Blow-up in Nonlinear Schroedinger Equations-I A General Review

Abstract

The general properties of a class of nonlinear Schroedinger equations: iut + p:∇∇u + f(|u|2)u = 0 are reviewed. Conditions for existence, uniqueness, and stability of solitary wave solutions are presented, along with conditions for blow-up and global existence for the Cauchy problem.

Stabilization of nonlinear excitations by disorder

“…This model has also a close relation to higher-dimensional models ͑for the continuum equation see Ref. 16 and Ref. 17 for the discrete model͒ and studying it allows us to predict the behavior of higher-dimensional systems.…”

“…In the opposite case the excitation will collapse. 16 From Eq. ͑41͒ we obtain that the nonlinear frequency ⌳ in the case of the quintic NLS equation has the form ⌳ϭ ͩ 336 ͱ 420 ͪ 4/3 3Ϫ4aN 2 /3 2 ͑ 1Ϫ4aN 2 /3 2 ͒ 4/3 .…”

Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations

“…Generalizations to higher-order equations such as the Swift-Hohenberg model or more complicated models without a Lyapunov function would also be desirable, but from the work of Collet and Eckmann [56], Eckmann and Procaccia [57], and Aranson et al [33] we know that the behavior can be quite rich indeed, and it is not clear whether the concepts we have developed will be useful in pursuing these questions. Finally, the corresponding problems in two and higher spatial dimensions [58] pose even greater challenges for the years ahead.…”

Stability conditions for one-dimensional optical solitons in cubic-quintic-septimal media

“…(7) and (8) had to be solved numerically. We recall that the quintic-only medium also gives rise to the instability [9]. In the present case, the addition of the positive seventh-order term leads to additional strong self-focusing, which may result in a supercritical collapse, as shown below.…”