This talk is devoted to solitons and wave collapses which can be considered as two alternative scenarios pertaining to the evolution of nonlinear wave systems describing by a certain class of dispersive PDEs (see, for instance, review [1]). For the former case, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude waves, a process absent in systems with finitely many degrees of freedom. The framework of the nonlinear Schrodinger equation, the ZK equation and the three-wave system is used to show how the boundedness of the Hamiltonian H, and hence the stability of the soliton minimizing H can be proved rigorously using the integral estimate method 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.
This work was supported by the Russian Science Foundation (project no. 14-22-00174).[-]