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Change of Polarity for Periodic Waves in the Variable‐Coefficient Korteweg‐de Vries Equation
Author(s) -
Grimshaw Roger
Publication year - 2015
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12067
Subject(s) - polarity (international relations) , korteweg–de vries equation , amplitude , mathematical analysis , mathematics , sign (mathematics) , variable coefficient , critical point (mathematics) , physics , nonlinear system , quadratic equation , geometry , optics , quantum mechanics , chemistry , cell , biochemistry
We examine the variable‐coefficient Kortweg‐de Vries equation for the situation when the coefficient of the quadratic nonlinear term changes sign at a certain critical point. This case has been widely studied for a solitary wave, which is extinguished at the critical point and replaced by a train of solitary waves of the opposite polarity to the incident wave, riding on a pedestal of the original polarity. Here, we examine the same case but for a modulated periodic wave train. Using an asymptotic analysis, we show that in contrast a periodic wave is preserved with a finite amplitude as it passes through the critical point, but a phase change is generated causing the wave to reverse its polarity.