  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.
