Samokhin A. Nonlinear waves in layered media: solutions of the KdV-Burgers equation (abstract)

From Geometry of Differential Equations
Jump to: navigation, search

Speaker: Alexey Samokhin

Title: Nonlinear waves in layered media: solutions of the KdV-Burgers equation

Abstract:
The KdV-Burgers equation is used to model a behavior of a KdV-soliton which, while moving in non-dissipative medium encounters a barrier with dissipation. The layered media consist of layers with both dispersion an dissipation and layers without dissipation. In the latter case the waves are described by the KdV equation, while in the former - by the Kdv-Burgers one.

The dissipation results in reducing the soliton amplitude/velocity, and a reflection and refraction occur at the boundary(s) of a dissipative layer. In the case of a finite width barrier on the soliton path, after the wave leaves the dissipative barrier it retains a soliton form and a reflection wave arises as small and quasi-oscillations (a breather). Hence a soliton solution of the KdV equation, meeting a layer with dissipation, transforms somewhat similarly to a ray of light in the air crossing a semi-transparent plate.

Another case models a passage from non-dissipative half-space to a dissipative one. It is natural to expect each solution to behave as the one of the KdV at the first half-space and as solution of KdV-B at the second one. The process of transition from the soliton to the correspondent solution of the KdV-B is predictable. The transient wave in a dissipative media becomes a solitary shock which loses speed and decays to become nonexistent at t\rightarrow +\infty; and a reflected wave is seen in the non-dissipative half-space.

Other traveling wave solutions are also studied within this model.


Event: Local and Nonlocal Geometry of PDEs and Integrability, 8-12 October 2018, SISSA, Trieste, Italy.
The conference in honor of Joseph Krasil'shchik's 70th birthday.