Mathematics Colloquia and Seminars

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Description

In 1993 Rosenau and Hyman introduced a family of nonlinear dispersive equations with compactly supported soliton solutions, the so-called "compactons". These models are particularly interesting due to the local nature of their solutions, which serves as a caricature of a wide range of phenomena in nature. The non-smooth interfaces of the compactons and the strong nonlinearity of the equation present significant theoretical and numerical challenges. In this talk we will discuss two approaches for approximating compacton solutions. First, we will present a particle method that is based on an extension of the diffusion-velocity method of Degond and Mustieles to the dispersive framework. We prove the short-time existence and uniqueness of a solution to the resulting dispersion-velocity transport equation. In its present formulation, our particle method is suitable for equations with solutions that do not change their sign. This is a joint work with A. Chertock. We will also discuss a local discontinuous Galerkin method for approximating the solutions of more general nonlinear dispersive problems. For these methods we can prove certain stability results that correspond to conservation laws of the PDEs. This is a joint work with J. Yan and C.-W. Shu.