# University of Leeds Workshop on Integrable Systems, 16th/17th May, 2014

## Available Abstracts

### Abstract

In the series of recent publications of H.Braden and author ( H.W. Braden and V~Enolski, Remarks on the complex geometry of 3-monopole, 1--65, arXiv: math-ph/0601040; H.W. Braden and V.Enolski, On the tetrahedrally symmetric monopole, Comm.Math.Phys. {\bf 299} (2010) no.1,255-282.) a program was formulated to derive in closed form the algebro-geometric solutions to the Bogomolny equation within Atiyah-Drinfeld-Hitchin-Manin-Nahm construction.

Our method is based on the Nahm Ansatz that permits to resolve analytically the associate Weyl equation and calculate monopole fields. Namely the Higgs field and gauges associated to an algebraic curve subjected to the Hitchin constrains are obtained in terms of values of the Baker-Akhiezer function that Jacobian variable are fixed at certain half-period of the curve whilst the spectral variable is built in terms solutions of the Atiyah-Ward constraint that is an algebraic equation with coefficients expressed in terms of the monopole coordinates.

We are considering here a step of realization of the program in the case of charge two-monopole, when associated algebraic curve is an elliptic curve.

### Abstract

For any PDE satisfying some non-degeneracy conditions, we define a family of Lie algebras, which we call the fundamental Lie algebras of this PDE. Fundamental Lie algebras are defined in a coordinate-independent way and are new geometric invariants for PDEs.

Recall that, for every topological space $X$ and every point $a\in X$, one has the fundamental group $\pi_1(X,a)$. The above-mentioned Lie algebras are called fundamental, because their role for PDEs is somewhat similar to the role of fundamental groups for topological spaces.

In this talk, we will concentrate on the case of analytic (1+1)-dimensional (multicomponent) evolution PDEs. For such equations, fundamental Lie algebras are responsible for all zero-curvature representations (up to gauge equivalence) and can be described in terms of generators and relations. Using these algebras, one obtains some necessary conditions for S-integrability and necessary conditions for existence of Backlund transformations for such PDEs.

In our construction, jets of arbitrary order are allowed. In the case of low-order jets, fundamental Lie algebras generalize Wahlquist-Estabrook prolongation algebras.

In the structure of fundamental Lie algebras for KdV, Krichever-Novikov, nonlinear Schrodinger, (multicomponent) Landau-Lifshitz type equations, one finds infinite-dimensional subalgebras of Kac-Moody algebras and infinite-dimensional Lie algebras of certain matrix-valued functions on some algebraic curves. Using fundamental algebras, one obtains also an invariant meaning for algebraic curves related to some PDEs. Applications to proving non-existence of Backlund transformations between some equations will be discussed as well.

### Abstract

Let $Gr(N,M)$ be the real Grassmann manifold defined by the set of all $N$-dimensional subspaces of ${\mathbb R}^M$. Each point on $Gr(N,M)$ can be represented by an $N\times M$ matrix $A$ of rank $N$. If all the $N\times N$ minors of $A$ are nonnegative, the set of all points associated with those matrices forms the totally nonnegative part of the Grassmannian, denoted by $Gr(N,M)_{\ge 0}$.

In this talk, I start to give a realization of $Gr(N,M)_{\ge 0}$ in terms of the (regular) soliton solutions of the KP (Kadomtsev-Petviashvili) equation which is a two-dimensional extension of the KdV equation. The KP equation describes small amplitude and long waves on a surface of shallow water. I then construct a cellular decomposition of $Gr(N,M)_{\ge 0}$ with the asymptotic form of the soliton solutions. This leads to a classification theorem of all solitons solutions of the KP equation, showing that each soliton solution is uniquely parametrized by a derrangement of the symmetric group $S_M$. Each derangement defines a combinatorial object called the Le-diagram (a Young diagram with zeros in particular boxes). Then I show that the Le-diagram provides a complete classification of the ''entire'' spatial patterns of the soliton solutions coming from the $Gr(N,M)_{\ge 0}$ for asymptotic values of the time.

I will also present some movies of real experiments of shallow water waves which represent some of those solutions obtained in the classification problem.

Finally I will discuss an application of those results to analyze the Tohoku-tsunami on March 2011. The talk is elementary, and shows interesting connections among combinatorics, geometry and integrable systems.

### Abstract

We propose an extension of the Dubrovin-Zhang perturbative scheme to the study of normal forms for integrable viscous and dispersive scalar conservation laws (not necessarily Hamiltonian). We show evidence that in the viscous case normal forms are parametrised by one single functional parameter (the viscous central invariant).

In the dispersive case such normal forms are parametrised by infinitely many arbitrary functions that can be identified with the coefficients of the quasilinear part of the equation. More in general, we conjecture that two scalar integrable evolutionary PDEs having the same quasilinear part are Miura equivalent. This conjecture is also consistent with the tensorial behaviour of these coefficients under general Miura transformations.

### Abstract

tt*-geometry is a way to put an Hermitian structures on complex manifolds in such a way as to perserve various compatibility conditions, and the defining equations are given by certain integrable systems. Even in the simplest cases (2D Frobenius manifolds) the equations are transcendental - they are given by a specific solution to Painleve III.

These structures live on the small phase space - a finite dimensional manifold. In this talk it will be shown how, using ideas originating in TQFTs, these structures may be lifted to the so-called big phase space - an infinite dimensional manifold which is the arena for quantum cohomology with gravitational descendents and dispersive integrable systems.

Joint work with Liana David.

### Abstract

In this talk I will discuss some recent classification results for a family of Lax pairs and I will present related integrable systems of difference and differential--difference equations.

More precisely, discrete integrable systems along with a pair of local symmetries and a pair of non-local symmetries are systematically constructed. It is shown that some of these systems generalize known integrable difference equations, like Hirota's KdV and H1, H3$_0$ and Q1$_0$ from the ABS classification, to $N$ component systems. In particular, a new two-component system is presented and its reduction to a discrete Tzitzeica equation is given.

Moreover, the connection of the classified discrete systems and their non-local symmetries to the continuous 2D Toda system and its B\"{a}cklund transformation is analysed. Finally, the two- and the three-dimensional consistency of the derived systems is discussed.

In collaboration with Allan Fordy.