Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

Abstract

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve K m − 2 of arithmetic genus m − 2 , from which the corresponding Baker-Akhiezer function and meromorphic functions on K m − 2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

A (2 + 1)-Dimensional Integrable Breaking Soliton Equation and Its Algebro-Geometric Solutions

“…Algebro-geometric solutions can not only reveal the intrinsic structure of solutions, but also characterize the quasi-periodic behavior of nonlinear phenomena. Various approaches have been developed to obtain algebrogeometric solutions of soliton equations, such as the algebro-geometric approach [1], the nonlinearization of Lax pairs [2], the finite-order expansion of the Lax matrix [3], and so on [4][5][6][7][8][9][10][11][12][13][14].…”

“…On the one hand, based on the nonlinearization technique of Lax pairs and direct methods, many algebro-geometric solutions of (1 + 1)-dimensional [4][5][6], (2 + 1)dimensional [3,7], and differential-difference [7,8] soliton equations have been obtained [9,10]. On the other hand, algebro-geometric solutions are successfully extended from a single equation to a hierarchy [11][12][13].…”

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

“…Algebro-geometric solutions are an important class among exact solutions of soliton equations, which can be regarded as explicit solutions of the nonlinear integrable evolution equation and used to approximate more general solutions. Based on the nonlinearization technique of Lax pairs and direct method, many of algebro-geometric solutions of (1 + 1)-dimensional [1][2][3], (2 + 1)-dimensional [4,5], and differential-difference [5,6] soliton equations have been obtained, such as the Gerdjikov-Ivanov, modified Kadomtsev-Petviashvili, and Toda lattice equations [7][8][9]. The existence of infinitely many exact solutions is a reflection of this complete integrability.…”