Orbital stability of negative solitary waves (Q1038138)

The authors study the stability of the negative solitary-wave solutions of the generalized regularized long-wave equation \[ u_t+u_x+(u^p)_x-u_{xxt}=0, \] where \(p\geq2\) is a positive integer. This equation admits solitary-wave solutions of the form \(u(x,t)=\Phi_c(x-ct)\). If \(p\geq2\) is even and \(c<0\), one obtains negative solitary waves propagating to the left. These waves can be unstable even if \(p<5\). The main contribution of the paper is a numerical study of the stability of these negative solitary waves for both subcritical and supercritical \(p\) by means of a spectral discretization.