Fourier splitting and dissipation of nonlinear dispersive waves

Presented herein is a new method for analysing the long-time behaviour of s olutions of nonlinear, dispersive, dissipative wave equations. The method i s applied to the generalized Korteweg-de Vries equation posed on the entire real axis, with a homogeneous dissipative mechanism included. Solutions of such equations that commence with finite energy decay to zero as time beco mes unboundedly large. In circumstances to be spelled out presently, we est ablish the existence of a universal asymptotic structure that governs the f inal stages of decay of solutions. The method entails a splitting of Fourie r modes into long and short wavelengths which permits the exploitation of t he Hamiltonian structure of the equation obtained by ignoring dissipation. We also develop a helpful enhancement of Schwartz's inequality. This approa ch applies particularly well to cases where the damping increases in streng th sublinearly with wavenumber. Thus the present theory complements earlier work using centre-manifold and group-renormalization ideas to tackle the s ituation wherein the nonlinearity is quasilinear with regard to the dissipa tive mechanism.