On integrability of $O(3)$–model

A three-dimensional $O(3)$ model for a unit vector $\mathbf{n}(\mathbf{r})$ has numerous application in the field theory and in the physics of condensed matter. We prove that this model is integrable under some differential constraint, that is, under certain restrictions for the gradients of fields $\Theta(\mathbf{r})$, $\Phi(\mathbf{r})$ parametrizing the vector $\mathbf{n}(\mathbf{r})$). Under the presence of the differential constraint, the equations of the models are reduced to a one-dimensional sine-Gordon equation determining the dependence of the field $\Theta(\mathbf{r})$ on an auxiliary field $a(\mathbf{r})$ and to a system of two equations $(\nabla S)(\nabla S)=0$, $\Delta S =0$ for a complex-valued function $S(\mathbf{r})=a(\mathbf{r}) + \mathrm{i} \Phi(\mathbf{r})$. We show that the solution of this system provide all known before exact solutions of models, namely, two-dimensional magnetic instantons and three-dimensional structures of hedgehog type. We find an exact solution for the field $S(\mathbf{r})$ as an arbitrary implicity function of two variables, which immediately represents the solution for the fields $\Theta(\mathbf{r})$, $\Phi(\mathbf{r})$ in an implicit form. We show that the found in this way exact solution of the system for the field $S(\mathbf{r})$ leads one to exact solution of equations of $O(3)$–model in the form of an arbitrary implicit function of two variables.