Long-time dynamics of Korteweg-de Vries solitary waves over a variable bottom.

摘要

We study the dynamics of solitons of KdV-type equations arising in the theory of shallow water waves propagating over channels with variable bottom. Under a rescaling, these waves satisfy the variable bottom generalized Korteweg-de Vries (bKdV) equation ∂tu = -∂x( 62x u + f(u) - b(t, x)u) with f = u2 and b related to the varying channel depth. The main result of the thesis is concerned with solitary wave dynamics of the bKdV with f a general nonlinearity (modulo some constraints) and b a small, bounded and slowly varying function. We prove that under suitable conditions on b and f, the bKdV is globally well-posed. With the knowledge that solutions exist, we study the long time behaviour of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of a solitary wave, whose centre and scale evolve according to a certain dynamcal law involving the function b(t, x), plus an H 1( R )-small fluctuation. As motivation, we also describe how the bKdV equation appears in the field of water waves.