SOLITONS IN THE CAMASSA-HOLM SHALLOW-WATER EQUATION

We study the class of shallow water equations of Camassa and Hold deri ved from the Lagrangian, L = integral[1/2(phi(xxxx)-phi(x))phi(t)-1/2( phi(x))3 - 1/2phi(x)(phi(xx))2 - 1/2kappaphi(x)2] dx, using a variatio nal approach. This class contains ''peakons'' for kappa = 0, which are solitons whose peaks have a discontinuous first derivative. We derive approximate solitary wave solutions to this class of equations using trial variational functions of the form u(x, t) = phi(x) = A(t) exp[-b eta(t)\x - q(t)\2n] in a time-dependent variational calculation. For t he case kappa = 0 we obtain the exact answer. For kappa not-equal-to 0 we obtain the optimal variational solution. For the variational solut ion having fixed conserved momentum P = integral1/2(u2 + u(x)2) dx, th e soliton's scaled amplitude, A/P1/2, and velocity, q/P1/2, depend onl y on the variable z=kappa/square-root P. We prove that these scaling r elations are true for the exact soliton solutions to the Camassa-Holm equation.