Raphael Boll

CLASSIFICATION OF 3D CONSISTENT QUAD-EQUATIONS

We consider 3D consistent systems of six possibly different quad-equations assigned to the faces of a cube. The well-known classification of 3D consistent quad-equations, the so-called ABS-list, is included in this situation. The extension of these equations to the whole lattice ℤ3 is possible by reflecting the cubes. For every quad-equation we will give at least one system included leading to a Bäcklund transformation and a zero-curvature representation which means that they are integrable.

What is integrability of discrete variational systems

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analogue of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however, having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. A d-dimensional pluri-Lagrangian problem can be described as follows: given a d-form on an m-dimensional space (called multi-time, m>d), whose coefficients depend on a sought-after function x of m independent variables (called field), find those fields x which deliver critical points to the action functionals for any d-dimensional manifold Σ in the multi-time. We derive the main building blocks of the multi-time Euler–Lagrange equations for a discrete pluri-Lagrangian problem with d=2, the so-called corner equations, and discuss the notion of consistency of the system of corner equations. We analyse the system of corner equations for a special class of three-point two-forms, corresponding to integrable quad-equations of the ABS list. This allows us to close a conceptual gap of the work by Lobb and Nijhoff by showing that the corresponding two-forms are closed not only on solutions of (non-variational) quad-equations, but also on general solutions of the corresponding corner equations. We also find an example of a pluri-Lagrangian system not coming from a multi-dimensionally consistent system of quad-equations.

Non-symmetric discrete Toda systems from quad-graphs

For all non-symmetric discrete relativistic Toda-type equations, we establish a relation to 3D consistent systems of quad-equations. Unlike the more simple and better understood symmetric case, here the three coordinate planes of ℤ3 carry different equations. Our construction allows for an algorithmic derivation of the zero curvature representations and yields analogous results also for the continuous time case.

Multi-time Lagrangian 1-forms for families of Bäcklund transformations: Toda-type systems

General Lagrangian theory of discrete one-dimensional integrable systems is illustrated by a detailed study of Backlund transformations for Toda-type systems. Commutativity of Backlund transformations is shown to be equivalent to the consistency of the system of discrete multi-time Euler–Lagrange equations. The precise meaning of the commutativity in the periodic case, when all maps are double-valued, is established. It is shown that the gluing of different branches is governed by the so-called superposition formulas. The closure relation for the multi-time Lagrangian 1-form on solutions of the variational equations is proved for all Toda-type systems. Superposition formulas are instrumental for this proof. The closure relation was previously shown to be equivalent to the spectrality property of Backlund transformations, i.e., to the fact that the derivative of the Lagrangian with respect to the spectral parameter is a common integral of motion of the family of Backlund transformations. We relate this integral of motion to the monodromy matrix of the zero curvature representation which is derived directly from equations of motion in an algorithmic way. This serves as further evidence in favor of the idea that Backlund transformations serve as zero curvature representations for themselves.

On the Lagrangian structure of 3D consistent systems of asymmetric quad-equations

Recently, the first-named author gave a classification of 3D consistent 6-tuples of quad-equations with the tetrahedron property; several novel asymmetric 6-tuples have been found. Due to 3D consistency, these 6-tuples can be extended to discrete integrable systems on . We establish Lagrangian structures and closeness of discrete Lagrangian 2-forms for the class of discrete integrable systems involving equations for which some (but not all) of the biquadratics are non-degenerate. This class covers, among others, some of the above-mentioned novel systems.

Multi-time Lagrangian 1-forms for families of Bäcklund transformations. Relativistic Toda-type systems

We establish the pluri-Lagrangian structure for families of Backlund transformations of relativistic Toda-type systems. The key idea is a novel embedding of these discrete-time (one-dimensional) systems into certain two-dimensional (2D) pluri-Lagrangian lattice systems. This embedding allows us to identify the corner equations (which are the main building blocks of the multi-time Euler–Lagrange equations) with local superposition formulae for Backlund transformations. These superposition formulae, in turn, are key ingredients necessary to understand and to prove commutativity of the multi-valued Backlund transformations. Furthermore, we discover a 2D generalization of the spectrality property known for families of Backlund transformations. This result produces a family of local conservations laws for 2D pluri-Lagrangian lattice systems, with densities being derivatives of the discrete 2-form with respect to the Backlund (spectral) parameter. Thus, a relation of the pluri-Lagrangian structure with more traditional integrability notions is established.

On the variational interpretation of the discrete KP equation

We study the variational structure of the discrete Kadomtsev-Petviashvili (dKP) equation by means of its pluri-Lagrangian formulation. We consider the dKP equation and its variational formulation on the cubic lattice \(\mathbb Z^{N}\) as well as on the root lattice \(Q(A_{N})\). We prove that, on a lattice of dimension at least four, the corresponding Euler-Lagrange equations are equivalent to the dKP equation.

On Bianchi permutability of Bäcklund transformations for asymmetric quad-equations

We prove the Bianchi permutability (existence of superposition principle) of Bäcklund transformations for asymmetric quad-equations. Such equations and their Bäcklund transformations form 3D consistent systems of a priori different equations. We perform this proof by using 4D consistent systems of quad-equations, the structural insights through biquadratics patterns and the consideration of super-consistent eight-tuples of quad-equations on decorated cubes.