Shallow Water Wave Systems

Abstract

In this article we study various systems that represent the shallow water wave equation vxxt+αvvt−βvx∂x‐1(vt) −vt−vx = 0,where (∂x−1f)(x)=∫x∞f(y) dy, and α and β are arbitrary, nonzero, constants. The classical method of Lie, the nonclassical method of Bluman and Cole [J. Math. Mech. 18:1025 (1969)], and the direct method of Clarkson and Kruskal [J. Math. Phys. 30:2201 (1989)] are each applied to these systems to obtain their symmetry reductions. It is shown that for both the nonclassical and direct methods unusual phenomena can occur, which leads us to question the relationship between these methods for systems of equations. In particular an example is exhibited in which the direct method obtains a reduction that the nonclassical method does not.