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Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation
Author(s) -
Flaschka H.,
Forest M. G.,
McLaughlin D. W.
Publication year - 1980
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.3160330605
Subject(s) - korteweg–de vries equation , mathematics , invariant (physics) , mathematical analysis , eigenvalues and eigenvectors , inverse , representation (politics) , mathematical physics , integrable system , inverse scattering transform , inverse scattering problem , inverse problem , physics , quantum mechanics , geometry , law , nonlinear system , politics , political science
Inverse spectral theory is used to prescribe and study equations for the slow modulations of N ‐phase wave trains for the Korteweg‐de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single‐phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first‐order terms.