Linear integral equations, infinite matrices, and soliton hierarchies

Abstract

A systematic framework is presented for the construction of hierarchies of soliton equations. This is realised by considering scalar linear integral equations and their representations in terms of infinite matrices, which give rise to all (2+1)- and (1+1)-dimensional soliton hierarchies associated with scalar differential spectral problems. The integrability characteristics for the obtained soliton hierarchies, including Miura-type transforms, $\tau$-functions, Lax pairs as well as soliton solutions, are also derived within this framework.

25 pages, 1 table

References in corpus (1)

• Direct linearising transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations

Cited by in corpus (8)

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• Lagrangian 3-form structure for the Darboux system and the KP hierarchy
• On reductions of the discrete Kadomtsev--Petviashvili-type equations
• A variational perspective on continuum limits of ABS and lattice GD equations
• On nonautonomous differential-difference AKP, BKP and CKP equations
• Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies
• Linear integral equations and two-dimensional Toda systems
• ${\mathbb{Z}}_{\mathcal{N}}$ graded discrete integrable systems and Darboux transformations