We consider the L-critical quintic focusing nonlinear Schrödinger equation (NLS) on R. It is well known that H solutions of the aforementioned equation blow-up in finite time. In higher dimensions, for H spherically symmetric blow-up solutions of the L-critical focusing NLS, there is a minimal amount of concentration of the L-norm (the mass of the ground state) at the origin. In this paper we p...

The Mathieu partial differential equation (PDE) is analyzed as a prototypical model for pattern formation due to parametric resonance. After averaging and scaling, it is shown to be a perturbed nonlinear Schrödinger equation (NLS). Adiabatic perturbation theory for solitons is applied to determine which solitons of the NLS survive the perturbation due to damping and parametric forcing. Numerica...

In a recent paper [4], we showed that the phenomenon of resonant tunneling, well known in linear scattering theory, takes place for fast solitons of the Nonlinear Schrödinger (NLS) equation in the presence of certain large potentials. Here, we illustrate numerically this situation for the one dimensional cubic NLS equation with two classes of potentials, namely the ‘box’ potential and a repulsi...

We develop a modulation theory model based on a Lagrangian formulation to investigate the evolution of dark and gray optical spatial solitary waves for both the defocusing nonlinear Schrödinger (NLS) equation and the nematicon equations describing nonlinear beams, nematicons, in self-defocusing nematic liquid crystals. Since it has an exact soliton solution, the defocusing NLS equation is used ...

We consider the inhomogeneous biharmonic nonlinear Schrödinger equation iut+?2u+?|x|?b|u|?u=0,where ?=±1 and ?, b>0. In subctritical case, we improve global well-posedness result obtained in Guzmán Pastor (2020) for dimensions N=5,6,7 Sobolev space H2(RN). The fundamental tools to establish our results are standard Strichartz estimates related linear problem Hardy-Littlewood inequality. Results...

Complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory. In this article, we explicitly construct such eigenfunctions for a large class of integrable equations including the KdV, NLS and mKdV hierarchies. We establish the striking result that the linearization operators of all equations in the...

The investigation of nonlocal reverse space, time, and space–time integrable equations becomes an interesting hot topic. In the paper Ma Zhu (2016), we have shown that focusing nonlinear Schrödinger (NLS) equation, defocusing NLS equation their discrete versions are, respectively gauge equivalent to a Heisenberg ferromagnet (HF)-like modified HF-like corresponding cases. this paper, further sho...

In the 2-d setting, given an H solution v(t) to the linear Schrödinger equation i∂tv + ∆v = 0, we prove the existence (but not uniqueness) of an H solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂tu+∆u− |u|p−1u = 0 for nonlinear powers 2 < p < 3 and the existence of an H solution u(t) to the defocusing Hartree equation i∂tu +∆u − (|x|−γ ? |u|)u = 0 for interaction powers 1...

The split-step Fourier method for solving numerically nonlinear Schrödinger equations (NLS) is considered as NLS with rapidly varying coefficients. This connection is exploited to justify the split-step approximation using an averaging technique. The averaging is done up to the second order and it is explained why (in this context) symmetric split-step produces a higher order scheme. The same a...