Algebro-geometric integration of the Q1 lattice equation via nonlinear integrable symplectic maps

Abstract

The Q1 lattice equation, a member in the Adler-Bobenko-Suris list of 3D consistent lattices, is investigated. By using the multidimensional consistency, a novel Lax pair for Q1 equation is given, which can be nonlinearized to produce integrable symplectic maps. Consequently, a Riemann theta function expression for the discrete potential is derived with the help of the Baker-Akhiezer functions. This expression leads to the algebro-geometric integration of the Q1 lattice equation, based on the commutativity of discrete phase flows generated from the iteration of integrable symplectic maps.

Algebro-geometric solutions to the lattice potential modified Kadomtsev–Petviashvili equation

“…The approach has proved effective and this paper added one more important successful example. Reviewing the series of work [9,10,12,13,[60][61][62] and the present paper, there are several related problems that are interesting and remain open. Let us raise them below.…”

“…Based on a series of work [9,10,12,13,[60][61][62], in section 1 we have summarized a framework of the approach for constructing algebro-geometric solutions to multidimensionally consistent systems. The approach has proved effective and this paper added one more important successful example.…”

“…This implies the Poisson commutativity {F(λ), F(μ)} = 0, where F(λ) = det L(λ) (cf [60]) and the Poisson bracket is defined as…”

“…For fully discrete equations, such as the ABS equations, algebro-geometry solutions have been derived from several equations in a series of papers [9,10,12,13,[60][61][62] by Cao, Xu and their collaborators, which establish an algebro-geometric approach to periodic solutions for discrete equations that are multi-dimensionally consistent. This approach can be sketched as the following three steps.…”

“…where the second one is known as the Kaup-Newell (KN) spectral problem. In [60], to solve Q 1 (δ), spectral problem…”