Group analysis of general Burgers-Korteweg-de Vries equations

Abstract

The complete group classification problem for the class of (1+1)-dimensional $r$th order general variable-coefficient Burgers-Korteweg-de Vries equations is solved for arbitrary values of $r$ greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie reductions of equations from the class under consideration with respect to its equivalence group. Studying of alternative gauges for equation coefficients with equivalence transformations allows us not only to justify the choice of the most appropriate gauge for the group classification but also to construct for the first time classes of differential equations with nontrivial generalized equivalence group such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the class. For the subclass of equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we explicitly construct its extended generalized equivalence group in a rigorous way. The new notion of effective generalized equivalence group is introduced.

arXiv comment40 pages, 1 table, minor corrections

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• Classification of reduction operators and exact solutions of variable coefficient Newell-Whitehead-Segel equations
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• Equivalence groupoids and group classification of multidimensional nonlinear Schrödinger equations
• Mapping method of group classification
• Admissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
• On the ineffectiveness of constant rotation in the primitive equations and their symmetry analysis
• Lie reductions and exact solutions of dispersionless Nizhnik equation
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