Algebro-Geometric Solutions of a ( 2 + 1 )-Dimensional Integrable Equation Associated with the Ablowitz-Kaup-Newell-Segur Soliton Hierarchy

Abstract

The ( $2+1$ )-dimensional Lax integrable equation is decomposed into solvable ordinary differential equations with the help of known ( $1+1$ )-dimensional soliton equations associated with the Ablowitz-Kaup-Newell-Segur soliton hierarchy. Then, based on the finite-order expansion of the Lax matrix, a hyperelliptic Riemann surface and Abel-Jacobi coordinates are introduced to straighten out the associated flows, from which the algebro-geometric solutions of the ( $2+1$ )-dimensional integrable equation are proposed by means of the Riemann $\theta$ functions.