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Long‐time dynamics of KdV solitary waves over a variable bottom
Author(s) -
Dejak Steven I.,
Sigal Israel Michael
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20120
Subject(s) - korteweg–de vries equation , mathematics , variable (mathematics) , bounded function , function (biology) , variable coefficient , mathematical analysis , nonlinear system , type (biology) , scale (ratio) , mathematical physics , physics , geology , quantum mechanics , paleontology , evolutionary biology , biology
We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ∂ t u = −∂ x (∂ x 2u + 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. © 2005 Wiley Periodicals, Inc.