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.