Freezing Traveling and Rotating Waves in Second Order Evolution Equations
This work extends the freezing method to second-order evolution equations, providing a numerical approach for analyzing traveling and rotating waves, but the results are incremental and limited to specific damping conditions.
The paper implements the freezing method for second-order wave equations, converting them into partial differential algebraic equations to separate solution motion into a co-moving frame and a minimally varying profile. Numerical examples demonstrate feasibility for semilinear wave equations with sufficient damping, including traveling waves in 1D and rotating waves in 2D.
In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave.