Stability of Solitary Waves for Some Schrodinger-KdV Systems

Abstract

This dissertation addresses existence and stability results for a two-parameter family of solitary-wave solutions to systems in which an equation of nonlinear Schrodinger type is coupled to an equation of Korteweg-de Vries type. Such systems govern interactions between long nonlinear waves and packets of short waves, and arise in fluid mechanics and plasma physics. Our proof involves the characterization of solitary-wave solutions as minimizers of an energy functional subject to two independent constraints. To establish the precompactness of minimizing sequences via concentrated compactness, we develop a new method of proving the sub-additivity of the problem with respect to both constraint variables jointly. The results extend the stability results previously obtained by Chen (1999), Albert and Angulo (2003), and Angulo (2006).

In addition, we also study the stability of solitary-wave solutions to an equation of short and long waves by using the techniques of convexity type. We shall apply the concentration compactness method to show the relative compactness of minimizing sequences for a different variational problem in which functional involved are not conserved quantities, and then, we use conserved quantities which arise from symmetries via Noether's theorem to obtain a relationship that makes it possible to utilize the variational properties of the solitary waves in the stability analysis. We prove that the stability of solitary waves is determined by the convexity of a function of the wave speed. The method is based on techniques appeared in Cazenave and Lions (1982), Levandosky (1998), and Angulo (2003), along with a convexity lemma of Shatah (1983).