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A Note on Resonance and Nonlinear Dispersive Waves
Author(s) -
Ablowitz M. J.
Publication year - 1975
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 - 10.1002/sapm197554161
Subject(s) - gravitational singularity , nonlinear system , perturbation (astronomy) , resonance (particle physics) , perturbation theory (quantum mechanics) , mathematical analysis , dispersion (optics) , physics , nonlinear resonance , limit (mathematics) , asymptotic analysis , classical mechanics , mathematics , optics , quantum mechanics
The general theory for the slow dispersion of nonlinear wave trains, first studied by Whitham, is applied to a wave train, which, in the weakly nonlinear limit, exhibits resonant singularities. Numerical and perturbation methods are used to develop singly periodic solutions both away from and near all such critical values. Similarly, the equations governing the slow modulations of such a system are found by asymptotic analysis. The expansions are found to be valid so long as the wave train is sufficiently nonlinear. These ideas should be applicable to other problems where resonant singularities arise, in particular, multiphase modes.