A variational perspective on continuum limits of ABS and lattice GD equations

Abstract

A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows (in the continuous case) with a variational principle. Recently we developed a continuum limit procedure for pluri-Lagrangian systems, which we now apply to most of the ABS list and some members of the lattice Gelfand-Dickey hierarchy. We obtain pluri-Lagrangian structures for many hierarchies of integrable PDEs for which such structures where previously unknown. This includes the Krichever-Novikov hierarchy, the double hierarchy of sine-Gordon and modified KdV equations, and a first example of a continuous multi-component pluri-Lagrangian system.

References in corpus (12)

• Lax pair for the Adler (lattice Krichever-Novikov) System
• Soliton Solutions for ABS Lattice Equations: I Cauchy Matrix Approach
• Soliton Solutions for ABS Lattice Equations II: Casoratians and Bilinearization
• Lagrangian multiforms and multidimensional consistency
• Lagrangian multiform structure for the lattice Gel'fand-Dikii hierarchy
• What is integrability of discrete variational systems?
• On the Lagrangian structure of integrable hierarchies
• Linear integral equations, infinite matrices, and soliton hierarchies
• Rational Solutions to the ABS List: Transformation Approach
• Continuum limits of pluri-Lagrangian systems
• On reductions of the discrete Kadomtsev--Petviashvili-type equations
• Rational solutions to the ABS list: Degenerating approach

Cited by in corpus (6)

• Variational symmetries and pluri-Lagrangian structures for integrable hierarchies of PDEs
• Continuum limits of pluri-Lagrangian systems
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• A Revisit to the ABS H2 Equation
• On the Fourth-Order Lattice Gel'fand-Dikii Equations
• Eigenfunction equations of lattice KdV equations and connections to ABS lattice equations with a $\delta$ term