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Some Higher‐Order Nonlinear Evolution Equations
Author(s) -
Haberman Richard
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/sapm1975544275
Subject(s) - nonlinear system , mathematics , inverse scattering transform , mathematical analysis , eigenvalues and eigenvectors , isospectral , inverse scattering problem , independent equation , differential equation , initial value problem , partial differential equation , physics , inverse problem , quantum mechanics
Nonlinear evolution equations are generated which correspond to isospectral differential‐matrix eigenvalue problems. Although not discussed here, this procedure is appropriate for the analysis of the initial‐value solution by the inverse‐scattering method. In the 2 × 2 case the results obtained are consistent with other more general procedures. It yields the sine‐Gordon equation, the reduced Maxwell‐Bloch equations for self‐induced transparency, and also a type of PBBM equation for long nonlinear water waves. In the n × n case, equations are derived corresponding to a new class of triad and multitriad resonant nonlinear interaction of wave envelopes.