The Dissipation Of Nonlinear Dispersive Waves: The Case Of Asymptotically Weak Nonlinearity

. It is proved herein that certain smooth, global solutions of a class of quasi-linear, dissipative, dispersive wave equations have precisely the same leading order, long-time, asymptotic behavior as the solutions (with the same initial data) of the corresponding linearized equations. The solutions of the nonlinear equations are shown to be asymptotically self-similar with explicitly determined profiles. The equations considered have homogeneous nonlinearities and homogeneous dispersive and dissipative symbols. By relating these degrees of homogeneity to the leading order asymptotic behavior of the Fourier transform of the initial data (near k = 0), different classes of long-time asymptotic behavior are characterized. These results cover the case where dissipation is not asymptotically negligible (in comparison with dispersion), and where nonlinear effects are asymptotically negligible (in comparison with linear effects, i.e. dissipation and dispersion). They always hold for solutions ..