Journal ArticleDOI
On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations
Reads0
Chats0
TLDR
In this article, conditions on φ and F were given so that, for solutions with nonpositive energy, the following obtains: there exists a finite time T, estimable from above, such that ∥ grad u(t)∥ L 2 (n ) →+∞ as t→T −.Abstract:
Solutions to the Cauchy problem for the equation iu t =Δu+F(|u| 2 )u (x∈ n , t>0), u(x,0)=φ(x), are considered. Conditions on φ and F are given so that, for solutions with nonpositive energy, the following obtains: There exists a finite time T, estimable from above, such that ∥ grad u(t)∥ L 2 ( n ) →+∞ as t→T − . It is also shown that other L q ‐norms of a solution (including q=∞) blow up in finite time.read more
Citations
More filters
Journal ArticleDOI
Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle
Journal ArticleDOI
Nonlinear Schrödinger equations and sharp interpolation estimates
TL;DR: In this paper, a sharp sufficient condition for global existence for the nonlinear Schrodinger equation is obtained for the case σ = 2/N. This condition is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of Nirenberg and Gagliardo.
Journal ArticleDOI
Orbital stability of standing waves for some nonlinear Schrödinger equations
TL;DR: In this article, a general method for proving the orbital stability of standing waves in nonlinear Schrodinger equations arising in laser beams has been presented, for the special case of time-dependent Hartree equations.
Journal ArticleDOI
Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case
Carlos E. Kenig,Frank Merle +1 more
TL;DR: In this paper, it was shown that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller compared to that of a standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5.
Journal ArticleDOI
Ultrashort filaments of light in weakly ionized, optically transparent media
TL;DR: In this article, the authors present the landmarks of the 10-odd-year progress in this field, focusing on the theoretical modeling of the propagation equations, whose physical ingredients are discussed from numerical simulations.
References
More filters
Journal ArticleDOI
Instability and nonexistence of global solutions to nonlinear wave equations of the form _{}=-+ℱ()
Journal ArticleDOI
Saddle points and instability of nonlinear hyperbolic equations
TL;DR: In this article, conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time were investigated and sharp results were derived for certain classes of nonlinearities.
Journal ArticleDOI
Development of Singularities of Solutions of Nonlinear Hyperbolic Partial Differential Equations
TL;DR: In this paper, the authors give a simpler derivation of the same result based on an estimate given some years ago by the author, using the hodograph method. But the proof of Zabusky's result is based on the assumption that the anomaly in the partition of energy among the various modes does not exist.