Abstracts

Nonlinear dynamics in breathing-soliton lasers

Sonia Boscolo (Aston University)

Our recent work demonstrates that lasers operating in the breathing-soliton regime [1] offer a powerful platform for investigating complex synchronisation and chaotic dynamics in nonlinear systems, all within a single oscillator—eliminating the need for external sources or coupled systems. In these lasers, harmonics of the breathing frequency can lock to the cavity repetition rate through competition between the system’s two intrinsic frequencies. In this talk I will review the key findings of this research, which include higher-order Farey hierarchies and self-similar fractal dynamics (devil’s staircases) [2], nonstandard synchronisation domains (unusual Arnold tongues) [3], a novel modulated subharmonic regime bridging synchrony and desynchrony [4], and a new route to chaos via modulated subharmonics [5]. 

1. J. Peng, S. Boscolo, Z. Zhao, and H. Zeng, “Breathing dissipative solitons in mode-locked fiber lasers,” Sci. Adv., vol. 5, eaax1110, 2019.

2. X. Wu, Y. Zhang, J. Peng, S. Boscolo, C. Finot, and H. Zeng, “Farey tree and devil’s staircase of frequency-locked breathers in ultrafast lasers,” Nat. Commun., vol. 13, 5784, 2022.

3. X. Wu, J. Peng, B. Yuan, S. Boscolo, C. Finot, and H. Zeng, “Unveiling the complexity of Arnold’s tongues in a breathing-soliton laser,” Sci. Adv., vol. 11, eads3660, 2025. 

4. X. Wu, J. Peng, S. Boscolo, C. Finot, and H. Zeng, “Synchronization, desynchronization, and intermediate regime of breathing solitons and soliton molecules in a laser cavity,” Phys. Rev. Lett., vol. 131, 263802, 2023. 

5. H. Kang et al., “Observation of optical chaotic solitons and modulated subharmonic route to chaos in mode-locked laser,” Phys. Rev. Lett., vol. 133, 263801, 2024.

Tau-functions and the Hirota bilinear method: The universality of the KP hierarchy

Marcos Caso Huerta (University of Oviedo)

The Hirota bilinear method is one of the main tools employed in the construction of solutions of integrable systems. In contrast with other mainstream techniques such as inverse scattering and Darboux-dressing, it is not a spectral method, as it does not rely on the usual Lax representation of the system. This allows for the construction of multi-soliton, multi-breather, and multi-rogue wave solutions without requiring explicit knowledge of a Lax pair. Within this framework, we will introduce the method of tau-functions, which involves establishing a correspondence between the bilinear system resulting from applying the Hirota technique and those that arise from the KP or discrete KP hierarchies. This connection enables the construction of solutions of the original system by translating known solutions for KP. We will illustrate the method by applying it to the multicomponent Yajima-Oikawa-Newell model.

Based on joint work with Bao-Feng Feng, Sara Lombardo, Ken-ichi Maruno and Matteo Sommacal.

Understanding Set-Theoretic Solutions of the Yang–Baxter Equation Through Skew Braces

Dr Ilaria Colazzo (University of Leeds)

Skew braces are algebraic structures introduced to study set-theoretic solutions to the Yang–Baxter equation. They provide a framework that connects group-theoretic and ring-theoretic methods with the combinatorial properties of these solutions.

This talk will offer an introduction to skew braces and their main properties, with an emphasis on their role in constructing and analysing solutions to the Yang–Baxter equation. As an application, we will briefly outline how skew braces can be used to describe simple solutions.

Mathematical Modelling and Data Assimilation for the 2022 Ireland Meteotsunami

Dr Ardiansyah Fauzi (Northumbria University)

Meteotsunamis are long-period sea surface oscillations induced by atmospheric disturbances, often modeled using the nonlinear shallow water equations (NSWE). Their generation involves resonant interactions such as Proudman and Greenspan resonance, leading to wave amplification. While extensively studied in regions like the Mediterranean and the Great Lakes, meteotsunamis in the Celtic Sea remain poorly understood due to limited offshore observations. This study focuses on numerical modeling and offshore monitoring strategies for meteotsunamis using data assimilation (DA) techniques. We employ NSWE with atmospheric forcing to simulate meteotsunami dynamics and assess the impact of offshore monitoring networks. A key aspect of this research is the proposed deployment of offshore bottom pressure gauges (OBPGs) at strategic locations, where meteotsunamis are most likely to develop. These sensors provide sea-level data that can be assimilated into numerical models using ensemble-based DA methods to improve wavefield reconstruction and model accuracy. We apply this framework to the 2022 Ireland meteotsunami, using recorded air-pressure anomalies from automatic weather stations (AWSs) and tide gauge data to reconstruct the event and evaluate the role of resonance mechanisms in wave amplification. Additionally, we demonstrate how a densely deployed OBPG network combined with DA techniques can significantly enhance the accuracy of meteotsunami modeling and monitoring in the Celtic Sea. Our findings provide a mathematical framework for integrating offshore observations and data assimilation, improving our understanding of meteotsunami dynamics in underexplored regions.

Differential and Difference Painlevé equations in complex function theory

Prof Rod Halburd (UCL)

We will show how differential and difference Painlevé equations arise in the solution of some natural problems in complex function theory.  An example of such a problem is to find all meromorphic functions that have many poles and at each one of these poles there are only finitely many possibilities for the first few terms in the Laurent series expansion.  We will show how such problems arise in finding all meromorphic solutions of a differential or difference equations. 

Geometric approach for classifying quasi-Painlevé  Hamiltonian systems

Thomas Kecker (University of Portsmouth)

We study second-order differential equations and Hamiltonian systems of quasi-Painlevé type by a geometric approach, originally applied by K. Okamoto to the six Painlevé equations to obtain their spaces of initial conditions. Quasi-Painlevé equations, first so named by S. Shimomura, possess the property that all movable singularities of their solutions in the complex plane are algebraic poles (rather than ordinary poles as for the Painlevé equations). To obtain their spaces of initial conditions, a larger number of blow-ups is required than in the Painlevé case, and we keep track of them using intersection diagrams for the exceptional divisors arising in the process. We give examples of quasi-Painlevé systems and show how they can be related through bi-rational transformations, by comparing the irreducible components of their exceptional divisors. We then utilise the space of initial conditions to classify quasi-Painlevé Hamiltonian systems, in the case where the Hamiltonians are polynomial and at most quartic in their dependent variables. This presentation is based on the two articles [1] and [2], joint work with M. Dell’Atti.

[1] M. Dell’Atti and T. Kecker, Geometric approach for the identification of Hamiltonian systems of quasi-Painlevé type, J. Phys. A: Math. Theor.   58: 095202, 2025, (doi: 10.1088/1751-8121/adb819)

[2] M. Dell’Atti and T. Kecker, Spaces of initial conditions for quartic Hamiltonian systems of Painlevé and quasi-Painlevé type, preprint, 

arXiv: 2412.17135, 2025

Riding the Wave of Green Change: Advancing Wave-Based Analog Reservoir Computing for Sustainable AI

Dr Giulia Marcucci (University of Glasgow)

Wave-based reservoir computing stands as a promising analogue computing paradigm, offering the potential for efficient and sustainable AI hardware.  Various systems, including hydrodynamic, random, nonlinear, quantum, and multimodal approaches, have demonstrated robust physical computing capabilities, each exhibiting reliable input-output characteristics within complex operational domains.

In this presentation, I will explore the recent advancements in photonic and hydrodynamic reservoir computing and share the journey that led me to investigate the complexities of wave propagation, with the goal of understanding its pivotal role in facilitating machine learning. My objective is to convey three key takeaways: first, the imperative to align technological innovation with ecological responsibility to address climate challenges; second, the specific contributions I have made toward achieving this alignment; and third, the strategies I have employed to understand, measure, optimise, and control wave-based reservoir computing systems, with the overarching goal of fostering a brighter, greener future.

Elliptic automorphic Lie algebras and integrable systems

Dr Casper Oelen (Heriot-Watt University)

Automorphic Lie algebras are a class of infinite-dimensional Lie algebras over the complex numbers that naturally arise in integrable systems, in particular in the context of reduction of Lax pairs. They can be thought of as Lie algebras of meromorphic maps (usually with prescribed poles) from a compact Riemann surface $X$ into a finite-dimensional Lie algebra $\mathfrak{g}$ which are equivariant with respect to a finite group $G$ acting on $X$ and on $\mathfrak{g}$, both by automorphisms. Independently of their origins in integrable systems, they show up in algebra as examples of equivariant map algebras.

In this talk we will highlight some motivations from integrable systems to study these algebras. We will mainly focus on elliptic automorphic Lie algebras, which, for example, prominently appear in the context of Landau-Lifshitz type of equations. We show that well-known algebras, such as Holod's hidden symmetry algebra of the Landau-Lifshitz equation and the Wahlquist-Estabrook prolongation algebra of the same equation, admit a particularly simple description deriving from the automorphicity perspective.  

They turn out to be isomorphic to a current algebra $\mathfrak{sl}(2,\mathbb{C})\otimes R$, or to its direct sum with the two-dimensional abelian Lie algebra $\mathbb{C}^2$, in the latter case, where $R$ is a suitable ring of elliptic functions invariant under a particular action of the dihedral group $D_2$ of order 4.  This talk is based on joint work with Sara Lombardo and Vincent Knibbeler.