# Seminar talk, 1 March 2017

${\displaystyle u_{t}=2uu_{x}+u_{xxx}+\varepsilon \chi _{a,b}u_{xx}}$,
where ${\displaystyle \chi _{a,b}}$ is the characteristic function of the interval ${\displaystyle [a,b]}$, then the soliton ${\displaystyle 6a^{2}\mathrm {ch} ^{-2}(4a^{3}t+ax)}$ arrived from the right partially reflects at viscous barrier ${\displaystyle x\in [a,b]}$ 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.