Classification of conservation laws of shallow-water equations

Abstract

We carry out the complete classification of zero-order conservation laws of the classes of one- and two-dimensional shallow-water equations with variable bottom topography. We also find the complete equivalence group for the one-dimensional case, using the direct method, and for the two-dimensional case, using the algebraic method. Using conservation-law characteristics, we find all inequivalent cases of bottom topographies (up to the equivalence group), which give different spaces of conservation laws. Analogously, using additionally the method of furcate splitting, we solve the classification problem for conservation laws for the two-dimensional case.