Exact static solutions of a two-dimensional discrete φ⁴ model

Abstract

For a two-dimensional scalar discrete φ⁴ model we obtain several exact static solutions in the form of the Jacobi elliptic functions (JEF) with arbitrary shift along the lattice. The Quispel-Roberts-Thompson-type quadratic maps are identified for the considered two-dimensional model by using a JEF solution. We also show that many of the static solutions can be constructed iteratively from these quadratic maps by starting from an admissible initial value. The kink solution, having the form of tanh, is numerically demonstrated to be generically stable.