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 equation. Damping then occurs according to the momentum balance equation, and a shelf is generated behind the soliton. 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 Korteweg-de Vries equation. In the next order of the perturbation theory, a higher-order equation can be obtained, which in general includes cubic nonlinearity, fifth-order linear dispersion, and nonlinear dispersion. For certain environmental conditions when the quadratic nonlinear term is small, the cubic nonlinear term becomes the major term and it should be taken into account together with the quadratic nonlinear term. The corresponding equation is known as the extended Korteweg-de Vries equation, or the Gardner equation. Our purpose here is to consider this equation supplemented with a damping term. If this term is small, the damping of a solitary wave can be studied with the use of asymptotic methods developed for perturbed solitons. Our aim is to obtain some new nontrivial results for the damping of large-amplitude solitary waves.