Group invariant decompositions, elliptic and Painlevé solitons of the AKNS systems and NLS equations

Decomposition algorithms are powerful methods for solving nonlinear mathematical physics problems. In this paper, we propose a novel decomposition method based on group-invariant space-time rather than the natural space-time. Taking the Ablowitz-Kaup-Newell-Segur (AKNS) system as a simple illustrative example, we apply the method to find interacting solutions among different types of nonlinear waves, including solitary waves, periodic cnoidal waves, Painlevé waves, rational/irrational algebraic waves, error-function/Kummer waves, and so on.