Stability of mKdV?KdV Waves

Stability of travelling waves

An overview of various aspects related to the spectral and nonlinear stability of travelling-wave solutions to partial differential equations is given. The point and the essential spectrum of the linearization about a travelling wave are discussed as is the relation between these spectra, Fredholm properties, and the existence of exponential dichotomies (or Green’s functions) for the linear ope...

Stability of Travelling Waves

Linear stability of scroll waves.

A full linear stability of a straight scroll wave in an excitable medium is presented. The five eigenmode branches which correspond to deformation in the third dimension of the five main modes of two-dimensional (2D) spiral dynamics are found to play a dominant role. For untwisted scroll waves, modulations in the third dimension have stabilizing or destabilizing effects on the different modes d...

Stability of Relativistic Blast Waves

A spherical blast wave with relativistic velocity can be described by a similarity solution, that is used for theoretical models of gamma-ray bursts. We consider the linear stability of such a relativistic blast wave propagating into a medium with density gradient. The perturbation can also be expressed by a self-similar form. We show that the shock front is unstable in general, and we evaluate...

Nonlinear Stability of Semidiscrete Shock Waves

The orbital stability of possibly large semi-discrete shock waves is considered. These waves are traveling wave solutions of discrete in space and continuous in time systems of conservation laws, which constitute a class of Lattice Dynamical Systems. The underlying lattice xZis by nature not invariant by change of frame. Thus semi-discrete shock waves cannot really be transformed into stationar...