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

We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ∂tu = −∂x(∂ x2u + 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...

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Bibliographic Details
Published in:Communications on pure and applied mathematics 2006-06, Vol.59 (6), p.869-905
Main Authors: Dejak, Steven I., Sigal, Israel Michael
Format: Article
Language:English
Subjects:
Exact sciences and technology
Global analysis, analysis on manifolds
Mathematical analysis
Mathematical problems
Mathematics
Partial differential equations
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Waveform analysis
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Summary:We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ∂tu = −∂x(∂ x2u + 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(ℝ)‐small fluctuation. © 2005 Wiley Periodicals, Inc.
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.20120