Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation

Provided ν>0, solutions of the generalized regularized long wave-Burgers equation(*)ut+ux+P(u)x−νuxx−uxxt=0that begin with finite energy decay to zero as t becomes unboundedly large. Consideration is given here to the case where P vanishes at least cubically at the origin. In this case, solutions of (*) may be decomposed exactly as the sum of a solution of the corresponding linear equation and a higher-order correction term. An explicit asymptotic form for the L2-norm of the higher-order correction is presented here. The effect of the nonlinearity is felt only in the higher-order term. A similar decomposition is given for the generalized Korteweg–de Vries–Burgers equation(**)ut+ux+P(u)x−νuxx+uxxx=0.