Decomposition of the Discrete Ablowitz–Ladik Hierarchy

Abstract

The nonlinearization approach of Lax pairs is extended to the discrete Ablowitz-Ladik hierarchy. A new symplectic map and a class of new finite-dimensional Hamiltonian systems are derived, which are further proved to be completely integrable in the Liouville sense. An algorithm to solve the discrete Ablowitz-Ladik hierarchy is proposed. Based on the theory of algebraic curves, the straightening out of various flows is exactly given through the Abel-Jacobi coordinates. As an application, explicit quasi-periodic solutions for the discrete Ablowitz-Ladik hierarchy are obtained resorting to the Riemann theta functions.

Trigonal curves and algebro-geometric solutions to soliton hierarchies I

“…Denote by θ(z) the Riemann theta function associated with K g equipped with the above homology basis [60]: 22) where z = (z 1 , · · · , z g ) ∈ C g is a complex vector, and ·, · stands for the Hermitian inner product on C g :…”

Hyperelliptic function solutions with finite genus ������ of coupled nonlinear differential equations*

“…Successful methods include inverse scattering transform [1], Lie group [2], Darboux transformation [3], Hirota direct method [4], algebro-geometrical approach [5], et al The algebrogeometrical approach presents quasi-periodic or algebro-geometric solutions to many nonlinear differential equations, which were originally obtained on the Korteweg-de Vries (KdV) equation based inverse spectral theory and algebro-geometric method developed by pioneers such as Novikov, Dubrovin, Mckean, Lax, Its, Matveev, and co-workers [5][6][7][8][9][10] in the late 1970s. Recently, this theory has been extended to a large class of nonlinear integrable equations [11][12][13][14][15][16][17]. By virtue of Riemann theta function, we obtain some quasi-periodic wave solutions of nonlinear equations, discrete equations and supersymmetric equations [43][44][45][46][47][48].…”

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

“…These methods have been widely used to solve soliton equations associated with the 2 × 2 matrix spectral problems, by which finite genus solutions to a great deal of integrable models have been successfully constructed, for example, the KdV [11,27,5,7], the discrete AblowitzLadik [33,28,16], the AKNS [19,21], and the Toda lattice equations [7], etc. However, finite genus solutions for the soliton equations associated with the 3 × 3 matrix spectral problems, such as the coupled Sasa-Satsuma hierarchy, cannot be constructed by using these known methods because the corresponding algebraic curve has been changed from the hyperelliptic curve to the trigonal curve.…”

“…It is known that there have been several systematic methods to obtain the finite genus solutions of soliton equations such as algebro-geometric method, the inverse scattering transformation for periodic problem, the nonlinearization approach and others [11,27,5,7,33,28,16,29,6,19,21,3]. These methods have been widely used to solve soliton equations associated with the 2 × 2 matrix spectral problems, by which finite genus solutions to a great deal of integrable models have been successfully constructed, for example, the KdV [11,27,5,7], the discrete AblowitzLadik [33,28,16], the AKNS [19,21], and the Toda lattice equations [7], etc.…”