Cnoidal wave solutions to Boussinesq systems

Abstract

In this paper, two different techniques will be employed to study the cnoidal wave solutions of the Boussinesq systems. First, the existence of periodic travelling-wave solutions for a large family of systems is established by using a topological method. Although this result guarantees the existence of cnoidal wave solutions in a parameter region in the period and phase speed plane, it does not provide the uniqueness nor the non-existence of such solutions in other parameter regions. The explicit solutions are then found by using the Jacobi elliptic function series. Some of these explicit solutions fall in the parameter region where the cnoidal wave solutions are proved to exist, and others do not; so the method with Jacobi elliptic functions provides additional cnoidal wave solutions. In addition, the explicit solutions can be used in many ways, such as in testing numerical code and in testing the stability of these waves.

Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

“…The class of solutions described in terms of the Weierstrass elliptic functions is identical to that given in [10]. Two advantages are obtained from our approach:…”

“…Both solutions are examples of the cnoidal wave solutions constructed in [10]. As stated in the introduction, an alternate method for constructing the cnoidal wave solutions is discussed in Appendix B.…”

“…• Since we have followed the method of Conte and Musette [11], we learn that the solutions found in [10] are all the elliptic solutions of (B.1). This information could not be obtained by using an ansatz as in [10].…”

“…In [8] Chen, and in a subsequent paper [10] Chen, Chen, and Nguyen, were able to provide an explicit description for different branches of exact solutions of the various Boussinesq equations. All the solutions that they constructed are one-dimensional solutions, in the sense that in a suitably rotated frame the solutions depend only on the first spatial variable, x, and not on the second one, y.…”

“…In this paper we return to the result of [10] in order to examine the spectral stability of the different line solutions constructed there. We investigate the stability both with respect to onedimensional perturbations (depending only on x) and with respect to fully two-dimensional perturbations (characterized by a wave length in the y direction).…”