Asymptotic behavior in time of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion

TL;DR

This paper studies the long-term behavior of solutions to a nonlinear Schrödinger equation with higher order anisotropic dispersion, showing solutions tend to linear solutions over time.

Contribution

It proves the existence of solutions to the 4NLS that asymptotically behave like solutions to the linearized equation, advancing understanding of ultrashort laser pulse propagation.

Findings

Solutions to 4NLS exist and scatter to linear solutions.

Analysis of anisotropic fourth-order dispersion effects.

Mathematical proof of asymptotic behavior.

Abstract

We consider the asymptotic behavior in time of solutions to the nonlinear Schr"odinger equation with fourth order anisotropic dispersion (4NLS) which describes the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to (4NLS) which scatters to a solution of the linearized equation of (4NLS).