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

The coupled Sasa–Satsuma hierarchy: Trigonal curve and finite genus solutions

“…Based on the work of Matveev, Smirnov, Previato and McKean in the reduction theory of Riemann theta functions [30,31,36,39,32,2], Dickson and his collaborators proposed a unified framework which yields all algebro-geometric quasi-periodic solutions of the entire Boussinesq hierarchy related to the third-order operator [9,10]. Recently, this method was successfully generalized to solve soliton equations associated with 3 × 3 matrix spectral problems such as the modified Boussinesq, the Kaup-Kupershmidt, the coupled modified Korteweg-de Vries and the three-wave resonant interaction hierarchies based on the trigonal curves introduced by the characteristic polynomials of the Lax matrices [17,18,20,24]. In order to solve finite genus solutions of the coupled Sasa-Satsuma hierarchy, we shall introduce a trigonal curve with the aid of the Lax matrix.…”

Algebro-Geometric Solutions to a New Hierarchy of Soliton Equations

“…In [28], a unified framework was proposed which yields all algebro-geometric solutions of the entire Boussinesq hierarchy related to the third-order spectral problem. In [29,30], we give a systematic method to define the trigonal curve by the characteristic polynomial of Lax matrix and develop the framework to deal with soliton equations associated with the 3 × 3 matrix spectral problems, from which the algebro-geometric solutions of some entire hierarchies are obtained, namely, the modified Boussinesq hierarchy and the Kaup-Kupershmidt hierarchy.…”

“…The principle aim of the present paper is to construct algebro-geometric solutions of a new hierarchy of soliton equations associated with a 3 × 3 matrix spectral problem on the basis of the approaches used in [28][29][30] but different from spectral problems considered in [28][29][30]. An obvious difference is that there is a potential on the main diagonal, which is a big trouble for us to construct proper meromorphic functions.…”

A note on the quasi-periodic solutions of the modified Boussinesq hierarchy

“…In the previous paper [1], based on a trigonal curve related to the characteristic polynomial of Lax matrix associated with a 3 × 3 matrix spectral problem, we discussed algebro-geometric constructions of the modified Boussinesq hierarchy proposed by Fordy and Gibbons [2] and its quasi-periodic solutions. However, the theta function representations of solutions of every equation in the modified Boussinesq hierarchy are related to a Riccati equation, which is nonlinear.…”

“…However, the theta function representations of solutions of every equation in the modified Boussinesq hierarchy are related to a Riccati equation, which is nonlinear. The principal aim of the present paper is to improve the main result in [1], by which explicit quasi-periodic solutions of the entire modified Boussinesq hierarchy are deduced through introducing another meromorphic function and using its asymptotic properties and the algebrogeometric characters.…”