Functional representation of the Ablowitz-Ladik hierarchy

Abstract

The Ablowitz-Ladik hierarchy (ALH) is considered in the framework of the inverse scattering approach. After establishing the structure of solutions of the auxiliary linear problems, the ALH, which has been originally introduced as an infinite system of difference-differential equations is presented as a finite system of difference-functional equations. The representation obtained, when rewritten in terms of Hirota's bilinear formalism, is used to demonstrate relations between the ALH and some other integrable systems, the Kadomtsev-Petviashvili hierarchy in particular.

15 pages, LaTeX

References in corpus (4)

• Solution of the dispersionless Hirota equations
• Functional representation of the Ablowitz-Ladik hierarchy
• Toda Lattice Hierarchy and Zamolodchikov's Conjecture
• Analytic-bilinear approach to integrable hierarchies. I.Generalized KP hierarchy

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• From AKNS to derivative NLS hierarchies via deformations of associative products
• Nonlocal reductions of the Ablowitz-Ladik equation
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• Algebro-Geometric Finite-Gap Solutions of the Ablowitz-Ladik Hierarchy
• Nonassociativity and Integrable Hierarchies
• Functional representation of the negative AKNS hierarchy
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• The closeness of localised structures between the Ablowitz-Ladik lattice and Discrete Nonlinear Schrödinger equations II: Generalised AL and DNLS systems
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• Orbifold melting crystal models and reductions of Toda hierarchy
• Soliton Fay identities. II. Bright soliton case
• 'Universality' of the Ablowitz-Ladik hierarchy
• Solutions of matrix NLS systems and their discretisations: A unified treatment
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• Bäcklund transformations between the AKNS and DNLS hierarchies
• The Davey Stewartson system and the Bäcklund Transformations
• On a new integrable discretization of the derivative nonlinear Schrödinger (Chen-Lee-Liu) equation
• Explicit solutions for a nonlinear model on the honeycomb and triangular lattices
• Solitons of a vector model on the honeycomb lattice
• Functional representation of the negative DNLS hierarchy
• On Equivariant Gromov--Witten Invariants of Resolved Conifold with Diagonal and Anti-Diagonal Actions
• The AKNS Hierarchy Integrability Analysis Revisited: The Vertex Operator Approach and its Lie-Algebraic Structure
• Explicit Solutions for a Nonlinear Vector Model on the Triangular Lattice