Stable and unstable solitary-wave solutions of the generalized regularizedlong-wave equation

Investigated here are interesting aspects of the solitary-wave solutions of the generalized Regularized Long-Wave equation u(t) + u(x) + alpha (u(p))(x) - betau(xxt) = 0. For p > 5, the equation has both stable and unstable solitary-wave solution s, according to the theory of Souganidis and Strauss. Using a high-order ac curate numerical scheme for the approximation of solutions of the equation, the dynamics of suitably perturbed solitary waves are examined. Among othe r conclusions, we find that unstable solitary waves may evolve into several , stable solitary waves and that positive initial data need not feature sol itary waves at all in its long-time asymptotics.