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In this paper we construct a family of Hamilton-Jacobi separable non-linear$\mathbb{S}^1\times\mathbb{S}^1$ Sigma models for which the kink variety can beanalytically identified and for which the linear stability of the emergingkinks is ensured. Furthermore, a model with only one vacuum point is found,where all kinks are forced to be non-topological. The non-simply connectednessof the torus guarantees the global stability of all the non-topological kinksin these models.

Topological and non-topological kink families in non-linear $(\mathbb{S}^1\times \mathbb{S}^1)$-Sigma models