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A direct theory for the perturbed unstable nonlinear Schrödinger equation. (English) Zbl 0972.35148

Summary: A direct perturbation theory for the unstable nonlinear Schrödinger equation with perturbations is developed. The linearized operator is derived and the squared Jost functions are shown to be its eigenfunctions. Then the equation of linearized operator is transformed into an equivalent \(4\times 4\) matrix form with first-order derivative in \(t\) and the eigenfunctions into a four-component row. Adjoint functions and the inner product are defined. Orthogonality relations of these functions are derived and the expansion of the unity in terms of the four-component eigenfunctions is implied. The effect of damping is discussed as an example.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
82D10 Statistical mechanics of plasmas
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