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Acta Applicandae Mathematicae

, Volume 109, Issue 2, pp 315–359 | Cite as

Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

  • Roman O. Popovych
  • Michael Kunzinger
  • Homayoon Eshraghi
Article

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.

Keywords

Group classification of differential equations Group analysis of differential equations Equivalence group Admissible transformations Normalized classes of differential equations Lie symmetry Nonlinear Schrödinger equations 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Roman O. Popovych
    • 1
    • 2
  • Michael Kunzinger
    • 1
  • Homayoon Eshraghi
    • 3
    • 4
  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.Institute of Mathematics of NAS of UkraineKyivUkraine
  3. 3.Physics DepartmentIran University of Science and Technology (IUST)TehranIran
  4. 4.Institute for Studies in Theoretical Physics and Mathematics (IPM)TehranIran

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