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The generation, propagation, and extinction of multiphases in the KdV zero‐dispersion limit
Author(s) -
Grava Tamara,
Tian FeiRan
Publication year - 2002
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.10050
Subject(s) - hodograph , mathematics , euler's formula , korteweg–de vries equation , mathematical analysis , limit (mathematics) , zero (linguistics) , integrable system , dispersion (optics) , poisson distribution , nonlinear system , physics , quantum mechanics , linguistics , philosophy , statistics
We study the multiphases in the KdV zero‐dispersion limit. These phases are governed by the Whitham equations, which are 2 g + 1 quasi‐linear hyperbolic equations where g is the number of phases. We are interested in both the interaction of two single phases and the breaking of a single phase for general initial data. We analyze in detail how a double phase is generated from the interaction or breaking, how it propagates in space‐time, and how it collapses to a single phase in a finite time. The Whitham equations are known to be integrable via a hodograph transform. The crucial step in our approach is to formulate the hodograph transform in terms of the Euler‐Poisson‐Darboux solutions. Under our scheme, the zeros of the Jacobian of the transform are given by the zeros of the Euler‐Poisson‐Darboux solution. Hence, the problem of inverting the hodograph transform to give the Whitham solution reduces to that of counting the zeros of the Euler‐Poisson‐Darboux solution. © 2002 Wiley Periodicals, Inc.