Integrable Submodels of Nonlinear $\sigma$-models and Their Generalization

TL;DR

This thesis explores and generalizes integrable submodels of nonlinear sigma models, including the CP^1 and Grassmann models, demonstrating their integrability through conserved currents and symmetries, and extending them to higher-order equations using Bell polynomials.

Contribution

It introduces a unified framework for integrable submodels of nonlinear sigma models, generalizes them to higher-order equations, and reveals their deep connection with Bell polynomials.

Findings

Constructed infinite conserved currents for submodels.

Identified symmetries enabling exact solutions.

Established integrability of generalized higher-order submodels.

Abstract

In this thesis, we investigate various integral submodels and generalize them. In part I, we study the submodel of the nonlinear $\mathbf{C}P^1$-model and the related submodels in $(1+2)$ dimensions. In part II, we construct integrable submodels of the nonlinear Grassmann models in any dimension. We call them the Grassmann submodels. To show that our submodels are integrable, we construct an infinite number of conserved currents in two ways. One is that we make full use of the Noether currents of the nonlinear Grassmann models. The other is that we use a method of multiplier. Next we investigate symmetries of the Grassmann submodel. By using the symmetries, we can construct a wide class of exact solutions for our submodels. In part III, keeping some properties of our submodels, we generalize our submodels to higher-order equations. First we prepare the Bell polynomials and the generalized Bell polynomials which play the most important roles in our theory of generalized submodels. Next we generalize the $\mathbf{C}P^1$-submodel to higher-order equations. Lastly we generalize the Grassmann submodel to higher-order equations. By using the generalized Bell polynomials, we can show that the generalized Grassmann submodels are also integrable. As a result, we obtain a hierarchy of systems of integrable equations in any dimension which includes Grassmann submodels. These results lead to the conclusion that the integrable structures of our generalized submodels are closely related to some fundamental properties of the Bell polynomials.