Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation

Provided v > 0, solutions of the generalized regularized long wave-Burgers equation u(1) + u(x) + P(u)(x) - vu(xx) - u(xxt) = 0 that begin with finite energy decay to zero as t becomes unboundedly large. Consideration is given here to the case where P vanishes at least cubicall y at the origin. In this case, solutions of (*) may be decomposed exactly a s the sum of a solution of the corresponding linear equation and a higher-o rder correction term. An explicit asymptotic form for the L-2-norm of the h igher-order correction is presented here. The effect of the nonlinearity is felt only in the higher-order term. A similar decomposition is given for t he generalized Korteweg-de Vries-Burgers equation u(t) + u(x) + P(u)(x) - vu(xx) + u(xxx) = 0 (C) 2001 Elsevier Science B.V. All rights reserved.