Dispersive shock waves in periodic non-dispersive media
Background
In a series of papers dating back to LeVeque & Yong 2003, it has been shown that solutions of hyperbolic systems with periodic coefficients can exhibit behavior very similar to that of dispersive nonlinear PDEs. Specifically, a wide range of initial data can develop into solitary waves that propagate without changing shape and interact with each other only through a phase shift.
This behavior can be understood through homogenization via the use of multiple-scale perturbation theory, in which one assumes that the solution is slowly-varying compared to the medium. In practice, solitary waves tend to be only a few medium-periods wide, but this is enough for the theory to give accurate results.
In dispersive nonlinear wave equations, it is often the case that positive initial perturbations lead to solitary waves, while negative perturbations (dips) lead to dispersive shock waves. The latter consist of rapid oscillations that spread over time and separate two approximately constant states. Negative initial perturbations have not been studied for hyperbolic PDEs with periodic coefficients.
Research project
It is natural to ask: Can dispersive shock waves be observed in hyperbolic systems with periodic coefficients? If so, in what ways are they similar to or different from those arising in dispersive nonlinear wave equations?
Initial computational experiments show complicated behavior. It seems that the homogenization approach may not accurately describe these solutions, because the oscillations that arise may vary over a scale that is about the same as that of the medium.
This project would involve:
- Numerical simulations of hyperbolic systems with periodic coefficients and a range of initial data
- Analysis of the numerical experiments
- Analysis and numerical simulations of the homogenized equations for the system
- Comparison of all of these results with the existing theory of dispersive shock waves
