Bifurcations of solitons and their stability

In spite of the huge progress in studies on solitary waves in the seventies and eighties of the XX century as well as their practical importance, the theory of solitons is far from being complete. Only in 1989, Longuet-Higgins in his numerical experiments discovered one-dimensional solitons for gravity–capillary waves in deep water. These solitons essentially differed from those in shallow water where the KDV equation could be used. Being localized, these solitons, unlike the KDV solitons, contain many oscillations in their shape. The number of oscillations was found to increase while approaching the maximal phase velocity for linear gravity–capillary waves and simultaneously the soliton amplitude was demonstrated to vanish. In fact, it was the first time ever that the bifurcation of solitons was observed. eview discusses bifurcations of solitons, both supercritical and subcritical, with applications to fluids and nonlinear optics as well. The main attention is paid to the universality of soliton behavior and stability of solitons while approaching supercritical bifurcations. For all physical models considered in this review, solitons are stationary points of the corresponding Hamiltonian for the fixed integrals of motion, i.e., the total momentum, number of quasi-particles, etc. Two approaches are used for the soliton stability analysis. The first method is based on the Lyapunov theory and another one is connected with the linear stability criterion of the Vakhitov–Kolokolov type. The Lyapunov stability proof is maintained by means of application of the integral majorized inequalities being sequences of the Sobolev embedding theorem. This allows one to demonstrate the boundedness of the Hamiltonians and show that solitons, as stationary points, which realize the minimum (or maximum) of the Hamiltonian, are stable in the Lyapunov sense. In the case of unstable solitons, the nonlinear stage of their instability near the bifurcation point results in the distraction of the solitons due to the wave collapse.