# Dispersive hydrodynamics

Dispersive hydrodynamics has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media, encompassing both dynamic and stochastic aspects of wave propagation. Recent theoretical and experimental developments have opened up new areas for research, with intriguing open issues in both theory and applications. These include the understanding of fundamental regularisation mechanisms of hydrodynamic singularities via the generation of dispersive shock waves (DSWs) and related phenomena. Physical examples of dispersive hydrodynamic phenomena include undular bores on rivers, in the ocean and atmosphere, nonlinear diffraction patterns in optics and quantum fluids, turbulence in fibre lasers and superfluids. The dispersive hydrodynamics research in the MCNP group is centred around four major themes:

- Dispersive shock waves in integrable and non-integrable systems
- Integrable turbulence and soliton gas
- Extreme nonlinear wave phenomena
- Physical applications in fluid mechanics and nonlinear optics

The mathematics employed involve a synthesis of exact and asymptotic methods from soliton theory, hyperbolic conservation laws and Whitham modulation theory. The analytical developments are supported by careful numerical simulations and, when possible, comparisons with available experimental and observational data.

**Lead by: **Gennady El

**Key publications: **

- G. A. El and A. Tovbis, arXiv:1910.05732 (2019)
- M. D. Maiden, D. V. Anderson, N. A. Franco, G. A. El, and M. A. Hoefer, Phys. Rev. Lett.
**120**, 144101 (2018) - G. A. El and M. A. Hoefer, Physica D
**33**, 11-65 (2016) - G. A. El, E. G. Khamis, and A. Tovbis, Nonlinearity
**29**, 2798-2836 (2016)

**Funding: **EPSRC (2017-2010, Grant No. EP/R00515X/1); Dstl (2017-2020, Contract No DSTLX-1000116851); LumOptica (2019-2020, Contract Research Grant), the London Mathematical Society Research in Pairs Grants (2014, 2015, 2017, 2018, 2019), the Royal Society International Exchanges Scheme (2014-2016).

# Classical-to-quantum correspondence for many-body systems

This research aims to understand the transition from a classical system to its quantum analogue. This fundamental research domain, sometimes dubbed 'quantum chaos', has had numerous connections ranging from physics (nuclear physics, microwave cavities, transport in quantum dots) to mathematics (number theory, spectral theory, dynamical systems). In the physics community, there has been recently a renewed attention to apply these techniques for quantum systems composed of interacting particles, under the name of 'many-body quantum chaos'. I am interested in the more fundamental questions, which are raised when studying a quantum many-body system/model using a semiclassical perspective.

**Lead by: **Rémy Dubertrand

**Key publications: **

- R. Dubertrand and S. Müller, New J. Phys.
**18**, 033009 (2016) - I. García-Mata, O. Giraud, B. Georgeot, J. Martin, R. Dubertrand, and G. Lemarié, Phys. Rev. Lett.
**118**, 166801 (2017)

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# Modulation of nonlinear waves and dispersive shock waves

The Whitham modulation theory describes intermediate and long-time dispersive hydrodynamics, and has proved particularly effective in the determination of nonlinear waves such as the celebrated dispersive shock wave (DSW) solution. My research explores new developments and applications of the modulation theory, with the description of:

- DSWs in non-integrable systems (e.g. Serre fully nonlinear shallow water equations, generalized nonlinear SchrÃ¶dinger equation in nonlinear optics)
- New wave structures in systems with nonconvex dispersion (e.g. Benjamin-Bona-Mahony equation)
- Interactions between nonlinear waves such as the linear wave-DSW interaction identified as a new realisation of wave-mean flow interaction in shallow water systems.

**Lead by: **Thibault Congy

**Key publications: **

- T. Congy, G. A. El, and M. A. Hoefer, J. Fluid Mech.
**875**, 1145-1174 (2019) - T. Congy, G. A. El, M. A. Hoefer, and M. Shearer, Stud. Appl. Math.
**142**, 241-268 (2019)

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**# Nonlinear conservation laws

1) Classification of (multidimensional) nonlinear consevation laws realised via nonlinear partial differential equations (PDEs) of hydrodynamic type with viscosity, dispersion or dispersionless, universal behaviour of solutions and singularities. 2) Complete integrability and equations of state of mean field statistical mechanical models and random networks. 3) Integrability, Toda Lattice, KP and complexity in random matrix models.

**Lead by: **Antonio Moro, Costanza Benassi

**Key publications: **Papers

**Funding: **Leverhulme Trust Research Project Grant (2018-2021, PI: A. Moro; Grant Ref. RPG-2017-228) - Dressing methods and complexity reduction for integrable networks models; Royal Society International Exchanges (2017-2019) - PI: A. Moro; Co-I: G. Biondini - p-Stars and KP: soliton phase diagrams for complex systems.

# Nonlinear dynamics in delayed feedback systems

Cyclic rhythms are crucial for the proper functioning and robustness of many biological systems. These are often described using nonlinear differential equations which incorporate feedback mechanisms with inherent delays of various types. The characterisation of periodic solutions in such systems provides markers of accuracy for these rhythms as well as pathways for restoring them. A particular emphasis given to the glucose-insulin regulation system.

**Lead by: **Benoit Huard

**Key publications: **

- A. Bridgewater, B. Huard, and M. Angelova, J. Nonlinear Sci.
*(In press)*(2019) - B. Huard, A. Bridgewater, and M. Angelova, J. Theor. Biol.
**418**, 66 - 76 (2017) - B. Huard, M. Angelova, and J. Easton, Commun. Nonlinear Sci.
**26**, 211 - 222 (2015)

**Funding: **Newton Advanced fellowship from the British Academy of Sciences.

# Nonlinear waves and complex dynamics

- Nonlinear waves in ferromagnetic systems: Localised, propagating magnetic structures in 1- and 2-dimensional ferromagnetic systems at the nanometre length-scale. Classical, continuous Heisenberg ferromagnet equation. Landau-Lifshitz equation with uni-axial and bi-axial anisotropy.
- Integrability and linear stability of nonlinear waves: Linear stability analysis of scalar and multi-component, nonlinear partial differential equations of integrable type. Manakov system. Vector nonlinear Schrödinger equation. Three waves resonant interaction equation.
- Complex dynamics and Riemann surfaces: Understanding complex dynamics by means of an associated Riemann surface. Circular flow on a compact Riemann surface. Landen transformations. Slow chaos and parabolic dynamical systems.

**Lead by: **Matteo Sommacal

**Funding: **LMS, Research in Pairs - Scheme 4, "Instabilities of nonlinear waves by means of elementary algebraic-geometry", Grant Ref. 41808 (2019). LMS, Research in Pairs - Scheme 4, "Propagating, localised waves in ferromagnetic nanowires", Grant Ref. 41622 (2017).

# Magnetisation dynamics

Magnetic materials are typically associated with permanent magnets. However, the magnetisation is quite dynamic, and its motion can be used for a variety of applications. Our magnetism research theoretically and numerically explores nonlinear and dispersive effects in magnetisation dynamics to find more efficient communication, logic, and storage technologies. Current topics of research are:

- Spin transport via a spin hydrodynamics formulation of solid-state magnetisation dynamics
- Non-equilibrium magnetisation dynamics including dynamics of domain networks and nucleation of solitons
- Spin wave dispersion in reconfigurable artificial spin ice
- E. Iacocca et al. Nature Communications
**10**, 1756 (2019) - E. Iacocca, T. J. Silva, and M. A. Hoefer, Phys. Rev. Lett.
**118**, 017203 (2017) - E. Iacocca, S. Gliga, R. L. Stamps, and O. G. Heinonen, Phys. Rev. B
**93**, 134420 (2016)

**Lead by: **Ezio Iacocca

**Key publications: **

# Metamaterials and geophysics dynamics

- Nonlinear waves in optical and mechanical topological insulators
- Localized pattern formation in driven dissipative systems
- Mathematical models of meltwater features in cold environments

**Lead by: **Yiping Ma

**Key publications: **

- D. D. J. M. Snee and Y.-P. Ma, Ext. Mech. Lett.
**30**, 100487 (2019) - Y.-P. Ma, I. Sudakov, C. Strong, and K. M. Golden, New J. Phys.
**21**, 1367-2630 (2019) - Y.-P. Ma and E. Knobloch, Physica D
**337**, 1-17 (2016)

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**# Phase transitions in spin systems and random matrix models

Statistical mechanics is the field of mathematical physics describing the collective behaviour of macroscopic systems from the aggregated features of their microscopic components. A typical phenomenon observed in this context in various physical models is phase transitions, an abrupt change in the qualitative features of the system when its parameters (e.g. temperature, magnetic field) are tuned. In my research I focus on the investigation of phase transitions in various models, specifically lattice spin systems and random matrix models. Lattice spin systems provide a simplified description of the magnetic behaviour of solid materials, and some of them can be investigated using their relationship with a class of probabilistic models describing interacting random loops. Random matrix models present very distinctive phase transitions, and can be explored exploiting their underlying integrable structure and the tools provided by dispersive hydrodynamics.

**Lead by: **Costanza Benassi

**Key publications: **

- C. Benassi and A. Moro, arXiv:1903.11473 (2019)
- C. Benassi and D.Ueltschi, Commun. Math. Phys. doi:10.1007/s00220-019-03474-9 (2019)
- C. Benassi, J. Fröhlich, and D. Ueltschi, Ann. Henri Poincaré
**18**, 2831-2847 (2017)