The trigonal curve and the integration of the Hirota‐Satsuma hierarchy

Abstract

By introducing a trigonal curve  m−1 , which constructed from the characteristic polynomial of Lax matrix for the Hirota-Satsuma hierarchy, we present the associated Baker-Akhiezer function and algebraic functions carrying the data of the divisor. Then the Hirota-Satsuma equations are decomposed into the system of Dubrovin-type ordinary differential equations. Based on the theory of algebraic geometry, we obtain the explicit Riemann theta function representations of the Baker-Akhiezer function, the meromorphic function, and solutions for the Hirota-Satsuma hierarchy.−∞ f (x ′ )dx ′ under the decaying condition at infinity. As is well known, it is very important to search for quasiperiodic solutions for soliton equations, which can reveal the internal structure of the solutions and describe the quasiperiodic behavior of nonlinear phenomena in physical and engineering sciences. In a series of literatures, 11-20 several systematic methods were developed from which finite genus solutions for a lot of soliton equations associated with 2 × 2 matrix spectral problems have been obtained, such as the KdV, nonlinear Schrödinger, mKdV, and Toda lattice equations. However, the study of soliton equations associated with 3 × 3 matrix spectral problems is very few, which is also much more difficult and complicated. Fortunately, some progress has been made. Underlying a unified framework for solving the soliton hierarchy associated with the third-order differential operator, some famous equations have been discussed, such as the Boussinesq equation, 21-23 the Kaup-Kaupershmidt hierarchy, 24 the 3-wave resonant interaction hierarchy, 25 and others. [26][27][28] The outline of the present paper is as follows. In Section 2, we briefly recall the construction for the HS hierarchy associated with the 3 × 3 matrix spectral problem. In Section 3, an algebraic curve  m−1 of arithmetic genus m − 1 is introduced with the help of the characteristic polynomial of Lax matrix for the stationary HS equations, from which the stationary Baker-Akhiezer function and the associated meromorphic function on the curve are given. Next, the stationary HS equations are decomposed into the system of Dubrovin-type ordinary differential equations. Then, we present the explicit theta function representations of the stationary Baker-Akhiezer function, the meromorphic function, and the potentials u and v for the entire stationary HS hierarchy. Section 4 then extends the analyses of Section 3 to the time-dependent case.

Action-angle variables for the Lie–Poisson Hamiltonian systems associated with the Hirota–Satsuma modified Boussinesq equation

“…1 are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations Dai and Geng [5]. The explicit Riemann theta function representations of solutions for the Hirota-Satsuma modified Boussinesq hierarchy were studied in He et al [6].…”