Hamiltonian approach to the problem of wave collapse

Formation of singularities in the wave system within a finite time, or, in the other words, a wave collapse is one of the basic phenomena in nonlinear physics. The collapse plays an essential role in various fields of physics. Intense, localized, high-frequency waves can modify the medium in which they propagate via excitation of low frequency disturbances by their ponderomotive forces, for example. A common feature is that the collapse involves interaction between high and low-frequency waves coupled by nonlinear interactions that cause the refractive index of high-frequency waves to increase with increasing the wave intensity. The resulting tendency to focus into regions of high intensity can then lead to collapse. By using the Hamiltonian formalism for the analysis of stationary solutions, the author proves that the Hamiltonian is limited from below for the fixed plasmon number. This enables conclusions to be made about the possibility of existence of stable solutions