Integrable Systems 2019

Mathematically, we define a rogue wave to be a solution of an evolution equation that has a background plus a bulging part that is localised both in time and in space. The wave which satisfies this definition naturally “appears from nowhere and disappears without a trace”. One example of such function is a Peregrine breather which is a solution of the NLSE. However, this is not the only known example. There is a variety of solutions of the NLSE and other evolution equations that satisfy this criterion. They have features remarkably different from those of solitons. Several examples of rogue wave solutions and their unique properties will be considered in this talk.

There are various generalisations of the Temperley-Lieb algebra. Just as for the usual Temperley-Lieb algebra their representations can be used to construct Hamiltonians of quantum spin chains. I will discuss some particular examples.

In this talk I will report the results of the study of a 1D integrable alternating spin chain whose critical behaviour is governed by a CFT possessing a continuous spectrum of scaling dimensions. I will review both analytical and numerical approaches to analyzing the spectrum of low energy excitations of the model. It turns out that the computation of the density of Bethe states of the continuous theory can be reduced to the calculation of the connection coefficients for a certain class of differential equations whose monodromy properties are similar to those of the conventional confluent hypergeometric equation. The finite size corrections to the scaling are also discussed.

Solitons and breathers are known to model stationary and extreme localisations in nonlinear dispersive media. Indeed, a series of laboratory experiments, for instance in water waves, optics and BEC, confirmed the validity of the the uni-directional nonlinear Schrödinger equation (NLSE) to describe the spatio-temporal dynamics of such waves. In this study, we report observations of slanted, and thus, directional localized envelope soliton and breather dynamics in a water wave basin. The water surface displacement has been stereo-reconstructed using a marker-net, deployed at the center of the basin, and two synchronized high-speed cameras. The results are in very good agreement with the hyperbolic 2D+1 NLS, generally known to be not integrable. The results confirm for the first time that short-crested as well as slanted strongly-localised waves can be also described by a simplified nonlinear wave framework.

I will discuss the properties of discrete Liouville-type equations, such as existence of local integrals, linearisability, and finiteness of the chain of Laplace invariants. I will explore connections between the solitonic and the Liouville-type equations.

We employ the WKB method to study the scattering problem of the Korteweg-deVries equation in order to analytically characterize the Zabusky-Kruskal numerical experiment.  We obtain explicit asymptotic expressions for the number of solitons as well as their amplitudes.  We confirm the results by comparing them with recent shallow water experiments.  We then generalize our approach to study the defocusing nonlinear Schrodinger equation, and we apply the corresponding results to characterize some recent experiments in nonlinear optics.  This is joint work with Gino Biondini and Stefano Trillo.

Separation of variables of the Laplace equation (or the free Schrodinger equation) in spheroidal coordinates leads to the spheroidal wave equation.This is a confluent Fuchsian equation which is the confluent Heun equation and it is well known how to compute eigenvalues and eigenfunctions in this case. We show that the spectrum of the spheroidal wave equation has quantum monodromy, which means that the joint spectrum of the two commuting operators cannot be globally uniquely labelled by quantum numbers. We suspect that the spheroidal wave equation is the simples (confluent) Fuchsian equation that shows this defect. We analyse the corresponding classical Liouville integrable system and show that it is a semi-toric system with a non-degenerate focus-focus singularity, which causes the defect. Finally we show that this spheroidal harmonics integrable system is symplectically equivalent to the C. Neumman system of a particle moving on sphere under the influence of a harmonic potential

We derive the local surface tension for the asymmetric five vertex model as a function of both local horizontal and vertical magnetisation. As a consequence, limit shapes for 5V configurations in finite geometries can be determined via the Euler-Langrange equation, of which the general solution is parametrised by a single analytic function. This work is in collaboration with Rick Kenyon and Samuel Watson, and is an extension of general limit shape theory beyond the free fermion (dimer) case.

We construct the space of initial values in the sense of Okamoto for two planar map possessing a quartic and a sexitc invariant respectively. We use this construction to prove the integrability of this system, in the sense of algebraic entropy. From this perspective, the systems turns out to have certain unusual properties, not previously observed in integrable systems that are not linearisable.

Matrix models and Toda-type lattices have been an intriguing topic for years. In this talk, I’d like to present a novel Toda-type lattice which was found during thestudies of the Cauchy two-matrix model. With the help of Cauchy bi-orthogonalpolynomials, the Lax pair of the Toda-type lattice is provided and thus presents theintegrability. Moreover, I’d like to show an integrable discretization of this lattice bythe Hirota’s method and give a determinant solution to the discrete CKP equation. If time is permitted, I’d like to demonstrate how to obtain the C-Toda hierarchy fromthe 2d-Toda hierarchy. Some ongoing projects will be illustrated at the end.

Integrability criteria that have been enormously successful for second order mappings, such a singularity confinement or zero algebraic entropy, are often applied to lattice equations as though the latter were mere mappings. In this talk we will show that such a naive approach can (and does) lead to all sorts of contradictions and that considerable care is needed when using such methods to investigate the integrability of a given lattice equation.
More precisely: in this talk we show that the results of degree growth calculations for lattice equations strongly depend on the initial value problem that one chooses, either because of problems that arise in the past light-cone, or because of interferences in the future light-cone. Among the examples we treat are initial value problems for dKdV, discrete Liouville and dToda, for which the degree growth becomes exponential, in contrast to the common belief that discrete integrable equations must have polynomial growth and that linearizable equations necessarily have linear degree growth, regardless of the initial value problem one imposes. Finally, as a possible remedy for one of the observed anomalies, we also propose basing integrability tests that use growth criteria on the degree growth of a single initial value instead of all the initial values.

We present discrete Hamiltonians of the discrete Painlevé equations. This is essentially a generating function of canonical transformation.

The normalizer theory of Coxeter groups was developed by Howlett (1980), Brink and Howlett (1999).Here we look at how it can be used to understand some peculiar special cases of Sakai’s classification of discrete Painleve equations; and to clarify the relationships between different higher-dimensional generalisations of discrete integrable systems.

Based on the commutativity of scalar vector fields, an algebraic scheme is presented which leads to a privileged multi-dimensionally consistent 2n+2n-dimensional integrable partial differential equation with the associated eigenfunction constituting an infinitesimal symmetry. The “universal” character of this novel equation of vanishing Pfaffian type is demonstrated by retrieving and generalising to higher dimensions a great variety of well-known integrable equations such as the dispersionless KP and Hirota equations and various avatars of the heavenly equation governing self-dual Einstein spaces.

TBA.

In recent years, research on 4D Painlevé systems have progressed mainly from the viewpoint of isomonodromy deformation of linear equations. In this talk we study the geometric aspects of 4D Painlevé systems by investigating the space of initial conditions in Okamoto-Sakai’s sense, which was a powerful tool in the original 2D case. Especially, we focus on the Fuji-Suzuki-Tsuda system as an example, and construct its space of initial conditions. Its symmetries are also reconstructed by investigating the N\”oron-Severi bilattice.

Second order maximally superintegrable systems are to date classified in dimension 2 and 3 and for conformally flat geometries only. The methods used for these traditional classification results do however not carry over to higher dimension, as the underlying system of partial differential equations quickly grows and becomes tedious to handle, even if computer algebra is employed.
A new, algebraic-geometric approach has recently been suggested by Kress & Schoebel (2019), who successfully apply it to (non-degenerate) second order superintegrable systems on 2-dimensional Euclidean geometry. In a joint project with J. Kress (UNSW) and K. Schoebel (HTWK Leipzig), we have extended this approach to suit any dimension, and include non-Euclidean geometries. In this framework superintegrable systems are characterized by smooth (0,3)-tensor fields (structure tensors), governed by a simple set of algebraic equations. Particularly, for spaces of constant curvature, superintegrable systems are characterized by essentially one scalar function, with the algebraic conditions encoded in an affine connection with torsion.

In this presentation, we give hierarchies of q-discrete second, third and fourth Painlevé (qPII,qPIII,qPIV) equations. These hierarchies were derived by using geometric reductions/ staircase reductions of a multi-parametric integrablelattice equation. Their Lax pairs can be obtained directly from the staircase re-ductions.We then present a method to obtain Bäcklund transformations of the hierar-chies systematically. The key ingredients for this method are the consistencyaround the cube property and the staircase reductions. We are able to recreatethe known Bäcklund transformations for the first member of the qPII hierarchy. Special q-rational solutions of the hierarchies are constructed and corresponding functions that solve the associated linear problems are derived. We also study the symmetry groups of the hierarchies by using the geometric reductions.

Integrability criteria that have been enormously successful for second order mappings, such a singularity confinement or zero algebraic entropy, are often applied to lattice equations as though the latter were mere mappings. In this talk we will show that such a naive approach can (and does) lead to all sorts of contradictions and that considerable care is needed when using such methods to investigate the integrability of a given lattice equation.
More precisely: in this talk we revisit the singularity structure of Hirota’s discrete KdV (dKdV) equation and show that the equation possess singularities that are, strictly speaking, not confining. That this is possible for certain configurations involving infinitely many singularities has been known for some time, but that there are non-confined singularity patterns even for a single singularity is an important new observation.
Our discussion of the singularities for dKdV is based on an ARS-type approach (i.e. based on traveling wave reductions of dKdV) and we give a full classification of the singularities for such reductions.
In passing, we also show that the `express method’ we introduced for calculating the dynamical degree of second order mappings can also be successfully applied to the higher order mappings we obtained from dKdV.

We establish an infinite family of solutions in terms of the Weierstrass elliptic functions of the lattice Boussinesq systems by setting up an elliptic direct linearisation scheme. The dispersion relation for these solutions is characterized by “elliptic cube root of unity”. We will also introduce such solutions of continuous equations such as the KdV and KP equation, and the understanding of such solutions from Darboux transformation and bilinear approach.