P. M. Santini

Integrable three-dimensional lattices

Following the work of Zakharov and Shabat (1974), the authors derive nonlinear equations which are integrable through a discrete Gelfand-Levitan integral equation in two (resp one, resp zero) continuous and one (resp two, resp three) discrete variables.

Integrable lattices and their sublattices: From the discrete Moutard (discrete Cauchy-Riemann) 4-point equation to the self-adjoint 5-point scheme

We present the sublattice approach, a procedure to generate, from a given integrable lattice, a sublattice which inherits its integrability features. We consider, as illustrative example of this approach, the discrete Moutard 4-point equation and its sublattice, the self-adjoint 5-point scheme on the star of the square lattice, which are relevant in the theory of the integrable discrete geometries and in the theory of discrete holomorphic and harmonic functions (in this last context, the discrete Moutard equation is called discrete Cauchy-Riemann equation). Therefore an integrable, at one energy, discretization of elliptic two-dimensional operators is considered. We use the sublattice point of view to derive, from the Darboux transformations and superposition formulas of the discrete Moutard equation, the Darboux transformations and superposition formulas of the self-adjoint 5-point scheme. We also construct, from algebro-geometric solutions of the discrete Moutard equation, algebro-geometric solutions of...

Bäcklund transformations for nonlinear evolution equations in 2 + 1 dimensions

Abstract We obtain the general form of the Backlund transformations for the nonlinear evolution equations solvable through the Zakharov and Shabat method and, as an example, we write them down explicitly for the Kadomtsev-Petviashvili equation, the two-dimensional “three-wave” equation and the Davey-Stewartson equation.

A unified algebraic approach to integral and discrete evolution equations

The authors investigate the integrability properties of discrete systems and singular integral evolution equations, showing that they correspond to different reductions and different concrete realisations of the same abstract algebraic structures. In particular, they derive the bi-Hamiltonian formulation of prototype examples like the discrete sine-Gordon equation and the sine-Hilbert equation.

Asymptotic lattices and their integrable reductions: I. The Bianchi-Ernst and the Fubini-Ragazzi lattices

We review recent results on asymptotic lattices and their integrable reductions. We present the theory of general asymptotic lattices in 3 together with the corresponding theory of their Darboux-type transformations. Then we find a novel permutability theorem for Bianchi surfaces, which can be reinterpreted as a discrete version of the Bianchi-Ernst system and coincides with an equation recently introduced by Schief (Schief W K 2001 Stud. Appl. Math. 106 85-137). Using the well known connection between the Bianchi and Ernst systems, we also propose the discrete analogue of the Ernst system. Finally, we present the theory of the discrete analogues of isothermally asymptotic (Fubini-Ragazzi) nets together with their transformations.

The remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the Hopf–Cole transformation

We establish deep and remarkable connections among partial differential equations (PDEs) integrable by different methods: the inverse spectral transform method, the method of characteristics and the Hopf-Cole transformation. More concretely, 1) we show that the integrability properties (Lax pair, infinitely-many commuting symmetries, large classes of analytic solutions) of (2+1)-dimensional PDEs integrable by the Inverse Scattering Transform method (S-integrable) can be generated by the integrability prop- erties of the (1+1)-dimensional matrix Burgers hierarchy, integrable by the matrix Hopf- Cole transformation (C-integrable). 2) We show that the integrability properties i) of S-integrable PDEs in (1+1)-dimensions, ii) of the multidimensional generalizations of the GL(M, C) self-dual Yang Mills equations, and iii) of the multidimensional Calogero equations can be generated by the integrability properties of a recently introduced mul- tidimensional matrix equation solvable by the method of characteristics. To establish the above links, we consider a block Frobenius matrix reduction of the relevant matrix fields, leading to integrable chains of matrix equations for the blocks of such a Frobenius matrix, followed by a systematic elimination procedure of some of these blocks. The con- struction of large classes of solutions of the soliton equations from solutions of the matrix Burgers hierarchy turns out to be intimately related to the construction of solutions in Sato theory. 3) We finally show that suitable generalizations of the block Frobenius ma- trix reduction of the matrix Burgers hierarchy generates PDEs exhibiting integrability properties in common with both S- and C- integrable equations.

Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric interpretation of the Laplace transformations of the self-adjoint 7-point scheme is given and the corresponding novel integrable discrete three-dimensional system is constructed.

Nonlocality and the inverse scattering transform for the Pavlov equation

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is nonlocal, and the proper choice of integration constants should be the one dictated by the associated inverse scattering transform (IST). Using the recently made rigorous IST for vector fields associated with the so-called Pavlov equation , in this paper we establish the following. 1. The nonlocal term arising from its evolutionary form corresponds to the asymmetric integral . 2. Smooth and well-localized initial data evolve in time developing, for , the constraint , where . 3. Because no smooth and well-localized initial data can satisfy such constraint at , the initial () dynamics of the Pavlov equation cannot be smooth, although, because it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results should be successfully used in the study of the nonlocality of other basic examples of integrable dispersionless PDEs in multidimensions.

Recursion operator and bi‐Hamiltonian structure for integrable multidimensional lattices

The recursion operator and the bi‐Hamiltonian structure for an integrable two‐dimensional version of the Toda chain are algorithmically derived from the corresponding linear problem. The intimate relation between two‐dimensional theories and non‐Abelian one‐dimensional theories is emphasized. The analogies and differences between discrete and continuous integrable two‐dimensional systems are pointed out and discussed.

An example of ∂̄ problem arising in a finite difference context: Direct and inverse problem for the discrete analog of the equation ψxx+uψ=σψy

The direct and inverse spectral problem for the discrete analog of the equation ψxx+uψ=σψy is solved in the framework of ‘‘∂’’ theory. The time evolution of the spectral data for the simplest nonlinear differential‐difference equations associated to this linear problem is derived.