A numerical study of singularity formation in an Euler model of shallow water wave propagation: the case of an ill-posed Boussinesq equation

Abstract We numerically study the issue of finite-time existence of solutions associated with the initial value problem (IVP) of an ill-posed Boussinesq equation subject to small-amplitude sinusoidal initial disturbances over a 2π periodic domain. A spectral method in combination with the fourth-order Runge–Kutta method is used to perform numerical experiments using several time steps, several precision arithmetic calculations and filters with several cut-off levels. The calculations indicate that the solution blows up in a very short time tc, the time of singularity formation. The value of tc changes very little with the filter cut-off level. To validate our numerical results, we also analyse the evolution of Fourier coefficients through a perturbation analysis taking into account that the perturbation parameter ϵ (amplitude of the sinusoidal initial disturbance) is small. The location of the singularity where the solution blows up is found to be 3π/2, independent of the parameter ϵ. It also appears that the profile of the solution at the location of the singularity is parabolic, of the form η(x,t;ϵ)=η0(t;ϵ)+c1(t;ϵ)(x−3π/2)2 as the singularity forms, where tc∼1/ϵp,p∼0.28. The functions η0(t;ϵ) and c1(t;ϵ) are determined numerically and discussed. From numerical calculations, we find that c1(t;ϵ)∼(tc−t)−δ(ϵ) as t→tc where ϵ dependent δ is the rate of singularity formation. We obtain the following estimates for δ: δ=2.46 for ϵ=0.10 and δ=2.67 for ϵ=0.025.