Admissible Transformations and Normalized Classes of Nonlinear Schrodinger Equations

作者：Popovych Roman O^*; Kunzinger Michael; Eshraghi Homayoon

来源：Acta Applicandae Mathematicae, 2010, 109(2): 315-359.

DOI：10.1007/s10440-008-9321-4

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

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

• 出版日期2010-2