Seminar talk, 1 March 2017

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Speaker: Alexey Samokhin

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

If waves are described by the equation

[math]\displaystyle{ u_t=2uu_x+u_{xxx}+\varepsilon\chi_{a,b}u_{xx} }[/math],

where [math]\displaystyle{ \chi_{a,b} }[/math] is the characteristic function of the interval [math]\displaystyle{ [a,b] }[/math], then the soliton [math]\displaystyle{ 6a^2\mathrm{ch}^{-2}(4a^3t+ax) }[/math] arrived from the right partially reflects at viscous barrier [math]\displaystyle{ x\in[a,b] }[/math] 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.