# Modulation of localized solutions in quadratic-cubic nonlinear Schr\"odinger equation with inhomogeneous coefficients

## TL;DR

This paper investigates exact localized solutions in a quadratic-cubic nonlinear Schrödinger equation with inhomogeneous coefficients, transforming it into an autonomous form and verifying stability through numerical simulations.

## Contribution

It introduces a method to find localized solutions in inhomogeneous nonlinear Schrödinger equations and analyzes their stability.

## Key findings

- Localized solutions are successfully derived using a specific ansatz.
- Numerical simulations confirm the stability of the modulated solutions.
- The approach enables understanding of inhomogeneous nonlinear wave dynamics.

## Abstract

We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schr\"odinger equation with inhomogeneous nonlinearities. Using a specific ansatz, we transform the nonautonomous nonlinear equation into an autonomous one, which engenders composed states corresponding to solutions localized in space, with an oscillating behavior in time. Direct numerical simulations are employed to verify the stability of the modulated solutions against small random perturbations.