Blow-up phenomena in a one-dimensional, bidirectional model equation for small-amplitude waves

Abstract

The purpose of this paper is to determine numerically the existence of blow-up in finite time of a one-dimensional, bidirectional equation that has been proposed for the study of wave propagation in shallow waters and the deformation of viscoelastic materials, subject to smooth initial conditions.

An implicit, time-linearized, finite difference method is used to solve the nonlinear wave equation in a truncated domain subject to homogeneous Dirichlet boundary conditions, and smooth initial conditions of both the Gaussian type and those corresponding to the exact solution of the inviscid, generalized regularized-long wave equation (RLW).

A large set of numerical experiments on the effects of the linear and nonlinear drift, dispersion and viscosity coefficients and relaxation time, and the amplitude and width of the Gaussian conditions and the amplitude of those corresponding to the initial conditions of the inviscid RLW, on blow-up are reported. The results of these experiments indicate that, for the same initial mass, if there exists blow-up, it occurs at longer times for Gaussian conditions due to the larger width of these conditions and its corresponding smaller initial kinetic energy. Blow-up has also been found to depend on the power of the nonlinear drift, and the viscosity coefficient and relaxation time. Comparisons with other previously published results that made used of nonsmooth or even discontinuous initial conditions indicate that the blow-up time decreases as the smoothness of the initial conditions is decreased, in accord with the fact that the stretching energy increases as the gradient of the solution increases.

The blow-up of the solution to a one-dimensional, bidirectional, nonlinear wave equation that models wave propagation in shallow waters and the deformation of viscoelastic materials, subject to two types of smooth initial conditions has been studied as a function of the parameters that characterize the linear and nonlinear drift, dispersion, viscosity and relaxation time, and the amplitude and width of the initial Gaussian conditions. Two conditions based on the growth of the local solution and the positiveness of a time-dependent function that depends on the kinetic and stretching energies have been used to determine the blow-up time.