Solitons and collapses: two evolution scenarios of nonlinear wave systems

Two alternative scenarios pertaining to the evolution of nonlinear wave systems are considered: solitons and wave collapses. For the former, it suffices that the Hamiltonian be boundedfrombelow(orabove),andthenthesolitonrealizingits minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude waves, a pro- cess absent in systems with finitely many degrees of freedom. The framework of the nonlinear Schrequation and the three-wave system is used to show how the boundedness of the Hamiltonian—andhencethestabilityofthesolitonminimizing it—can be proved rigorously using the integral estimate meth- od based on the Sobolev embedding theorems. Wave systems with the Hamiltonians unbounded from below must evolve to a collapse, which can be considered as the fall of a particle in an unbounded potential. The radiation of small-amplitude waves promotes collapse in this case.