# Stepsize-adaptive integrators for dissipative solitons in cubic-quintic complex Ginzburg-Landau equations

## TL;DR

This survey evaluates adaptive exponential integrators for solving stiff cubic-quintic complex Ginzburg-Landau equations, focusing on accuracy near pulsating and exploding solitons with multiple time scales.

## Contribution

It introduces and compares stepsize-adaptive exponential integrator schemes, including new formulas and a comoving frame approach for improved performance.

## Key findings

- Adaptive integrators effectively handle multiple time scales.
- The comoving frame improves phase rotation resolution.
- Thorough numerical comparisons demonstrate method strengths.

## Abstract

This paper is a survey on exponential integrators to solve cubic-quintic complex Ginzburg-Landau equations and related stiff problems. In particular, we are interested in accurate computation near the pulsating and exploding soliton solutions where different time scales exist. We explore stepsize-adaptive variations of three types of exponential integrators: integrating factor (IF) methods, exponential Runge-Kutta (ERK) methods and split-step (SS) methods, and their embedded versions for computation and comparison. We present the details, derive formulas for completeness, and consider seven different stepsize-adaptive integrating schemes to solve the cubic-quintic complex Ginzburg-Landau equation. Moreover, we propose using a comoving frame to resolve fast phase rotation for better performance. We present thorough comparisons and experiments in the one- and two-dimensional cubic-quintic complex Ginzburg-Landau equations.