Bifurcations of Solitary Waves
P.N. Lebedev Physical Institute 53 Leninsky Ave., Moscow, 119991, Russia
L.D. Landau Institute for Theoretical Physics 2 Kosygin Str., Moscow, 119334, Russia
CMLA, ENS Cachan, CNRS, PRES UniverSud 61 Av. President Wilson, F-94230 Cachan, France
Received June 25, 2008
The paper provides a brief review of the recent results devoted to bifurcations of solitary waves. The main attention is paid to the universality of soliton behavior and stability of solitons while approaching supercritical bifurcations. Near the transition point from supercritical to subcritical bifurcations, the stability of two families of solitons is studied in the framework of the generalized nonlinear Schrodinger equation. It is shown that one-dimensional solitons corresponding to the family of supercritical bifurcations are stable in the Lyapunov sense. The solitons from the subcritical bifurcation branch are unstable. The development of this instability results in the collapse of solitons. Near the time of collapse, the pulse amplitude and its width exhibit a self-similar behavior with a small asymmetry in the pulse tails due to self-steepening.
Mathematics Subject Classification 2000: 37K50, 70K50