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

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).