Some algebro-geometric solutions for the coupled modified Kadomtsev-Petviashvili equations arising from the Neumann type systems

Abstract

A treatment is described for getting some algebro-geometric solutions of the coupled modified Kadomtsev-Petviashvili (cmKP) equations and a hierarchy of 1+1 dimensional integrable nonlinear evolution equations (INLEEs) by using the Neumann type systems through three steps: (i) according to the nonlinearization of Lax pair, the cmKP equations and the 1+1 dimensional INLEEs are reduced to a family of compatible Neumann type systems on a symplectic submanifold, whose involutive solutions give rise to the finite parametric solutions of the INLEEs in both 2+1 and 1+1 dimensions; (ii) from the holomorphic differentials and the Abel map on a hyperelliptic curve of Riemman surface, the Abel-Jacobi variables are introduced to straighten out the Neumann type flows, the 1+1 and 2+1 dimensional flows giving the Abel-Jacobi solutions; (iii) based on the Riemann theorem and the trace formulas, the Jacobi inversion is applied to the straightened flows for getting some new algebro-geometric solutions of the cmKP equations and the 1+1 dimensional integrable hierarchy.