Admissible Transformations and Normalized Classes of Nonlinear Schroedinger Equations

Abstract

The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques are proposed. Using these, we exhaustively describe admissible point transformations in classes of nonlinear (1+1)-dimensional Schroedinger equations, in particular, in the class of nonlinear (1+1)-dimensional Schroedinger equations with modular nonlinearities and potentials and some subclasses thereof. We then carry out a complete group classification in this class, representing it as a union of disjoint normalized subclasses and applying a combination of algebraic and compatibility methods. Moreover, we introduce the complete classification of (1+2)-dimensional cubic Schroedinger equations with potentials. The proposed approach can be applied to studying symmetry properties of a wide range of differential equations.

42 pages, extended version, the group classification of (1+2)-dimensional cubic Schroedinger equations with potentials was added

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• Symbolic Computation of Equivalence Transformations and Parameter Reduction for Nonlinear Physical Models
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• Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries
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• Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem
• Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients
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• Generalization of the algebraic method of group classification with application to nonlinear wave and elliptic equations
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• Group classification and exact solutions of variable-coefficient generalized Burgers equations with linear damping
• Lie symmetries of two-dimensional shallow water equations with variable bottom topography
• Algebraic method for finding equivalence groups
• Extended symmetry analysis of generalized Burgers equations
• Complete point symmetry group of the barotropic vorticity equation on a rotating sphere
• Group classification of variable coefficient generalized Kawahara equations
• Inverse problem on conservation laws
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• Equivalence groupoid of a class of variable coefficient Korteweg--de Vries equations
• Reduction operators and exact solutions of generalized Burgers equations
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• Equivalence groupoids of classes of linear ordinary differential equations and their group classification
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• Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
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• Conservation laws and hierarchies of potential symmetries for certain diffusion equations
• Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equations
• Group-invariant solutions of semilinear Schrodinger equations in multi- dimensions
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• On the Study of the General Group Classification of Systems of Linear Second-Order Ordinary Differential Equations
• Conservative parameterization schemes
• Equivalence groupoids and group classification of multidimensional nonlinear Schrödinger equations
• Point and contact equivalence groupoids of two-dimensional quasilinear hyperbolic equations
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• Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
• Symmetry classification of variable coefficient cubic-quintic nonlinear Schrödinger equations
• Normalized classes of generalized Burgers equations
• Extended symmetry analysis of (1+2)-dimensional fine Kolmogorov backward equation
• Group analysis of variable coefficient generalized fifth-order KdV equations
• Enhanced group classification of Benjamin-Bona-Mahony-Burgers equations
• Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients
• Solutions and reductions for radiative energy transport in laser-heated plasma
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• Group analysis of Benjamin-Bona-Mahony equations with time dependent coefficients
• Invariant parameterization of geostrophic eddies in the ocean
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• Group-theoretical analysis of variable coefficient nonlinear telegraph equations
• Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity