Riemann Problems for Nonlinear Dispersive Equations

par Michael Shearer (North Carolina State University)

mardi 28 mars 2017, 14:00 → 15:00 Europe/Paris

salle Fokko du Cloux (ICJ, UCBL - La Doua, Bât. Braconnier)

Dispersive shock waves (DSW) are oscillatory solutions of nonlinear dispersive equations such as the KdV equation. The structure of a DSW is described as a modulated wave train joining a leading edge solitary wave to a trailing oscillatory tail of linear waves. This structure is well described by the Whitham equations, a system of three nonlinear conservation laws modeling amplitude, frequency and wave speed. In this talk, I describe the connection to dissipative-dispersive equations, specifically, the modified KdV-Burgers equation, u_t+u^2.u_x=\mu u_{xx} +\beta u_{xxx} in which the constant coefficients \mu >= 0 and \beta measure dissipation and  dispersion.  Much can be learned from the structure of solutions of initial value problems with Riemann initial data, in which u(x,0) is piecewise constant with a single jump. When \mu>0 the solutions are easily related to shock waves and rarefaction waves for the conservation law u_t+u^2.u_x=0. However, with \mu=0, the solutions involve DSWs. I show how the two cases are related, discuss the limit \mu > 0+, and demonstrate time scales over which different wave structures appear. The construction of DSWs turns out to contain subtleties related to the presence of undercompressive traveling waves for the \mu>0 case, and to the construction of shock-rarefaction wave solutions of the conservation law, due to the non-convex flux. The BBM equation u_t+u.u_x= u_{xxt}, is also dispersive, and an additional conservation law is enough to allow formulation of Whitham equations. The Riemann problem exhibits two notable phenomena. Numerical simulations reveal the persistence and decay of expansion shocks, which are analyzed with matched asymptotics. By appealing to Riemann invariants, a similar analysis applies to expansion shocks for the related Boussinesq system of equations for water waves. Finally, I discuss numerical simulations and beginning analysis of implosion, in which a DSW meets a singularity in the dispersion relation, corresponding to a loss of genuine nonlinearity in the Whitham equations. This is joint work with Gennady El and Mark Hoefer.