APDE seminar@BCAM: Existence and interactions of solitary waves in the Zakharov Water Waves system under a slowly variable bottom

Abstract

The Zakharov Water Waves system (ZWW) models the evolution of an inviscid irrotational fluid with free surface. These are characterized by a quasilinear system for the free surcase and the fluid potential at the free boundary. In the finite flat bottom case, Amick-Kirchgässner proved the existence of small solitary waves. However, in practical situations, the bottom is always non-constant. 

In this work, we deal with the generalized solitary wave problem for the ZWW system with surface tension and a non-flat bottom, in one dimension, in the form of a slowly varying (in space) bottom. Our main result establishes that, under suitable conditions on the variation of the bottom, such a generalized nonlinear wave exists and interacts with the bottom in a well-defined fashion, surviving the weak long interaction and exiting the interaction region with well-defined final scaling and shift parameters. The techniques used in the proof of the main result are extensions of the construction of a multi-soliton like solution, introduced by Ming, Rousset and Tzvetkov, and the interaction of solitary waves and different media for KdV. However, the ZWW case presents a considerable amount of new challenges, including: shape derivatives of Dirichlet-Neumann and Neumann-Neumann boundary operators, the quasilinear character of the model, and the lack of a suitable asymptotic stability theory for solitary waves