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Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

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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 Schrödinger equations, in particular, in the class of nonlinear (1+1)-dimensional Schrödinger 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 Schrödinger equations with potentials. The proposed approach can be applied to studying symmetry properties of a wide range of differential equations.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090, Wien, Austria

    Roman O. Popovych & Michael Kunzinger

  2. Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kyiv, 4, Ukraine

    Roman O. Popovych

  3. Physics Department, Iran University of Science and Technology (IUST), Narmak, Tehran, Iran

    Homayoon Eshraghi

  4. Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box: 19395-5531, Tehran, Iran

    Homayoon Eshraghi

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  1. Roman O. Popovych
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  2. Michael Kunzinger
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Popovych, R.O., Kunzinger, M. & Eshraghi, H. Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations. Acta Appl Math 109, 315–359 (2010). https://doi.org/10.1007/s10440-008-9321-4

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