Painleve solitons of AKNS system and irrational algebraic solitons of NLS equations

Abstract

A novel symmetry decomposition approach is introduced to derive the so-called ``Painlev\'e solitons'' of the Ablowitz-Kaup-Newell-Segur (AKNS) system. These Painlev\'e solitons propagate against a background governed by a Painlev\'e transcendent, establishing a fundamental generalization of the well-known elliptic solitons concept. We demonstrate that while elliptic solitons arise from the combination of translation invariance and square eigenfunction symmetry, a \textit{different} symmetry combination-scaling invariance, Galilean invariance, and square eigenfunction symmetry-generates ``Painlev\'e IV solitons'' for the AKNS system. This discovery represents a significant theoretical advance in integrable systems theory. By selecting special solutions of the Painlev\'e IV equation, we obtain explicit forms of several previously unknown classes of solutions for the AKNS system and the nonlinear Schr\"odinger (NLS) equation: irrational algebraic solitons, rational algebraic solitons, and parabolic cylindrical function solitons. These results dramatically expand the known solution landscape of one of the most important integrable models in mathematical physics, with broad implications for nonlinear wave phenomena across multiple physical disciplines including optics, Bose-Einstein condensates, and fluid dynamics.