Existence Of Periodic Traveling Wave Solutions Research Articles

Existence of periodic traveling-wave solutions for a nonlinear Schrödinger system: A topological approach

In this paper, the existence of periodic traveling-wave solutions for a nonlinear Schrodinger system is established using the topological degree theory for positive operators. The method guarantees existence of periodic solutions in a parameter region in the period and phase speed plane.

On the periodic Schrödinger–Boussinesq system

We study the local and global well-posedness of the periodic boundary value problem for the nonlinear Schrödinger–Boussinesq system. The existence of periodic traveling-wave solutions as well as the orbital stability of such solutions are also considered.

Cnoidal wave solutions to Boussinesq systems

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.

Existence of periodic travelling-wave solutions of nonlinear, dispersive wave equations

For general classes of nonlinear, dispersive wave equations, we discuss the existence of the analogue of the cnoidal-wave solutions of the Korteweg–de Vries equation. The solutions in question have travelling and periodic features. The topological-degree theory of positive operators on Banach spaces is applied.