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Mathematics of Complex and Nonlinear Phenomena (MCNP)

Mathematical theories underpinning different classes of phenomena possess a universal character, with the same mathematical equations appearing to govern phenomena of different nature. The Mathematics of Complex and Nonlinear Phenomena can therefore be seen as the seed for cross-fertilisation between different research areas and disciplines. 

Complexity and nonlinearity are often concomitant features of real-world phenomena that are commonly associated to the notion of disorder and chaos.

Unexpectedly, disordered structures and chaotic behaviours observed from fluid dynamics, classical and quantum physics to biological systems, economic and social sciences can be related, under suitable conditions, to the emergence of beautiful ordered and stable structures, as well as coherent, cooperative or cyclic behaviours.

The formation of tsunamis and rogue waves, laser pulses, turbulence of accretion disks around black holes, phase transitions from ordered to disordered states of matter are just some examples where disorder/chaos and order/coherence arise as two sides of the same coin.

Most amazingly, mathematical theories underpinning such different classes of phenomena turn out to possess a universal character so that the same mathematical equations appear to govern phenomena of (seemingly) different nature. Hence, the Mathematics of Complex and Nonlinear Phenomena (MCNP) can be seen naturally as the seed for cross-fertilisation between different research areas and disciplines.

Northumbria’s MCNP Research Group conducts interdisciplinary research aiming at impacting on challenges posed by a fast-changing world. It is engaged in ground-breaking research in the hottest areas of modern Mathematics, Physics, Biology and other sciences, including multiple international collaborations with other leading experts.

MCNP research activity focuses mainly on the advancement of mathematical methods for the extensive study of nonlinear partial differential equations and dynamical systems. This ranges from their classification to the development of analytical and numerical techniques to calculate or extract qualitative and quantitative information on their solutions, e.g. asymptotic methods, multiscale analysis, nonlinear stability analysis, bifurcation theory and methods of integrable systems.

The findings of the MCNP research group are concerned with the study of mechanisms for the formation and propagation of nonlinear classical and quantum waves, coherent and localised structures, dynamic emergence of instabilities and singularities under different regimes of dispersion and dissipation, diffraction, interference, delays, phase transitions, classical and quantum chaos, geometric structures associated to differential equations and Riemann surfaces.

Applications that are currently under investigation include the solutions of models for ferromagnetic nanostructures, topological insulators, climate components, magnetohydrodynamics, biological phenomena as cyclic rhythms in glucose regulation, liquid crystals and complex networks.

The MCNP research group has successfully received funding from EPSRC, Leverhulme Trust, London Mathematical Society. 

Group leadership and contacts

Contact: Prof. Gennady El

Academic Staff


Newton Institute: Dispersive Hydrodynamics

The group is part of the organising committee of a 6-month programme on Dispersive hydrodynamics: mathematics, simulation and experiments, with applications in nonlinear waves to be held at the Newton Institute in Cambridge, 4 July - 16 December 2022.
The scientific program is organised around four themes:
  • Modulation theory and dispersive shock waves
  • Analysis of dispersive systems
  • Statistical mechanics, integrability and dispersive hydrodynamics
  • Physical applications
For more details:

New horizons in dispersive hydrodynamics

Virtual bridging workshop aims at exploring novel, cross-disciplinary connections of dispersive hydrodynamics within the four main themes of the above forthcoming 6-month programme at the Newton Institute. Will be held 21st June to 2nd July 2021.
Details at

4th IMA Conference on Nonlinearity and Coherent Structures

Online conference on nonlinearity in applied Mathematics, mathematical Analysis, fluid dynamics, engineering and physics. To be held 7 - 9 July 2021.
Details at

Research seminars

Seminars take place at 3pm in room ELA002 (Ellison building, 9a on campus map).

9 February 2022
Dr Davide Proment (East Anglia). Thermalisation and heat transport in anharmonic chains.
I will present theoretical and numerical results on the thermalisation and transport properties of the seminal Fermi-Pasta-Ulam-Tsingou (FPUT) model, that is a one-dimensional system formed of N identical masses connected by nonlinear quadratic (α-FPUT) or cubic (β-FPUT) springs.
I will first explain why the weakly nonlinear α-FPUT system shows a very long thermalisation time-scale, scaling as the eight power of the nonlinearity of the system [1]. I will then move to the β-FPUT system coupled to thermal baths at its two ends, and address how the heat transport properties depend on the number of masses N. Specifically, I will present numerical results and theoretical arguments supporting the coexistence of ballistic and Fourier regimes that rules out any fractional diffusion argument to explain the observed anomalous heat conduction [2].

[1] Onorato, Miguel, et al. "Route to thermalization in the α-Fermi–Pasta–Ulam(-Tsingou) system." Proceedings of the National Academy of Sciences 112.14 (2015): 4208-4213.
[2] Dematteis, Giovanni, et al. "Coexistence of Ballistic and Fourier Regimes in the β Fermi-Pasta-Ulam-Tsingou Lattice." Physical Review Letters 125.2 (2020): 024101.

23 February 2022
Dr Joel Tabak (Exeter). Flexible coordination of neuronal rhythms by the duration of synaptic coupling.
Our nervous system incorporates many rhythmic circuits that control critical behaviours such as breathing, moving, and sleeping. These behaviours require flexible but precise coordination between neuronal oscillators. Changing the coordination between component oscillators enables one neuronal circuit to produce many patterns of activity, such as walk, trot, gallop. Current thinking is that the coordination between neuronal oscillators is tuned either by altering the coupling strength between oscillators, or by altering the free-running frequencies of the oscillators relative to each other.
This talk will describe a newly identified mechanism to alter the coordination between oscillators: changing the duration of the coupling influence between oscillators. I will show that changing the duration of influence changes the phase difference between oscillators linearly. A simple formula describes this effect, and applies to all the most widely used neuronal oscillator models.
This mechanism may apply to a wide array of oscillators, for example circadian oscillators. It may shed light on why some species exhibit seasonal changes in behaviour, and whether people can experience jet lag when traveling north/south.

9 March 2022
Prof Onno Bokhove (Leeds). Oblique interactions of extreme water waves.
The interactions of crossing seas with waves coming from two or three directions will be investigated. Water waves can be modelled in a hierarchy of mathematical and numerical models of varying complexity. These models include the commonly-used potential-flow models and approximations thereof, such as the unidirectional Kadomtsev-Petviashvili equation (KPE) and the bi-directional, higher-order Benney-Luke equations (BLE).
Exact solutions of the KPE will be presented with a fourfold maximum amplification for (solitary) waves coming from two directions and a ninefold maximum amplification, at one point in space and time, for (solitary) waves coming from three directions. Subsequently, these exact solutions will serve as seed, at a suitable initial time, for numerical simulations of the higher-order BLE. Simulations are undertaken with compatible, geometric numerical schemes, adhering to the variational structure of the basic equations. Hence, the question is to what extent these exact but idealised extreme-wave solutions can survive in more realistic settings or will fall apart due to dispersion or other mechanisms. What do you think: will we be able to reach the ninefold wave amplification in more realistic calculations or in reality?
The work was inspired by the following soliton-splash demonstrations:

30 March 2022
Prof Tamara Grava (Bristol and SISSA). Gibbs ensemble for Integrable Systems, a case study: the discrete nonlinear Schrodinger equation.
We consider discrete integrable systems with random initial data. In particular we consider the defocusing nonlinear Schrodinger equation in its integrable version, that is called Ablowitz Ladik lattice.
In the random initial data setting the Lax matrix of the Ablowitz Ladik lattice turns into a random matrix that is related to the circular beta-ensemble at high temperature. We obtain the density of states of the random Lax matrix, when the size of the matrix goes to infinity, by establishing a mapping to the one-dimensional log-gas. The density of states is obtained via a particular solution of the double-confluent Heun equation.
Joint work with Guido Mazzuca.

27 April 2022
Dr Stefan Ruschel (Auckland). Nonlinear Dynamics of Time-Delay Systems.
In this talk, I will present several analytical and numerically obtained results on the nonlinear dynamics of delay differential equations, including a characterisation of the eigenvalue spectrum of steady states with large delay and temporally localized solutions. Understanding the role of time-delays is crucial, here, as time-delays can dramatically alter the behaviour of a system possibly leading to oscillations, complex dynamical behaviour, and chaos. I will motivate the need to further improve these methods with the consequences and challenges posed by examples in Mathematical Biology, Climate dynamics, and Nonlinear Optics.

25 May 2022
Dr Arseni Goussev (Portsmouth). Quantum backflow: Overview and recent developments.
In quantum mechanics, the direction of the flow of probability density is not necessarily the same as that of momentum. One manifestation of this counter-intuitive fact is the so-called quantum backflow effect, in which the probability of finding a particle "on the left" increases with time despite the particle's momentum pointing "to the right." In my talk, I will review the quantum backflow effect and show its relation to other physical situations characterised by classically forbidden probability flow.

Contact for more information
Remy Dubertrand: remy.dubertrand(AT)

Research themes

Dispersive hydrodynamics

Dispersive wave interaction 1

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).

Integrability and Complexity

Integrability and Complexity are often viewed as antinomic concepts. The notion of Integrability is associated to systems that exhibit some form of regularity and therefore are "simple enough" to be solved analytically. The idea of Complexity arises in a variety of contexts where a system shows features such as chaos, disorder, instability, randomness.

The modern theory of integrable systems demonstrates that it is indeed possible to use analytical techniques to capture and describe complex behaviours in nonlinear and many-body systems. The research focuses on the method of differential identities as a unified approach to the solution of a general class of statistical mechanical systems, from classical "simple" systems, which exhibit Ising type phase transitions, to systems where complex behaviours emerge. Key features of a variety statistical mechanical models, e.g. critical and collective behaviours, scaling properties, can be understood and explained in terms of scaling properties and regularisation mechanisms of singularities of nonlinear integrable systems. Main research topics are:
  • Partition functions of statistical mechanical systems, random graph and random matrix models.
  • Singular limits of nonlinear integrable partial differential equations and multiscale asymptotics.
  • Classical and dispersive shocks of order parameters.
  • Classification of nonlinear integrable partial differential equations and normal forms.

Lead by: Antonio Moro

Key publications:
  • C. Benassi and A. Moro, Phys. Rev. E 101, 052118 (2020)
  • G. De Matteis, F. Giglio, and A. Moro, Annals of Physics 396 386-396 (2018)
  • B. Dubrovin, T. Grava, C. Klein, and A. Moro, J. Nonlinear Sci., 25 631-707 (2015)
  • A. Moro, Annals of Physics. 343 49-60 (2014)
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.

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)

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)

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)
  • New wave structures in systems with nonconvex dispersion (e.g. Benjamin-Bona-Mahony equation)
  • Interactions between nonlinear waves within the framework of wave-mean flow interaction

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)
  • S. K. Ivanov, A. M. Kamchatnov, T. Congy, and N. Pavloff, Phys. Rev. E 96, 062202 (2017)

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. 30, 737-766 (2020)
  • 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).

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, Phys. Rev. E 101, 052118 (2020)
  • 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)

Research relationships

This group is a part of the University’s multi-disciplinary research into the theme of Extreme Environments.

To view research papers emanating from this group, please click here to view Northumbria Research Link, our open access repository of research output from Northumbria University.


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