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Long‐time dynamics of KdV solitary waves over a variable bottom
Authors:Steven I Dejak  Israel Michael Sigal
Abstract:We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ?tu = ??x(?urn:x-wiley:00103640:media:CPA20120:tex2gif-stack-1u + 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 H1(?)‐small fluctuation. © 2005 Wiley Periodicals, Inc.
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