Mathematical Institute, University of Göttingen, and Max Planck Institute for Dynamics and Self-Organization
Title of talk: Binary Darboux Transformations in Bidifferential Calculus
Abstract: In the framework of bidifferential calculus, we apply a universal binary Darboux transformation result to obtain infinite families of exact solutions of several integrable equations. This includes the selfdual Yang-Mills equation, a generalization of Hirota's bilinear difference equation, the matrix (potential) KP, and matrix versions of the Davey-Stewartson and the two-dimensional Toda lattice equations. This is based on joint work with Aristophanes Dimakis and Folkert Müller-Hoissen.
- Aristophanes Dimakis
University of the Aegean, Chios, Greece
Title of talk: Soliton interactions: tropical limit and combinatorics
Abstract: In a "tropical limit" (Maslov dequantization), the interaction of KdV solitons is described by piecewise linear graphs in two-dimensional space-time. These formally resemble Feynman diagrams of quantum field theory. There are incoming and outgoing "particles" (tropical limit of solitons) that interact via exchange of virtual particles. Extending previous work ( J. Phys. Conf. Series 482 (2014) 012010 ), we explore the combinatorial structures ruling these interactions.
- Georgi Grahovski
Department of Mathematical Sciences, University of Essex, Colchester, UK
Title of talk: On Integrable Discretisations for Grassmann-Extended Nonlinear Schrödinger Equations
Abstract: Integrable discretisations for a class of coupled nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting discrete systems will have Lax pairs provided by the set of two consistent Darboux transformations.
Finally, Yang-Baxter maps for the Grassmann-extended NLS equation will be presentes. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional Yang-Baxter maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. Their Liouville integrability will be briefly discussed.
Based on a joint work with A. V. Mikhailov and S. G. Konstantinou-Rizos.
- Benoit Huard
University of Northumbria, Newcastle, UK
Title of talk: Riemann waves in inhomogeneous hydrodynamic-type systems
Abstract: The integrability properties of multidimensional dispersionless systems have been put into light by several methods in the recent years, amongst others Lax representations and inverse scattering transform for vector fields on one hand and hydrodynamic reductions and Painlev\'e reductions on the other hand. Related inhomogeneous hydrodynamic systems of the Gibbons-Tsarev type appear to possess similar features, especially Lax representations as was shown by Sokolov and Odeskii (2013). We investigate the construction of Riemann-invariant solutions for systems with inhomogeneous part and the role of these solutions in indicating integrability. In particular, we study the symmetries of generalised systems of the Gibbons-Tsarev type and compare with the integrability conditions for their associated multidimensional homogeneous systems.
- Nils Kanning
Humboldt University, Berlin, Germany
Title of talk: Graßmannian Integrals, Matrix Models and Scattering Amplitudes
- Antonio Moro
University of Northumbria, Newcastle, UK
Title of talk: Nonlinear conservation laws, asymptotics and phase transitions
Abstract: A phase transition denotes a drastic change of state of a complex system due to a continuous change of model parameters.
Inspired by the theory of nonlinear conservation laws and the theory of classical shock waves that find their deep roots in the original Riemanns work we propose a novel approach, in the framework of statistical mechanics, that allows to naturally extend the equilibrium thermodynamics of mean field models to the critical region.
We show how a general effective statistical mechanical model combined with the assumption on the existence of a unique solution to the Stiltjes moment problem allows to explicitly construct the extension of the partition function inside the critical region given the equation of state outside the criticality.
The theory is shown at work in the case of the classical van der Waals equation of state. We calculate the global partition function of the van der Waals model, providing the exact mathematical description of discontinuities of the order parameter within the phase transition region, the universal form of the equations of state and explaining the occurrence of triple points in terms of the dynamics of nonlinear shock wave fronts.
- Folkert Müller-Hoissen
Max Planck Institute for Dynamics and Self-Organization, Göttingen
Title of talk: Simplex and Polygon equations
Abstract: Higher Bruhat orders (Manin and Schechtman 1986) are known to underly simplex equations, which generalize the famous Yang-Baxter equation in an attempt to construct higher-dimensional quantum integrable systems or solvable models of statistical mechanics. We describe a decomposition of any higher Bruhat order, which contains a corresponding "higher Tamari order" (higher Stasheff-Tamari order). These higher Tamari orders determine a hierarchy of equations that generalize the well-known pentagon equation and are connected by the same kind of "integrability", which is at work in case of the simplex equations. This is based on joint work with Aristophanes Dimakis (arXiv:1409.7855).
- Cornelia Schiebold
Mid Sweden University, Sundsvall, Sweden
Title of talk: Multiple-pole solutions of the Nonlinear Schrödinger equation
Abstract: We will start by a short resume on an operator theoretic approach to the Nonlinear Schrödinger equation, with the aim to motivate a solution formula which gives a unified access to the multiple-pole solutions. The main result is a complete asymptotic description of these solutions, which was so far only achieved for cases of low complexity by Olmedilla. After an overview on the geometric and algebraic ingredients of the proof, we will conclude by a discussion of cases of higher degeneracy and a comparison with the situation for the KdV equation.
- Konrad Schöbel
University of Jena, Germany
An operad structure on separation coordinates
Abstract: We reveal a rich algebraic geometric structure behind Kalnins & Miller's famous classification of separation coordinates on spheres. Namely, we show that the space of separation coordinates on the n-dimensional sphere modulo isometries is naturally parametrised by the moduli space of stable genus zero curves with n+2 marked points. As a consequence we reinterpret Kalnins & Miller's construction of separation coordinates in terms of Stasheff polytopes and the natural operad structure on these moduli spaces.
- Nikola Stoilov
Mathematical Institute, University of Göttingen, Germany
Title of talk: Dispersionful Version of WDVV Associativity Equations
Abstract: The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations arise as the conditions of associativity of an algebra in an N dimensional space. These equations are fully integrable for any N. Furthermore, the compatibility conditions for the WDVV equation can be written as a hydrodynamic type system of PDEs, which possesses a bi-Hamiltonian structure. This structure allows us to construct members of the positive and negative parts of the hierarchy. Exploiting a special transformation, together with this hierarchy, we are able to construct dispersionfull version of the above equations.
Abstract: In the past years, there have been tremendous advances in the field of planar maximally supersymmetric Yang-Mills scattering amplitudes. At tree-level they were formulated as integrals over Graßmann manifolds. Furthermore, an integrable structure was exposed by showing that the amplitudes are invariant under the Yangian of the superconformal algebra psu(2,2|4). We construct Yangian invariant Graßmannian integrals for oscillator representations of the non-compact superalgebra u(p,q|r). We show that these integrals generalize Brezin-Gross-Witten matrix models. In addition, they contain the R-matrix of the symmetry algebra, which generates integrable spin chains. Lastly, we recover a Graßmannian integral for scattering amplitudes by applying a Bargmann transformation to change basis from oscillator to spinor helicity variables.