Long-time dynamics of KdV solitary waves over a variable bottom

We study the 2 variable-bottom, generalized Korteweg-de Vries (bKdV) equation partial derivative(1)u = -partial derivative(x)(partial derivative(2)(x)u + f(u) - b(t,x)u), where f is a nonlinearity and b is a small, bounded, and slowly varying function related to the varying depth of a channel of water. Many variable-coefficient KdV-type equations, including the variable-coefficient, variable-bottom KdV equation, can be rescaled into the bKdV. We study the long-time behavior 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 the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b(t,x) plus an H(1)(R)-small fluctuation. (c) 2005 Wiley Periodicals, Inc.