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

TL;DR

This paper introduces a new symmetry-based method to derive Painlevé solitons of the AKNS system, revealing novel solution classes like irrational algebraic solitons for NLS equations, with broad implications in nonlinear physics.

Contribution

It presents a novel symmetry decomposition approach that generalizes elliptic solitons to Painlevé solitons and discovers new solution classes for integrable systems.

Findings

Derived Painlevé solitons propagating against Painlevé transcendent backgrounds.

Identified explicit forms of irrational algebraic, rational algebraic, and parabolic cylindrical solitons.

Expanded the solution landscape of the AKNS and 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.