Stability and computation of interacting nonlinear waves
Talk by Prof. Dr. Wolf-Jürgen Beyn
Speaker invited by: Prof. Dr. Stefan A. Sauter
Date: 04.09.14 Time: 11.00 - 11.45 Room: Y27H28
We consider the numerical solution of semilinear time-dependent partial differential equations that support the propagation of nonlinear waves such as traveling, rotating or spiral waves. Depending on initial data, the solutions of such systems often show multiple patterns which are composed of several waves that either travel towards each other and collide (strong interaction) or miss each other and depart (weak interaction). Such multiple patterns look like linear superpositions of single waves which, however, cannot be true in a strict sense due the nonlinear character of the system. We suggest a numerical method that allows to handle several coordinate frames in which the single patterns can stabilize while still keeping their full nonlinear interaction.The procedure is adaptive in the sense that the position of the single frames is not prescribed a-priori but determined during computation. Several numerical experiments illustrate the method for multi-fronts in one space dimension and for multiple spinning solitons in two space dimensions. The approach extends an earlier method (called the freezing method), which allows to stabilize dynamic patterns in a single co-moving frame. For the case of weak interaction in one space dimension, we present a theoretical result which shows stability with asymptotic phase for the decomposition system.