Asymptotics for a C1-version of the KdV Equation

Abstract

We consider KdV-type equations with nonlinearities u , κ ∈ (1, 5), and small dispersion ε. The first result consists in the conclusion that, in the leading term with respect to ε, the solitary waves in this model interact like KdV solitons. Next it turned out that there exists a very interesting scenario of instability in which the short-wave soliton remains stable whereas a small long-wave part, generated by perturbations of original equation, turns to be unstable, growing and destroying the leading term. At the same time, such perturbation can eliminate the collision of solitons. Numerical simulations confirming the results are also presented.