On the solutions of the $CP^{1}$ model in $(2+1)$ dimensions

TL;DR

This paper applies group theory to analyze the $CP^{1}$ model in (2+1) dimensions, reducing its equations to simpler forms and exploring their solutions, revealing insights into the model's integrability and soliton properties.

Contribution

It introduces a group-theoretic reduction method for the $CP^{1}$ model's equations, providing explicit solutions and analyzing integrability aspects.

Findings

Many equations solved explicitly using elementary, elliptic, and Painlevé functions

Some reduced equations lack the Painlevé property, indicating non-integrability

The model exhibits stable solitons despite non-integrability

Abstract

We use the methods of group theory to reduce the equations of motion of the $CP^{1}$ model in (2+1) dimensions to sets of two coupled ordinary differential equations. We decouple and solve many of these equations in terms of elementary functions, elliptic functions and Painlev{\'e} transcendents. Some of the reduced equations do not have the Painlev{\'e} property thus indicating that the model is not integrable, while it still posesses many properties of integrable systems (such as stable ``numerical'' solitons).