Nonlinear waves in two spatial dimensions: Modulation theory, dispersive shocks and applications
When
Where
Speaker:
Gino Biondini
Affiliation:
State University of New York at Buffalo, Department of Mathematics
Title:
Nonlinear waves in two spatial dimensions: Modulation theory, dispersive shocks and applications
Abstract:
The talk is aimed at presenting an overview of recent results on the development and application of Whitham modulation theory for two-dimensional nonlinear evolution equations, focusing primarily on the Kadomtsev-Petviashvili (KP) equation.
After a brief review of some background material, the first part of the talk will discuss the recently-derived Whitham modulation equations for the KP equation, their properties, and the applications of these modulation equations to characterize (i) the stability of the cnoidal wave solutions of KP, (ii) the temporal evolution of soliton stems, and (iii) the oblique interactions between soliton stems, rarefaction waves (RWs) and dispersive shock waves (DSWs).
The second part of the talk will discuss the temporal dynamics generated by wedge-shaped initial conditions in the KP equation. Various asymptotic wave patterns are identified, classified and characterized in terms of the incidence angle and the amplitude of the initial step, which can give rise to either subcritical or supercritical configurations, including the generalization to DSWs of the Mach reflection and expansion of viscous shocks and line solitons, and an eightfold amplification of the amplitude of an obliquely incident flow upon a wall at the critical angle.
Time permitting, the last part of the talk will discuss the analysis of Riemann problems for the stationary reduction of the KP equation and its application to describe steady shallow water flow past an obstacle, as well as the refraction of line solitons past the resulting RWs and DSWs.