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Weyl functions, the inverse problem and special solutions for the system auxiliary to the nonlinear optics equation. (English) Zbl 1151.35105

Summary: A Borg-Marchenko-type uniqueness theorem (in terms of the Weyl function) is obtained here for the system auxiliary to the \(N\)-wave equation. A procedure to solve the inverse problem is used for this purpose. The asymptotic condition on the Weyl function, under which the inverse problem is uniquely solvable, is completed by a new and simple sufficient condition on the potential, which implies this asymptotic condition. The evolution of the Weyl function is discussed and the solution of an initial-boundary-value problem for the \(N\)-wave equation follows. Explicit solutions of an inverse problem are obtained. The system with a shifted argument is treated.

MSC:

35R30 Inverse problems for PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
78A10 Physical optics
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