Decomposition of a 2 + 1-dimensional Volterra type lattice and its quasi-periodic solutions

A 2 + 1-dimensional Volttera type lattice is proposed. Resorting to the nonlinearization of Lax pair, the 2 + 1-dimensional Volttera type lattice is decomposed into the known 1+1-dimensional differential-difference equations. The relation between a new 2 + 1-dimensional differential-difference equation, certain 1+1-dimensional continuous evolution equations and the known 1+1-dimensional differential-difference equations is discussed. Based on finite-order expansion of the Lax matrix, we introduce elliptic coordinates, from which the two 2 + 1-dimensional differential-difference equations are separated into solvable ordinary differential equations. The evolution of various flows is explicitly given through the Abel–Jacobi coordinates. Quasi-periodic solutions for the two 2 + 1-dimensional differential-difference equations are obtained.