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Toward a General Solution of the Three‐Wave Partial Differential Equations
Author(s) -
Martin Ruth A.,
Segur Harvey
Publication year - 2016
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.12133
Subject(s) - partial differential equation , integrable system , mathematics , inverse scattering transform , mathematical analysis , inverse scattering problem , boundary value problem , inverse problem , space (punctuation) , wave equation , scattering , series (stratigraphy) , differential equation , physics , computer science , paleontology , optics , biology , operating system
The three‐wave, resonant interaction equations appear in many physical applications. These partial differential equations (PDEs) are known to be completely integrable, and have been solved with initial data that decay rapidly in space, using inverse scattering theory. We present a new way to solve these equations, which makes no use of inverse scattering theory, and which can be used with a wide variety of boundary conditions. A “general solution” of these PDEs would involve six free, real‐valued functions of space. At this time, our “nearly general solution” accepts five free, real‐valued functions of space, and embeds them in convergent series in a deleted neighborhood of a pole.