Admissible Transformations and Normalized Classes of Nonlinear Schrödinger Equations

• Published in: Acta applicandae mathematicae Vol. 109; no. 2; pp. 315 - 359
• Main Authors: Popovych, Roman O., Kunzinger, Michael, Eshraghi, Homayoon
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.02.2010; Springer Nature B.V
• Subjects: Algebra; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Classification; Computational Mathematics and Numerical Analysis; Differential equations; Independent variables; Inverse problems; Mathematical analysis; Mathematics; Mathematics and Statistics; Ordinary differential equations; Partial Differential Equations; Physics; Probability Theory and Stochastic Processes; Studies; Symmetry; Theory of relativity; Variables; Admissible transformations; Normalized classes of differential equations; Group analysis of differential equations; Group classification of differential equations; Equivalence group; Nonlinear Schrödinger equations; Lie symmetry
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-008-9321-4

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.