# Remarks on proofs of conservation laws for nonlinear SchrÃ¶dinger equations

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By applying a special BÃ¤cklund transformation, a general variable separation solution for the (2 + 1)-dimensional nonlinear SchrÃ¶dinger equation is derived. In addition to some types of the usual localized excitations, such as dromions, lumps, ring solitons, oscillated dromions, and...

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No abstract available.

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This paper is concerned with the nonlinear SchrÃ¶dinger equation with an unbounded potential iÏ†t = - Î”Ï† + V(x)Ï† - Î¼âˆ£Ï†âˆ£P-1Ï† - Î»âˆ£Ï†âˆ£q-1Ï†, x âˆˆ â„N, t â‰¥ 0, where Î¼ > 0, Î» > 0, and 1

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We consider the nonstationary Schrodinger equation with the potential being a perturbation of a generic one-dimensional potential by means of a decaying two-dimensional function in the framework of the extended resolvent approach. We give the corresponding modification of the Jost and...

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Darboux transformations in one variable form the basis for the factorization methods and have numerous applications to geometry, nonlinear equations and SUSY quantum mechanics. In spite of this wide range of applications the theory of Darboux transformations in two variables and its elegant...

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As a milestone method, the inverse scattering transformation is also known as the nonlinear Fourier transformation for solving nonlinear partial differential equations analytically. The equivalent integral equations play a crucial role for the inverse scattering transformation. In this paper,...

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The arguments leading to a nonlinear generalization of the SchrÃ¶dinger equation in the context of the maximum uncertainty principle are reviewed. The exact and perturbative properties of that equation depend on a free regulating/interpolating parameter Î·, which can be fixed using...