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Cauchy problem for multidimensional coupled system of nonlinear Schrödinger equation and generalized IMBq equation. (English) Zbl 0940.35181
The author studies the multidimensional coupled system of a nonlinear complex Schrödinger equation and a generalized IMBq equation, i.e., the system \(i\varepsilon _t+\nabla ^2\varepsilon -u\varepsilon =0\), \(u_{tt}-\nabla ^2u -a\nabla ^2u_{tt}=\nabla ^2f(u) + \nabla ^2 (|\varepsilon |^2)\). For a periodic boundary value problem (on an \(n\)-dimensional cube), the existence and uniqueness of the generalized local solution and the classical local solution are proved. For the Cauchy problem, the author proves existence, uniqueness and regularity of the generalized local solution and the classical local solution.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35D05 Existence of generalized solutions of PDE (MSC2000)
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35K55 Nonlinear parabolic equations
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