Hasimoto [10] introduced the map from vortex filament solutions of Euler’s equations for incompressible fluids in the local induction approximation to solutions of the nonlinear Schrödinger equation and he showed vortex filament equation is equivalent nonlinear Schrödinger equation. After this discovering of Hasimoto, several authors [1, 5, 9, 12, 13, 14, 15, 17, 20, 21, 22, 23, 24] studied the...

We consider a cubic nonlinear wave equation with highly oscillating cubic coefficient and wave packet initial data. Using a regularization step of the initial data, we give a low regularity justification of the Nonlinear Schrödinger equation as the envelope equation.

It is natural to apply multiple-length-scale methods to the study of optical-fiber transmission because the key length scales span 13 orders of magnitude and cluster in three main groups. At the lowest scale, corresponding to micrometers, the full set of Maxwell’s equations should be used. At the intermediate scale, corresponding to the range from one centimeter to tens of meters, the coupled n...

The stationary nonlinear Schrödinger equation, or Gross–Pitaevskii equation, is studied for the cases of a single delta potential and a delta–shell potential. These model systems allow analytical solutions, and thus provide useful insight into the features of stationary bound, scattering and resonance states of the nonlinear Schrödinger equation. For the single delta potential, the influence of...

We discuss, with illustrations, some physical significances of fifth-order nonlinear susceptibility for pulse dynamics in monomode optical fibres. The amplitude dynamic governing equation is the cubic-quintic nonlinear Schrödinger equation, (CQNLSE), which has soliton properties similar to the cubic nonlinear Schrödinger equation, (CNLSE), based on solutions by a variational method. Some differ...

For the semi-classical limit of the cubic, defocusing nonlinear Schrödinger equation with an external potential, we explain the notion of criticality before a caustic is formed. In the sub-critical and critical cases, we justify the WKB approximation. In the super-critical case, the WKB analysis provides a new phenomenon for the (classical) cubic, defocusing nonlinear Schrödinger equation, whic...

A model describing wave propagation in optically modulated waveguide arrays is proposed. In the weakly guided regime, a two-dimensional semidiscrete nonlinear Schrödinger equation with the addition of a bulk diffraction term and an external "optical trap" is derived from first principles, i.e., Maxwell equations. When the nonlinearity is of the defocusing type, a family of unstaggered localized...

We consider an extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms with variable coefficients. We demonstrate its integrability, provide its Lax pair, and, applying the Darboux transfromation, present its first and second order soliton solutions. The equation and its solutions have two free parameters. Setting one of these parameters to zero...

Using the Fermi Golden Rule analysis developed in [CM], we prove asymptotic stability of asymmetric nonlinear bound states bifurcating from linear bound states for a quintic nonlinear Schrödinger operator with symmetric potential. This goes in the direction of proving that the approximate periodic solutions for the cubic Nonlinear Schrödinger Equation (NLSE) with symmetric potential in [MW] do ...

A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapping to a well-known ordinary differential equation, as for example Airy’s equation. In this paper we generalize the mapping procedure to the nonlinear Schrödinger equation or Gross-Pitaevskii equation describing the macroscopic wave function of a Bose-Einstein condensate. The nonlinear Schrödinger e...