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Linear stability of solitary waves of the Green‐Naghdi equations
Author(s) -
Li Yi A.
Publication year - 2001
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.1
Subject(s) - mathematics , eigenvalues and eigenvectors , mathematical analysis , nonlinear system , perturbation (astronomy) , stability (learning theory) , amplitude , physics , quantum mechanics , machine learning , computer science
Abstract We investigate the eigenvalue problem obtained from linearizing the Green‐Naghdi equations about solitary wave solutions. Unlike weakly nonlinear water wave models, the physical system considered here has nonlinearity in its highest derivative term. This results in more detailed asymptotic analysis of the eigenvalue problem in the presence of a large parameter. Combining the technique of singular perturbation with the Evans function, we show that for solitary waves of small amplitude, the problem has no eigenvalues of positive real part and the Evans function is nonvanishing everywhere except the origin. This fact then leads to the linear stability of these solitary waves. © 2001 John Wiley & Sons, Inc.