An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

Abstract

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.

Algebro-Geometric Solutions and Their Reductions for the Fokas-Lenells Hierarchy

“…Thus, ψ 1 and ψ 1 share the same singularities and zeros, and the Riemann-Roch-type uniqueness result (cf. Lemma 3.4 [6]) then proves that ψ 1 and ψ 1 coincide up to normalization. By (3.62) one infers from the right-hand side of (3.65) that…”

Algebro-geometric solutions for the two-component Camassa–Holm Dym hierarchy

“…, r, are the constants introduced in the definition of F r (z). Hence, one infers 14) for some constantsẽ r,0 ∈ C. Recalling (4.17)-(4.29), the analog of Theorem 4.4 in the stationary case then reads as follows. Theorem 6.3.…”

“…A systematic approach, proposed by Gesztesy and Holden to construct algebro-geometric solutions for integrable equations, has been extended to the whole (1+1) dimensional integrable hierarchy, such as the AKNS hierarchy, the Camassa-Holm (CH) hierarchy etc. [11]- [14]. Recently, we investigated algebro-geometric solutions for the modified CH hierarchy and the Degasperis-Procesi hierarchy [19,20].…”

Algebro-geometric solutions for the generalized nonlinear Schrödinger hierarchy

“…As a degenerated case of the algebro-geometric solution, the elliptic function solution and multi-soliton solution may be worked out [1,22,25]. The inverse spectral theory has been extended to the whole (1+1) dimensional integrable hierarchy by Gesztesy and Holden, such as the AKNS hierarchy, the Camassa-Holm hierarchy, and so on [9,10,11,12]. In recent years, Fan and his co-authors [15,16,17,18,29,30,31] investigated the Ruijsenaars-Toda Hierarchy, the Degasperis-Procesi hierarchy, derivative Burgers hierarchy, etc., and obtained their algebro-geometric solutions.…”