# Traveling waves of nonlinear Schr\"{o}dinger equation including higher order dispersions

## TL;DR

This paper derives solitary and periodic wave solutions for a nonlinear Schr"{o}dinger equation with higher order dispersions, revealing conditions for bounded solutions and describing pulse trains in optical fibers.

## Contribution

It introduces a method to obtain explicit elliptic and solitary wave solutions for the NLS equation with third and fourth order dispersion effects.

## Key findings

- Solitary wave solutions with sech^2 shape are derived as limits of periodic solutions.
- Periodic solutions form a one-parameter family depending on an integration constant.
- Bounded solutions exist only within specific parameter domains.

## Abstract

The solitary wave solution and periodic solutions expressed in terms of elliptic Jacobi's functions are obtained for the nonlinear Schr\"{o}dinger equation governing the propagation of pulses in optical fibers including the effects of second, third and fourth order dispersion. The approach is based on the reduction of the generalized nonlinear Schr\"{o}dinger equation to an ordinary nonlinear differential equation. The periodic solutions obtained form one-parameter family which depend on an integration constant $p$. The solitary wave solution with ${\rm sech}^2$ shape is the limiting case of this family with $p=0$. The solutions obtained describe also a train of soliton-like pulses with ${\rm sech}^2$ shape. It is shown that the bounded solutions arise only for special domains of integration constant.