Straightening out of the flows for the Hu hierarchy and its algebro-geometric solutions

Abstract

a b s t r a c tWith the help of the Lax matrix, the Hu hierarchy is decomposed into two systems of solvable ordinary differential equations. Based on the theory of algebraic curves, the Abel-Jacobi coordinates are introduced to straighten out all the flows related to the entire hierarchy. Algebro-geometric solutions for the entire Hu hierarchy are obtained by using asymptotic expansion of the meromorphic function and its Riemann theta function representation.

Algebro‐geometric constructions of the Wadati‐Konno‐Ichikawa flows and applications

“…It will be convenient to introduce the notion of a degree [8,17,18], deg. /, to effectively distinguish between homogeneous and nonhomogeneous quantities.…”

“…Seeking algebro‐geometric solutions of soliton equations is an important and interesting subject in recent years, because algebro‐geometric solutions describe the quasi‐periodic behavior of nonlinear phenomenon and reveal inherent structure mechanism of solutions. In a series of papers , algebro‐geometric solutions for many soliton equations associated with 2 × 2 or 3 × 3 matrix spectral problems have been obtained, such as the Korteweg‐de Vries KdV, nonlinear Schrödinger, modified KdV, discrete modified KdV, Boussinesq, sine‐Gordon, Toda lattice, and Camassa‐Holm equations. The main aim of this paper is to construct the explicit theta function representations of solutions for the entire Wadati–Konno–Ichikawa (WKI) hierarchy related to WKI equations which were first discovered by Wadati, Konno, and Ichikawa in 1979 .…”

Finite genus solutions for Geng hierarchy

“…In this paper we will concentrate primarily on constructing the finite genus solutions of the entire Geng hierarchy related to (1.2) based on the approaches in Refs. [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The finite genus solutions Co-published by Atlantis Press and Taylor & Francis…”

Explicit quasi-periodic solutions of the Kaup–Newell hierarchy

“…In this paper, we construct explicit quasi-periodic solutions of the entire Kaup-Newell hierarchy based on the approaches in Refs. [5,[9][10][11][12]31], especially, explicit quasi-periodic solutions of the coupled derivative nonlinear Schrödinger equations (1.1) are as follows…”