Inverse scattering method and vector higher order non-linear Schrödinger equation

Higher Order Asymptotics of the Modified Non-Linear Schrödinger Equation

Using the matrix Riemann-Hilbert factorisation approach for non-linear evolution systems which take the form of Lax-pair isospectral deformations, the higher order asymptotics as t→±∞ (x/t∼O(1)) of the solution to the Cauchy problem for the modified non-linear Schrödinger equation, i∂tu+ 1 2 ∂ x u+ |u|u+is∂x(|u|u) = 0, s ∈ R>0, which is a model for non-linear pulse propagation in optical fibres...

Periodic waves of a discrete higher order nonlinear Schrödinger equation ∗

The Hirota equation is a higher order extension of the nonlinear Schrödinger equation by incorporating third order dispersion and one form of self steepening effect. New periodic waves for the discrete Hirota equation are given in terms of elliptic functions. The continuum limit converges to the analogous result for the continuous Hirota equation, while the long wave limit of these discrete per...

Higher Order Unitary Integrators for the Schrödinger Equation

Inverse scattering approach to coupled higher order nonlinear Schrödinger equation and N -soliton solutions

A generalized inverse scattering method has been applied to the linear problem associated with the coupled higher order nonlinear schrödinger equation to obtain it’s N -soliton solution. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. It has been shown that the coupled system admits two different class of solutions, characterised by t...

ṕ Estimates for the Schrödinger Equation on the Line and Inverse Scattering for the Nonlinear Schrödinger Equation with a Potential ∗

In this paper I prove a L − L estimate for the solutions of the one–dimensional Schrödinger equation with a potential in Lγ where in the generic case γ > 3/2 and in the exceptional case (i.e. when there is a half–bound state of zero energy) γ > 5/2. I use this estimate to construct the scattering operator for the nonlinear Schrödinger equation with a potential. I prove moreover, that the low–en...