A numerical study of singularity formation in an Euler model of shallow water wave propagation: The case of illposed Boussinesq equation

We numerically study an initial value problem for an Euler model of shallow water wave propagation with an initial periodic small data. This model, known as illposed or bad Boussinesq equation, is one of the fundamental models for dispersive water waves in the weakly nonlinear regime. This equation admits a Lax pair formulation and supports waves including solitons traveling in both the right and left directions. This equation, also known as “nonlinear string equation” also models other physical phenomena, including the propagation of ion-sound waves in a plasma, the dynamics of the anharmonic lattice in the Fermi-Pasta-Ulam problem, and generally nonlinear lattice waves in the continuum limit.

We compute solutions of this equation with an initial periodic small data using a spectral method in combination with the fourth-order Runge-Kutta method. The calculations indicate that the solution blows up in a very short time at a fixed location. We also find that the local profile of the solution at the singularity is parabolic as the singularity forms and the time of singularity formation is proportional to some positive power of the inverse of size of the disturbance. We find that these results are universal in the sense that these are independent of the size of initial small sinusoidal disturbance.