Complete group classification of shallow water equations

Abstract

The thesis is devoted to the symmetry analysis of the shallow water equations with variable bottom topography in dimensions one and two. We find the generalized equivalence groups for the classes of one- and two-dimensional shallow water equations with variable bottom topography using the automorphism-based version of the algebraic method. It turns out that for both the classes, the generalized equivalence groups coincide with the corresponding usual equivalence groups. In the case of dimension one, we also compute the generalized equivalence group of the natural reparametrization of the class and then compare the computation with the analogous calculations for the original class. Specific attention is paid to the class of two-dimensional shallow water equations with variable bottom topography. We carry out the complete group classification for this class up to its equivalence transformations using the modern method of furcate splitting. This class is neither normalized in the usual sense nor in the generalized sense. In other words, it possesses admissible transformations that are not induced by elements of the equivalence group, and such admissible transformations establish additional equivalences among classification cases. For a number of pairs of classification cases we either prove that these cases are not equivalent to each other with respect to point transformations or indicate the associated point transformations for pairs of equivalent cases.