# On The Darboux B\"acklund Transformation of Optical Solitons with Resonant and Nonresonant Nonlinearity

## TL;DR

This paper explores the Darboux-Bäcklund transformation for optical solitons governed by coupled equations with resonant and nonresonant nonlinearities, enabling explicit multi-soliton solutions in nonlinear optics.

## Contribution

It extends the Darboux-Bäcklund transformation formalism to a coupled system modeling optical solitons with resonant and nonresonant nonlinearities, allowing for comprehensive N-soliton solutions.

## Key findings

- Derived explicit N-soliton solutions for the coupled system.
- Applied the Darboux-Bäcklund transformation separately to different parts of the Lax operator.
- Enhanced understanding of soliton dynamics in resonant and nonresonant nonlinear optical systems.

## Abstract

Solitons in nonlinear optics holds a special role both in theoretical and experimental studies. Several types of evolution equations are seen to govern different situation of physical relevance. One such is the existence of both resonant and nonresonant situation in optical fibre. The corresponding evolution equation was devised by Doktorov et. al., which consists of a forced NLS equation along with two other equations for population difference and polarization. Here, we have followed an earlier formulation of Neugebauer for Darboux-B\"acklund transformation for this coupled systems. This formalism has the advantage that one can write the N-soliton solution altogether.An important difference with the usual non-linear system is that all the field variables are not present in both part of Lax operator. So we are to apply the DT to both part separately.