Well-Posedness for a Coupled System of Kawahara/KdV Type Equations

Abstract

We consider the initial value problem (IVP) associated to a coupled system of Kawahara/KdV type equations. We prove the well-posedness results for given data in a Gevrey spaces. The proof relies on estimates in space-time norms adapted to the linear part of the equations. In particular, estimates in Bourgain spaces are proven for the linear and nonlinear terms of the system and the main result is obtained by a contraction principle. The class of system in view contains a number of systems arising in the modeling of waves in fluids, stability and instability of solitary waves and models for wave propagation in physical systems where both nonlinear and dispersive effects are important. The techniques presented in this work were based in Grujic and Kalisch, who studied the Gevrey regularity for a class of water-wave models and the well-posedness of a IVP associated to a general equation, whose the initial data belongs to Gevrey spaces.