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Section: New Results

Singular solutions of dispersive systems

Participants : S. Gavrilyuk, B. Nkonga, K-M Shyue, L. Truskinovsky.

We study a dispersive regularization of p-system. The governing equations are the Euler- Lagrange equations for a Lagrangian depending not only on the velocity and density, but also on the first material derivative of density. Such regularization arises, in particular, in the modeling of waves in solids, in bubbly fluids as well as in the theory of water waves. We show that such terms are not always regularizing. The solution can develop shocks even in the presence of dispersive terms. In particular, we construct such a shock solution that con- nects a constant state to a periodic wave train. The corresponding shock speed coincides with the velocity of the wave train. The generalized Rankine-Hugoniot relations (jump relations) are also obtained. The numerical evidence of the existence of such shocks is demonstrated in the case of the Serre-Green-Naghdi equations describing long surface water waves. In particular, it has been shown that such waves can dynamically be formed