A Numerical Study of Nonlinear Dispersive Wave Models with SpecTraVVave

In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. We describe a Python package which computes traveling-wave solutions of equations of the form \(u_t + [f(u)]_{x} + \mathcal{L} u_x = 0\), where \(\mathcal{L}\) is a self-adjoint operator, and \(f\) is a real-valued function with \(f(0) = 0\). We use a continuation method coupled with a spectral projection. For the Whitham equation, we obtain numerical evidence that the main bifurcation branch features three distinct points of interest: a turning point, a point of stability inversion, and a terminal point corresponding to a cusped wave. We also found that two solitary waves may interact in such a way that the smaller wave is annihilated. We also observed that bifurcation curves of periodic traveling-wave of the Benjamin solutions may cross and connect high up on the branch in the nonlinear regime.This research was supported by the Research Council of Norway on grant no. 213474/F20 and by the J C Kempe Memorial Fund on grant no. SMK-1238