The 2019 edition of Integrable Systems in Newcastle will take place on the 5th and 6th of April at Northumbria University, hosted by the MCNP group at the Department of Mathematics, Physics and Electrical Engineering. The workshop aims at exploring new connections between integrability and physics and promote interactions between researchers in both areas.

The event is sponsored by the LMS Joint Research Group Classical and Quantum Integrability.

People interested in attending the workshop should contact directly the Organising Committee before the 24th of March.

- Panagiota Adamopoulou (Heriot-Watt University)
- Agis Athanassoulis (University of Dundee)
- Costanza Benassi (Northumbria University)
- Vincent Caudrelier (University of Leeds)
- Thibault Congy (Northumbria University)
- Marta Dell'Atti (Northumbria University)
- Carlos Dibaya (University of Leeds)
- Clare Dunning (University of Kent)
- Gennady El (Northumbria University)
- Rod Halburd (UCL)
- Benoit Huard (Northumbria University)
- Antonio Moro (Northumbria University)
- Simone Paleari (University of Milan)
- Georgios Papamikos (University of Leeds)
- Giacomo Roberti (Northumbria University)
- Oleg Senkevich (Northumbria University)
- Matteo Sommacal (Northumbria University)
- Matteo Stoppato (University of Leeds)
- Benoit Vicedo (York University)

In this talk I will present some recent results regarding the construction of the vector modified KdV hierarchy, based on the Drinfeld-Sokolov scheme. I will discuss how Darboux transformations play a central role in this construction, and then use the dressing method in order to generate solutions for the whole hierarchy.

The Alber equation has been proposed as a model for stochastic ocean waves and their interaction with a homogeneous background wavefield. The main problem is to determine whether the background wavefield (represented by its power spectrum) can feed energy into inhomogeneous disturbances or not. In particular, a nonlinear “eigenvalue relation” is known to control the possible linear instability of given wave spectra -- i.e. it is known that if there exist eigenvalues then there exist instabilities in the linearised problem. We present three main results: first, we establish for the first time the well-posedness of the fully nonlinear problem. Second, we show Landau damping for stable spectra, i.e. if there are no eigenvalues then any inhomogeneities disperse, vanishing in time, despite the presence of a background wavefield with infinite energy. Third, we fully characterise stable and unstable spectra. We find that typical real-life spectra are stable, but come close to the unstable region. Thus Landau damping seems to actually be the limiting factor on how narrow wave spectra can become in the ocean before modulation instability appears. Moreover, the observed robustness of stationary and homogeneous spectra in the ocean is finally explained. Some formal computations relating the fundamental scalings of Rogue Waves with the eigenvalues for unstable spectra will be briefly discussed.

We show that Hermitian Matrix Models support, in the thermodynamic regime, the occurrence of a new type of phase transition characterised by dispersive regularisation of the order parameter near the critical point. Using the identification of the partition function with a particular solution of the even Toda hierarchy, we argue that the singularity is resolved via the onset of a multi-dimensional dispersive shock described by an integrable flow in the space of coupling constants. This analysis explains the origin and mechanism leading to the emergence of chaotic behaviours observed in $M^6$ matrix models and extends its validity to even nonlinearity of arbitrary order. Based on a joint work with Antonio Moro.

A dispersive shock wave (DSW) is an expanding, modulated nonlinear wavetrain that connects two disparate hydrodynamic states, and can be viewed as a dispersive counterpart to the dissipative, classical shock. DSWs have raised a lot of interest in the recent years, due to the growing recognition of their fundamental nature and ubiquity in physical applications, examples being found in oceanography, meteorology, geophysics, nonlinear optics, plasma physics and condensed matter physics. Although well-established methods, such as the Whitham modulation theory, have proved particularly effective for the determination of DSW solutions of certain nonlinear wave equations, a universal description of these objects is still lacking.

The nonlinear Schrödinger (NLS) equation and the Whitham modulation equations both describe slowly varying, locally periodic nonlinear wavetrains, albeit in differing amplitude-frequency domains. Taking advantage of the overlapping asymptotic regime that applies to both the NLS and Whitham modulation descriptions, we developed a universal analytical description of DSWs generated in Riemann problems for a broad class of integrable and nonintegrable nonlinear dispersive equations. The proposed method extends DSW fitting theory that prescribes the motion of a DSW’s edges into the DSW’s interior, that is, this work reveals the DSW structure. I will present this new method and illustrate its efficacy by considering various physically relevant examples. This talk is based on joint work with Gennady El, Mark Hoefer and Michael Shearer.

Wronskian determinants of classical orthogonal polynomials have a number of applications in mathematical physics. For example, they are

- building blocks of certain rational solutions of the Painleve equations;
- the main classes of examples of exceptional orthogonal polynomials;
- certain solutions of the Kadomtzev-Petviashvili hierarchy of partial differential equations in integrable systems.

Natural analogues of differential and difference equations with solutions defined on functions fields over finite fields will be discussed. Analogues in this setting of important special functions such as the exponential, gamma and hypergeometric functions have been discovered by Carlitz, Goss, Thakur and others. We will discuss integrable analogues of certain differential and difference equations in this context and the role played by singularity analysis.

In many contexts, extension of a theory to infinite dimensions represents both a challenge and a source interesting problems and results. It has been the case in the field of integrable systems, e.g. with the discovery of the integrability of PDEs like the celebrated KdV and NLS equations.

In the neighboring field of quasi integrable systems, although it is far from being complete, a relevant extension of many classical results has been made. Surprisingly, the case of a finite but arbitrarily large number of degrees of freedom appears to be even more difficult. It is indeed a well known limit of the classical results of perturbation theory (like KAM or Nekhoroshev theorem) to suffer a bad dependence on the number of degrees of freedom, often resulting in void or non applicable statements in the thermodynamic limit.

It is thus an important challenge in Hamiltonian dynamics the development of a suitably adapted perturbation theory for Hamiltonian systems in such a limit. Indeed, motivated by the problems arising in the foundations of Statistical Mechanics, it is relevant to consider also large systems with non vanishing energy per particle (which corresponds to a non zero temperature in the physical model).

I will try to present some results in the direction of the aforementioned goals. The model considered are nonlinear chains, in particular finite but arbitrarily large Klein-Gordon chain, with periodic boundary conditions.

We construct an extensive adiabatic invariant in the thermodynamic limit. Given a fixed and sufficiently small value of the coupling constant $a$, the evolution of the adiabatic invariant is controlled up to times scaling as $\beta^{1/a}$ for any large enough value of the inverse temperature $\beta$. The time scale becomes a stretched exponential if the coupling constant is allowed to vanish jointly with the specific energy. The adiabatic invariance is exhibited by showing that the variance along the dynamics, i.e. calculated with respect to time averages, is much smaller than the corresponding variance over the whole phase space, i.e. calculated with the Gibbs measure, for a set of initial data of large measure. All the perturbation constructions and the subsequent estimates are consistent with the extensive nature of the system.

Joint results with Antonio Giorgilli and Tiziano Penati.

I will begin by reviewing how classical integrable sigma models can be recast as classical Gaudin models associated with affine Kac-Moody algebras, or affine Gaudin models for short. One usually thinks of the Gaudin model as a spin chain, since it can be obtained from a certain limit of the XXX spin chain. When viewed in this way, integrable sigma models are essentially described by affine Gaudin models with a single site. It is then very natural, in this formalism, to consider affine Gaudin models with arbitrarily many sites. I will then go on to show that such multi-site classical Gaudin models can be used to construct new relativistic classical integrable field theories that couple together an arbitrary number of integrable sigma models. This talk is based on joint work, arXiv:1811.12316 and to appear, with F. Delduc, M. Magro and S. Lacroix.

- Holiday Inn Express (20min walk, click here for map)
- County hotel (15min walk, click here for map)
- Premier Inn, Newcastle City Centre (15min walk, click here for map)
- Sandman Signature Newcastle Hotel (15min walk, click here for map)

- Holiday Inn (15min walk, click here for map)
- The Adelphi Bed & Breakfast (25min walk, click here for map)

- Costanza Benassi
*costanza.benassi at northumbria.ac.uk* - Thibault Congy
*thibault.congy at northumbria.ac.uk* - Benoit Huard
*benoit.huard at northumbria.ac.uk* - Antonio Moro
*antonio.moro at northumbria.ac.uk*

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