No abstract

Collapsing solutions in the Cauchy problem of the nonlinear Schrödinger equation i ∂tψ + ∇2ψ +‖ ψ‖pψ = 0 (x∈ℛd) are considered in the so-called critical case pd=4, where d is the spatial dimension. A stability theorem for radial collapse is presented which proves that the formation of the singularity remains ‘‘close’’ to the self-similar collapsing solution with a spatial profile given by the ground state solitary wave, provided the energy H{ψ}<0. An analogous result is given for the scalar Zakharov equations and a generalized Korteweg–de Vries equation ∂tu + (p + 1)up ∂xu + ∇2 ∂xu = 0.

It is discussed whether steady-state-stimulated Brillouin scattering (SBS) can exist in homogeneous plasmas. For the case when the damping of the ion-sound waves is more important than their convection, the general solutions of the associated perturbed equations are presented and the influence of various boundary conditions is discussed. Steady-state SBS in semi-infinite plasmas is nearly always unstable. The same is true for SBS with reflective boundary conditions, although the physical mechanism is different. By a Hopf bifurcation a (regular) nonstationary state (limit cycle) can appear with a possible consecutive transition to chaos.

No abstract

No abstract