We study the focusing NLS equation in $\mathbb{R}^N$ mass-supercritical and energy-subcritical (or intercritical ) regime, with $H^1$ data at mass-energy threshold $\mathcal{ME}(u\_0)=\mathcal{ME}(Q)$, where $Q$ is ground state. Previously, Duyckaerts–Merle studied behavior of solutions $H^1$-critical case, dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In Duycka...

In this note we present a new derivation of Bäcklund transformations for the nonlinear Schroedinger equation. We discuss conserved quantities related with transformation and its connection inverse scattering method. Besides, construct quantum analog defined by Baxter’s Q-operator.

A generalized nonlinear Schrödinger equation with polynomial Kerr nonlinearity and non-Kerr terms of an arbitrarily higher order is investigated. This model can be applied to the femtosecond pulse propagation in highly-nonlinear optical media. We introduce a new chirping ansatz given as an expansion in powers of intensity of the light pulse and obtain both linear and nonlinear chirp contributio...

Motivated by recent experimental results, we demonstrate that the ubiquitous pulse propagation equation based on a single generalized nonlinear Schrödinger equation is incomplete and inadequate to explain the formation of the so called negative-frequency resonant radiation emitted by optical solitons. The origin of this deficiency is due to the absence of a peculiar nonlinear coupling between t...

Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schrödinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presen...

In the present short communication, we revisit nonlinearity management of the timeperiodic nonlinear Schrödinger equation and the related averaging procedure. By means of rigorous estimates, we show that the averaged nonlinear Schrödinger equation does not blow up in the higher dimensional case so long as the corresponding solution remains smooth. In particular, we show that the H norm remains ...

The numerical approximation of the solution of the time-dependent Schrödinger equation arising in ultrafast laser dynamics is discussed. The linear Schrödinger equation is reduced to a computationally tractable, lower dimensional system of nonlinear partial differential equations by the multi-configuration time-dependent Hartree-Fock method. This method serves to approximate the original wave f...

A system of semi-discrete coupled nonlinear Schrödinger equations is studied. To show the complete integrability of the model with multiple components, we extend the discrete version of the inverse scattering method for the single-component discrete nonlinear Schrödinger equation proposed by Ablowitz and Ladik. By means of the extension, the initial-value problem of the model is solved. Further...