Principal chiral models on non-semisimple groups (Q2766277)

Integrable models in two dimensions are important theoretical laboratories for investigating possible phenomena of nonlinear theories in higher dimensions. The main topic of the paper is classification of models on the two- and three-dimensional nonsemisimple groups that admit Lax formulation -- i.e., that can be written using a pair -- called Lax pair -- of differential operators of certain type, called Lax operators. The possibility of such formulation imposes the condition of constance on the coefficients of the field differential equation. If the group is Abelian, the problem is extremely simplified, and in certain coordinates the free model is obtained. For the two-dimensional non-Abelian case, the equations of motion are obtained, and so are the Lax operators, but only for a specific metric the Lax formulation is equivalent to the equations of motion. The classification of three-dimensional solvable Lie groups is made, and in each case, the class of models admitting Lax formulation is obtained.