Lattice and q-difference Darboux-Zakharov-Manakov systems via $\bar{\partial }$-dressing method

Abstract

A general scheme is proposed for introduction of lattice and q-difference variables to integrable hierarchies in frame of $\bar{\partial }$-dressing method . Using this scheme, lattice and q-difference Darboux-Zakharov-Manakov systems of equations are derived. Darboux, Bäcklund and Combescure transformations and exact solutions for these systems are studied.

8 pages, LaTeX, to be published in J Phys A, Letters.

References in corpus (2)

• Geometry and Integrability of Topological-Antitopological Fusion
• Geometry of 2d topological field theories

Cited by in corpus (38)

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• Quantum geometry of 3-dimensional lattices
• Quadratic reductions of quadrilateral lattices
• Desargues maps and the Hirota-Miwa equation
• The B-quadrilateral lattice, its transformations and the algebro-geometric construction
• The C-(symmetric) quadrilateral lattice, its transformations and the algebro-geometric construction
• Discrete KP equation with self-consistent sources
• Non-commutative lattice modified Gel'fand-Dikii systems
• Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrodinger operator
• The pentagon relation and incidence geometry
• Super-tetrahedra and super-algebras
• Asymptotic lattices and their integrable reductions I: the Bianchi and the Fubini-Ragazzi lattices
• Discrete Levy Transformations and Casorati Determinant Solutions of Quadrilateral Lattices
• Integrable Systems and Discrete Geometry
• Spherical geometry and integrable systems
• Discrete line complexes and integrable evolution of minors
• Lagrangian 3-form structure for the Darboux system and the KP hierarchy
• Quantization of three-wave equations
• Generalized integrable hierarchies and Combescure symmetry transformations
• Möbius invariant integrable lattice equations associated with KP and 2DTL hierarchies
• Discrete Darboux system with self-consistent sources and its symmetric reduction
• Darboux transformations for linear operators on two dimensional regular lattices
• Soliton Fay identities. I. Dark soliton case
• Discrete integrable systems and deformations of associative algebras
• Integrable discrete nets in Grassmannians
• Linear problems and Bäcklund transformations for the Hirota-Ohta system
• Geometric algebra and quadrilateral lattices
• Second-order integrable Lagrangians and WDVV equations
• WDVV and DZM
• On $\tau$-function of the quadrilateral lattice
• Geometry of quadrilateral nets: second Hamiltonian form
• Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
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• Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations