Algebro-geometric solution of the 2+1 dimensional Burgers equation with a discrete variable

The quasiperiodic solution of the 2+1 dimensional Burgers equation with a discrete variable is obtained through three steps: (a) decomposition into a symplectic map plus two finite-dimensional Hamiltonian systems; (b) straightening out of both the discrete and the continuous flows in the Jacobian variety; (c) inversion into the original variables. Inner relation with the modified Kadomtsev–Petviashvili equation is presented. The explicit theta function solutions for these two 2+1 integrable models are given.