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Finding eigenvalue problems for solving nonlinear evolution equations. (English) Zbl 1079.37509

Summary: The problem of determining what nonlinear evolution equations are exactly solvable by inverse scattering techniques is simplified by considering a linear limit. By linearizing a given eigenvalue problem and the associated time evolution operator, it is possible to determine the class of linearized dispersion relation(s) of the exactly solvable nonlinear evolution equations. Examples are given to illustrate the method.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35G20 Nonlinear higher-order PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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