The effect of dissipation on solutions of the generalized Korteweg—de Vries equation

Abstract

Recent numerical simulations of the generalized Korteweg—de Vries equation u t + u p u x + u xxx = 0 indicate that for p⩾4, smooth solutions of the initial-value problem may form singularities in finite time. It is the purpose of this paper to ascertain what effect dissipation has on the instability of solitary waves and the associated blow-up phenomena that are related to this singularity formation. Two different dissipative mechnisms are appended to ( ∗) in our study, a Burgers-type term − δu xx and a simple, zeroth-order term δu. For both of these types of dissipation, it is found that for small values of the positive parameters δ and σ, solutions continue to form singularities in finite time. However, for given initial data u 0, it appears there are critical values δ c and σ c such that if δ > δ c or σ > σ c , the solution associated with u 0 is globally defined and decays as t → +∞. In the case wherein the singularity formation is averted by larger values of δ or σ, a simple analysis shows the solution to approach its mean value exponentially fast. Theoretical analysis in the case when u 0 is a perturbed solitary wave leads to a conjecture about how δ c and σ c depend on the amplitude and spread of u 0. The numerical simulations indicate the analysis to be surprisingly sharp in predicting the qualitative dependence of δ c and σ c on u 0.