Elliptic Solutions of Boussinesq Type Lattice Equations and the Elliptic $$N{\textrm{th}}$$ Root of Unity

AbstractWe establish an infinite family of solutions in terms of elliptic functions of the lattice Boussinesq systems by setting up a direct linearisation scheme, which provides the solution structure for those equations in the elliptic case. The latter, which contains as main structural element a Cauchy kernel on the torus, is obtained from a dimensional reduction of the elliptic direct linearisation scheme of the lattice Kadomtsev–Petviashvili equation, which requires the introduction of a novel technical concept, namely the ‘elliptic cube root of unity’. Thus, in order to implement the reduction we define, more generally, the notion of elliptic$$N{\textrm{th}}$$ N th root of unity, and discuss some of its properties in connection with a special class of elliptic addition formulae. As a particular concrete application we present the class of elliptic multi-soliton solutions of the lattice Boussinesq systems.