Algebro-geometric constructions of the modified Boussinesq flows and quasi-periodic solutions

The modified Boussinesq hierarchy associated with the 3×3 matrix spectral problem is derived with the help of Lenard recursion equations. Based on the characteristic polynomial of Lax matrix for the modified Boussinesq hierarchy, we introduce an algebraic curve Km−1Km−1 of arithmetic genus m−1m−1, from which we establish the associated Baker–Akhiezer function, meromorphic function and Dubrovin-type equations. The straightening out of various flows is exactly given through the Abel map. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic function, and in particular, that of solutions for the entire modified Boussinesq hierarchy.

► The modified Boussinesq hierarchy is derived via Lenard recursion equations. ► The associated Baker–Akhiezer function and meromorphic function are established. ► The straightening out of modified Boussinesq flows is given through the Abel map. ► Quasi-periodic solutions for the entire modified Boussinesq hierarchy are obtained.