Self-similar solutions of equations of the nonlinear Schrödinger type
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A study is made of self-similar solutions of an entire family of one-dimensional integrable dynamic systems of the nonlinear Schrödinger equation type. This family is reduced to one of three canonical forms corresponding to a Toda chain, a Volterra chain, or to the Landau-Lifshitz model, which can also be reduced to three self-similar systems coupled by Miura transformations with the fourth Painleve equation. A commutative representation is constructed for this equation. A relationship is established between the poles of the rational solutions of the fourth Painleve equation and the steady-state distribution of the electric charges in a parabolic potential. A self-similar solution is constructed for the spin dynamics. An exact solution is obtained for the nonlinear Schrödinger equation with variable dispersion (optical soliton).
KeywordsSoliton Elementary Particle Electric Charge Canonical Form Equation Type
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