaning [1]. However, as we mentioned above, in previous works [7, 10–12], the SBO system has been studied from a mathematical point of view, establishing well-posedness and that travelling wave solutions for the SBO system are possible when the parameter takes both signs. This is an interesting mathematical fact, and thus we will conduct some numerical experiments which illustrate the family of travelling wave solutions for both and .

This paper is organized as follows. In Section 2, we present a brief review of well-known exact solutions of the SBO system for . In Section 3, we explain the numerical methodology used to find approximations to travelling wave solutions of the SBO system. In Section 4, we compute some periodic and nonperiodic travelling wave solutions of the system, analyze the influence of model’s parameters on the geometry of periodic and nonperiodic travelling waves, and explore the collision of two right-going solitary waves of the SBO system. Finally, Section 5 contains the conclusions of our work.

#### 2. Computing Exact Travelling Wave Solutions

For the sake of completeness, in this section, we present a brief review of a class of exact travelling wave solutions of system (1) (periodic and non-periodic) which are well known in the literature when (see [7, 10, 13] for more details and analytical results on existence of travelling wave solutions for ). These analytical special cases are important because they can be used as initial points for computing branches of solutions when , through Newton’s iteration and numerical continuation, for example.

In the nonperiodic case and , we can construct a family of exact travelling wave solutions in the form (3) of system (1). In first place, note that, for , the real-valued functions must satisfy the systemwhere the tildes denote differentiation with respect to the variable .

We recall that and their derivatives decay to zero at infinity. Thus we can integrate (5) to getAfter substituting the value of into (4), we obtainBy multiplying the previous equation by , we get Therefore,Thus, by integrating (9) and using again the decaying properties of the functions , we obtainLooking for solutions of (11) in the form , where are real constants to be determined, we have thator, equivalently,

Using the identity , we arrive atFrom this equation we can conclude thatOn the other hand, evaluating (15) at ,Substituting the value of the constant in the previous equation, we arrive atTherefore,On the other hand, in the periodic case and with , we also have that must also satisfy system (4)-(5). By integrating (5), we getwhere is a constant. Here we assume that . Thus, after substituting this expression for into (4), we arrive atTherefore, we obtainThen integrating the last equation, we arrive atwhere is a constant. Suppose that , are positive real numbers and is negative. Furthermore, let , (, ) be the roots of the polynomial of the right side of the previous equations; then we can writeand thusTherefore,Without less of generality, we may assume that , and consider the change of variable . Substituting (24), we get

Let . ThusNow let us define . Thenand thusSubstituting this expression into (28), we obtainWe can simplify the previous equation to getAssuming that , then and integrating the previous equation where . Using from the definition of the Jacobian elliptic function , we get and hence . In terms of the original variables , we obtain the dnoidal wave solutions corresponding to system (11)-(12):whereSince the fundamental period of the dnoidal function is , whereit follows that has fundamental period given byWe point out that, due to the factor in the component of the travelling wave solution (3), we have the conditionfor some .

#### 3. Methodology for Approximation of Travelling Wave Solutions of the Full SBO System

For , we recall that if is a travelling wave solution in the formof system (1), then are real-valued functions which must satisfy the following equations:In first place, we are interested in finding approximations to even solutions with period , of system (1); let us introduce truncated cosine expansions for and :whereand let us introduce analogous expressions for the coefficients . By substituting expressions (41) into (40), evaluating them at the collocation pointsand using the property of the Hilbert transformwe obtain the system of nonlinear equations,for , which can be written in the formwhere the coefficients are the unknowns. We point out that the property (44) of the Hilbert transform makes the numerical solution of system (40) with the spectral collocation proposed in the present paper easier. This is the motivation for the choice of this numerical strategy.

Nonlinear system (46) can be solved by Newton’s iteration. Computation of the cosine series in (41) and the integrals in (42) is performed using the FFT (Fast Fourier Transform) algorithm. The Jacobian of the vector field is approximated by the second-order accurate formulawhere and . We stop Newton’s iteration when the relative error between two successive approximations and the value of the vector field are smaller than . We use the travelling wave solutions given in Section 2, when , for initializing the Newton’s iterative scheme explained above.

#### 4. Description and Discussion about the Numerical Results

In the following, we present some numerical experiments performed by using the numerical scheme described in the previous section.

*Experiment Set 1 (Periodic Waves).* In the first numerical experiment we set , , and , and the fundamental period is . The iterative Newton’s procedure is initialized withThe result of this computer simulation is presented in Figure 1. To check that we have computed really a periodic travelling wave of the SBO system (1), we run the spectral Crank-Nicolson numerical solver to approximate the solutions to this system, introduced by the author in [14], with time step , and FFT points; the spatial computational domain is the interval with , and the initial values arewith being the profiles displayed in Figure 1. The result of this computer simulation at is displayed in Figure 2, superimposed with the expected position of the travelling wave given by (39). We observe a good accordance with a maximum error of between the profiles of the modulus of components and , corroborating that, in fact, we have an approximation of a travelling wave solution to system (1). The same verification was performed successfully for other periodic travelling waves computed for different values of the parameter , displayed in Figures 3, 4, and 5 for , , and , respectively.