Seminar talk, 1 March 2017

Speaker: Alexey Samokhin

Title: Reflections of soliton on viscous barrier and the degradation of conserved quantities for the KdV

Abstract:
If waves are described by the equation

$u_{t}=2uu_{x}+u_{xxx}+\varepsilon \chi _{a,b}u_{xx}$,

where $\chi _{a,b}$ is the characteristic function of the interval $[a,b]$, then the soliton $6a^{2}\mathrm {ch} ^{-2}(4a^{3}t+ax)$ arrived from the right partially reflects at viscous barrier $x\in [a,b]$ and partially passes through in the form of a soliton of smaller velocity and amplitude. The process in some degree is described by the so-called balance laws, which are the evolution of conservation laws for KdV.