Solitary Waves in Nonlinear Beam Equations: Stability, Fission and Fusion

We continue work by the second author and co-workers on solitary wave solutions of nonlinear beam equations and their stability and interaction properties. The equations are partial differential equations that are fourth-order in space and second-order in time. First, we highlight similarities between the intricate structure of solitary wave solutions for two different nonlinearities; a piecewise-linear term versus an exponential approximation to this nonlinearity which was shown in earlier work to possess remarkably stable solitary waves. Second, we compare two different numerical methods for solving the time dependent problem. One uses a fixed grid discretization and the other a moving mesh method. We use these methods to shed light on the nonlinear dynamics of the solitary waves. Early work has reported how even quite complex solitary waves appear stable, and that stable waves appear to interact like solitons. Here we show two further effects. The first effect is that large complex waves can, as a result of roundoff error, spontaneously decompose into two simpler waves, a process we call fission. The second is the fusion of two stable waves into another plus a small amount of radiation.