Using the KdV conserved quantities in problems of splitting of initial data and reflection / refraction of solitons in varying dissipation and/or dispersion media

Abstract: An arbitrary compact-support initial datum for the Korteweg-de Vries equation asymptotically splits into solitons and a radiation tail, moving in opposite direction. We give a simple method to predict the number and amplitudes of resulting solitons and some integral characteristics of the tail using only conservation laws.

A similar technique allows to predict details of the behavior of a soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The process is described with a special type generalized KdV-Burgers equation $u_t=(u^2+f(x)u_{xx})_x$.

The transmitted wave either retains the form of a soliton (though of different parameters) or scatters a into a number of them. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapor saturation/condensation area. Some rough estimations for a prediction of an output are given using the relative decay of the KdV conserved quantities; in particular a formula for a number of solitons in the transmitted signal is given.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

( slides | video )

Geometry of differential equations seminar