Lmn Equipe

Research project

Integrability of non-strictly hyperbolic conservation laws in dispersive hydrodynamics

Project summary

The project is concerned with extending of the classical concept of integrability of hydrodynamic type systems to non-strictly hyperbolic systems with applications to the description of dispersive shock waves (DSWs). DSWs are dynamic nonlinear wave structures replacing classical shocks in media with dispersion-dominated mechanisms of the gradient catastrophe regularisation. DSWs have recently attracted much interest in fluid dynamics, nonlinear optics and condensed matter physics. Their mathematical description is achieved in the framework of the Whitham modulation theory. The existing, classical theory of DSWs involves analysis of the Whitham equations which are strictly hyperbolic and genuinely nonlinear. However, there is an increasing evidence of importance of DSWs in nonlinear dispersive systems described by the so- called modified equations, involving higher order nonlinearities. The Whitham modulation systems associated with such modified dispersive equations are not strictly hyperbolic, and their integrability properties have not been explored so far. The development of the effective methods for integration of non-strictly hyperbolic Whitham systems is the main objective of the proposed project. The project involves an international team of researchers in different fields (mathematical physics, integrable systems, fluid dynamics), which have complementary expertise crucial for the project’s success.