Singular reduction operators in two dimensions

Abstract

The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to first-order ODEs are are exhaustively described. As examples, properties of singular reduction operators of (1+1)-dimensional evolution and wave equations are studied. It is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in-depth prior study of the corresponding singular vector fields.

22 pages, minor corrections

References in corpus (4)

• A precise definition of reduction of partial differential equations
• Symmetry reduction and exact solutions of the Navier-Stokes equations
• Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation
• Reduction Operators of Linear Second-Order Parabolic Equations

Cited by in corpus (11)

• Generalized conditional symmetries of evolution equations
• Extended symmetry analysis of generalized Burgers equations
• Point and generalized symmetries of the heat equation revisited
• Singular reduction modules of differential equations
• Reduction operators and exact solutions of generalized Burgers equations
• Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equations
• Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system
• Classification of reduction operators and exact solutions of variable coefficient Newell-Whitehead-Segel equations
• Reduction operators of Burgers equation
• Analytical and numerical studying of the perturbed Korteweg--de Vries equation
• Periodic structures described by the perturbed Burgers-Korteweg-de Vries equation