Damping of large-amplitude solitary waves

Soliton damping in weakly dissipative media has been studied for several decades, usually using the asymptotic theory of the slowly-varying solitary wave solution of the Korteweg-de Vries (KdV) equation. Damping then occurs according to the energy balance equation, and a shelf is generated behind the soliton. Various examples of this process have been given by Ott & Sudan (1970) Pelinovsky (1971), Karpman & Maslov (1977), Grimshaw (1981), Grimshaw (1983), Newell (1985), and Smyth (1988). However, for many cases in nonlinear wave dynamics, such as for internal solitary waves in the ocean and atmosphere, the nonlinearity is not so weak as is implied by the KdV equation. In the next order of the perturbation theory, a higher-order KdV equation can be obtained, which in general includes cubic nonlinearity, fifth-order linear dispersion, and nonlinear dispersion. The contribution of the various high-order terms depends on the parameters of the model (e.g. density stratification and shear flow), and sometimes several of them may be more important. For instance for internal waves in a two-layer fluid the quadratic nonlinear term vanishes when the pycnocline lies in the middle of fluid (in the Boussinesq approximation) and then it is the cubic nonlinear term which is the most important, so that the higher-order equation reduces to the modified KdV equation. In the absence of any damping term, this last equation is fully integrable, as is of course the KdV equation. For certain environmental conditions when the quadratic nonlinear term is small, the cubic nonlinear term becomes the major from the high-order terms and it should be taken into account together with the quadratic nonlinear term. The corresponding equation