Shallow water wave systems

In this article we study various systems that represent the shallow water w ave equation upsilon(xxt) + alpha upsilon upsilon(t) - beta upsilon(x)partial derivative (x)(-1) (upsilon(t)) -upsilon(t) - upsilon(x) = 0, (1) where (partial derivative(x)(-1)f)(x) = integral(x)(infinity)f(y) dy, and a lpha and beta are arbitrary, nonzero, constants, The classical method of Li e, 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 reduction s. It is shown that for both the nonclassical and direct methods unusual ph enomena can occur, which leads us to question the relationship between thes e methods for systems of equations. In particular an example is exhibited i n which the direct method obtains a reduction that the nonclassical method does not.