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.
Members

Staff   Research fellows   Research students  
Prof Gennady El Dr Costanza Benassi Adam Bridgewater
Dr Arseni GoussevDr Thibault CongyMarta dell'Atti
Dr Benoit HuardDavid Graham
Dr Oleg KirillovJoris Labarbe
Dr Yiping MaGiacomo Roberti
Dr Antonio MoroMarzia Romano
Dr Matteo SommacalOleg Senkevich
David Snee
Research projects





Dr Costanza Benassi
  • Nonlinear integrable systems and random matrix models.

  • Statistical mechanics and lattice spin systems.

  • Probabilistic representations of lattice models.

Collaborators: Dr. A. Moro, Prof. D. Ueltschi.








Dr Thibault Congy
  • Wave excitations in dispersive hydrodynamics (e.g. dispersive shock waves).

  • Integrable turbulence in the framework of the nonlinear Schrödinger equation.

Collaborators: G. El (Northumbria), M. Hoefer (University of Colorado, Boulder), M. Shearer (North Carolina State University).

Personal page: https://tcongy.github.io/








Prof Gennady El
  • Dispersive shock waves in integrable and non-integrable systems.

  • Integrable turbulence and soliton gas with applications in fibre optics.

  • Modulational instability and rogue waves: semi-classical theory.

Collaborators: M. Hoefer (University of Colorado, Boulder), M. Shearer (North Carolina State University), M. Bertola (Concordia University and SISSA), T. Congy (Northumbria), G. Roberti (PhD student, Northumbria), S. Randoux (University of Lille), P. Suret (University of Lille), A. Tovbis (University of Central Florida).

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

Full details in the Dispersive Hydrodynamics page.








Dr Arseni Goussev
  • Quantum wave packet dynamics (exactly solvable systems, diffraction in time, quantum time mirrors, Loschmidt echo, quantum chaos, rotating wave packets, quantum motion in the presence of absorption).

  • Dynamics of magnetic domains (Landau-Lifshitz-Gilbert equation in various geometries and topologies).

  • Semiclassical theory of chemical reactions (reactive scattering in phase space, transition state theory)
Funding: EPSRC (Ref. EP/K024116/1), 2013-2016.

Personal page: https://sites.google.com/site/aogscifi




Dr Benoit Huard
  • Periodic solutions in functional differential equations with application to glucose regulation.
    Collaborators: Maia Angelova (Deakin), Ruben Fossion (UNAM), Jiaxu Li (Louisville).
    Funding: This research is funded by a Newton Advanced fellowship from the British Academy of Sciences (R. Fossion).

  • Integrability and Riemann wave interactions in multidimensional quasilinear systems of the first order.
    Collaborators: Michel Grundland (Montreal), E.V. Ferapontov (Loughborough), Vladimir Novikov (Loughborough).
Further details: Personal page.








Dr Yiping Ma
  • Nonlinear waves in optical and mechanical topological insulators.

  • Localized pattern formation in driven dissipative systems.

  • Mathematical models of meltwater features in cold environments.

Further details: Personal website.








Dr Antonio Moro
Nonlinear Conservation laws and applications
  • 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.

  • Complete integrability and equations of state of mean field statistical mechanical models and random networks.

  • Integrability, Toda Lattice, KP and complexity in random matrix models.
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.

For more details click here.








Dr Matteo Sommacal
  • Nonlinear waves in ferromagnetic systems
    Short description: 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.

    Collaborators: Francesco Demontis and Cornelis van der Mee (University of Cagliari, Italy); Mark Hoefer (University of Colorado at Boulder, CO USA); Sara Lombardo (Loughborough University, UK); Giovanni Ortenzi (Università di Milano Bicocca, Italy); Thomas Silva (National Institute of Standards and Technology, Boulder, CO USA).

  • Integrability and linear stability of nonlinear waves
    Short description: Linear stability analysis of scalar and multi-component, nonlinear partial differential equations of integrable type. Manakov system. Vector nonlinear Schroedinger equation. Three waves resonant interaction equation.
    Collaborators: Antonio Degasperis (University of Rome "La Sapienza", Italy); Sara Lombardo (Loughborough University, UK).

  • Complex dynamics and Riemann surfaces
    Short description: 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.
    Collaborators: Francesco Calogero and Paolo Maria Santini (University of Rome "La Sapienza", Italy); David Gomez-Ullate (Universidad Complutense de Madrid, Spain).
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).




Research students

Current
Completed
Research seminars

The group runs the weekly Mathematics and Mathematical Physics series of research seminars and colloquia. Full details can be found at the seminars page.
Recent publications

2019

Kraych, Adrien, Suret, Pierre, El, Gennady and Randoux, Stephane (2019) Nonlinear evolution of the locally induced modulational instability in fiber optics. Physical Review Letters, 122. pp. 054101. ISSN 0031-9007

Haspolat, Emrah, Huard, Benoit and Angelova, Maia (2019) Deterministic and stochastic models of Arabidopsis Thaliana flowering. Bulletin of Mathematical Biology, 81 (1). pp. 277-311. ISSN 0092-8240

2018

Congy, Thibault, El, Gennady, Hoefer, Mark and Shearer, Michael (2018) Nonlinear Schrödinger equations and the universal description of dispersive shock wave structure. Studies in Applied Mathematics. ISSN 0022-2526 (In Press)

Demontis, Francesco, Lombardo, Sara, Sommacal, Matteo, van der Mee, Cornelis and Vargiu, F. (2018) Effective generation of closed-form soliton solutions of the continuous classical Heisenberg ferromagnet equation. Communications in Nonlinear Science and Numerical Simulation, 64. pp. 35-65. ISSN 1007-5704

De Matteis, Giovanni, Giglio, Francesco and Moro, Antonio (2018) Exact equations of state for nematics. Annals of Physics, 396. pp. 386-396. ISSN 0003-4916

Randoux, Stéphane, Suret, Pierre, Chabchoub, Amin, Kibler, Bertrand and El, Gennady (2018) Nonlinear spectral analysis of Peregrine solitons observed in optics and in hydrodynamic experiments. Physical Review E, 98 (2). 022219. ISSN 2470-0045

Degasperis, Antonio, Lombardo, Sara and Sommacal, Matteo (2018) Integrability and linear stability of nonlinear waves. Journal of Nonlinear Science, 28 (4). pp. 1251-1291. ISSN 0938-8974

Goussev, Arseni, Reck, Phillipp, Moser, Florian, Moro, Antonio, Gorini, Cosimo and Richter, Klaus (2018) Overcoming dispersive spreading of quantum wave packets via periodic nonlinear kicking. Physical Review A (PRA), 98. 013620. ISSN 2469-9926

Demontis, Francesco, Ortenzi, Giovanni and Sommacal, Matteo (2018) Heisenberg ferromagnetism as an evolution of a spherical indicatrix: localized solutions and elliptic dispersionless reduction. Electronic Journal of Differential Equations, 2018 (106). pp. 1-34. ISSN 1072-6691

Maiden, Michelle D., Anderson, Dalton V., Franco, Nevil A., El, Gennady and Hoefer, Mark A. (2018) Solitonic Dispersive Hydrodynamics: Theory and Observation. Physical Review Letters, 120 (14). ISSN 0031-9007

Demontis, Francesco, Ortenzi, Giovanni, Sommacal, Matteo and van der Mee, Cornelis (2018) The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. II. IST and closed-form soliton solutions. Ricerche di Matematica. ISSN 0035-5038 (In Press)

Demontis, Francesco, Ortenzi, Giovanni, Sommacal, Matteo and van der Mee, Cornelis (2018) The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. I. Direct and inverse scattering theory. Ricerche di Matematica. ISSN 0035-5038 (In Press)

Sprenger, Patrick, Hoefer, Mark and El, Gennady (2018) Hydrodynamic optical soliton tunneling. Physical Review E, 97 (3). ISSN 2470-0045

Reck, Phillipp, Gorini, Cosimo, Goussev, Arseni, Krueckl, Viktor, Fink, Mathias and Richter, Klaus (2018) Towards a quantum time mirror for nonrelativistic wave packets. New Journal of Physics, 20. 033013. ISSN 1367-2630

El, Gennady, Nguyen, Lu Trong Nguyen and Smyth, Noel (2018) Dispersive shock waves in systems with nonlocal dispersion of Benjamin–Ono type. Nonlinearity, 31 (4). pp. 1392-1416. ISSN 0951-7715

El, Gennady, Hoefer, Mark A. and Shearer, Michael (2018) Stationary Expansion Shocks for a Regularized Boussinesq System. Studies in Applied Mathematics, 140 (1). pp. 27-47. ISSN 0022-2526

2017

Suret, Pierre, El, Gennady, Onorato, Miguel and Randoux, Stéphane (2017) Rogue waves in integrable turbulence: semi-classical theory and fast measurements. In: Nonlinear Guided Wave Optics: A testbed for extreme waves. IOP Publishing, 12-1-12-32. ISBN 978-0-7503-1460-2

Gaididei, Yuri, Goussev, Arseni, Kravchuk, Volodymyr, Pylypovskyi, Oleksandr, Robbins, Jonathan, Sheka, Denis, Slastikov, Valeriy and Vasylkevych, Sergiy (2017) Magnetization in narrow ribbons: curvature effects. Journal of Physics A: Mathematical and Theoretical, 50. p. 385401. ISSN 1751-8113

Hoefer, Mark, El, Gennady and Kamchatnov, Anatoly (2017) Oblique Spatial Dispersive Shock Waves in Nonlinear Schrödinger Flows. SIAM Journal on Applied Mathematics, 77 (4). pp. 1352-1374. ISSN 0036-1399

Tikan, Alexey, Billet, Cyril, El, Gennady, Tovbis, Alexander, Bertola, Marco, Sylvestre, Thibaut, Gustave, Francois, Randoux, Stephane, Genty, Goëry, Suret, Pierre and Dudley, John M. (2017) Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation. Physical Review Letters, 119 (3). 033901. ISSN 0031-9007

Goussev, Arseni (2017) Rotating Gaussian wave packets in weak external potentials. Physical Review A (PRA), 96. 013617. ISSN 1050-2947

Randoux, Stéphane, Gustave, François, Suret, Pierre and El, Gennady (2017) Optical Random Riemann Waves in Integrable Turbulence. Physical Review Letters, 118 (23). ISSN 0031-9007

Reck, Phillipp, Gorini, Cosimo, Goussev, Arseni, Krueckl, Viktor, Fink, Mathias and Richter, Klaus (2017) Dirac quantum time mirror. Physical Review B, 95 (16). p. 165421. ISSN 2469-9950

Huard, Benoit, Bridgewater, Adam and Angelova, Maia (2017) Mathematical investigation of diabetically impaired ultradian oscillations in the glucose-insulin regulation. Journal of Theoretical Biology, 418. pp. 66-76. ISSN 0022-5193

El, Gennady, Hoefer, Mark and Shearer, Michael (2017) Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws. SIAM Review, 59 (1). pp. 3-61. ISSN 0036-1445

Chesnokov, A. A., El, Gennady, Gavrilyuk, S. L. and Pavlov, M. V. (2017) Stability Of Shear Shallow Water Flows with Free Surface. SIAM Journal on Applied Mathematics, 77 (3). pp. 1068-1087. ISSN 0036-1399

2016

El, Gennady and Hoefer, Mark (2016) Dispersive shock waves and modulation theory. Physica D: Nonlinear Phenomena, 333. pp. 11-65. ISSN 0167-2789

Giglio, Francesco, Landolfi, Giulio and Moro, Antonio (2016) Integrable extended van der Waals model. Physica D: Nonlinear Phenomena, 333. pp. 293-300. ISSN 0167-2789

Tovbis, Alexander and El, Gennady (2016) Semiclassical limit of the focusing NLS: Whitham equations and the Riemann–Hilbert Problem approach. Physica D: Nonlinear Phenomena, 333. pp. 171-184. ISSN 0167-2789

Bertola, Marco, El, Gennady and Tovbis, Alexander (2016) Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472 (2194). p. 20160340. ISSN 1364-5021

Onorato, Miguel, Proment, Davide, El, Gennady, Randoux, Stephane and Suret, Pierre (2016) On the origin of heavy-tail statistics in equations of the Nonlinear Schrödinger type. Physics Letters A, 380 (39). pp. 3173-3177. ISSN 0375-9601

El, Gennady, Khamis, Eduardo and Tovbis, Alex (2016) Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves. Nonlinearity, 29 (9). pp. 2798-2836. ISSN 0951-7715

Randoux, Stéphane, Suret, Pierre and El, Gennady (2016) Inverse scattering transform analysis of rogue waves using local periodization procedure. Scientific Reports, 6 (1). p. 29238. ISSN 2045-2322

Goussev, Arseni, Jalabert, Rodolfo, Pastawski, Horacio and Wisniacki, Diego (2016) Loschmidt echo and time reversal in complex systems. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374 (2069). p. 20150383. ISSN 1364-503X

El, Gennady, Hoefer, Mark and Shearer, Michael (2016) Expansion shock waves in regularized shallow-water theory. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472 (2189). p. 20160141. ISSN 1364-5021

Goussev, Arseni, Robbins, Jonathan, Slastikov, Valeriy and Tretiakov, Oleg (2016) Dzyaloshinskii-Moriya domain walls in magnetic nanotubes. Physical Review B (PRB), 93 (5). 054418. ISSN 1098-0121

2015

Barbier, Maximilien, Beau, Mathieu and Goussev, Arseni (2015) Comparison between two models of absorption of matter waves by a thin time-dependent barrier. Physical Review A (PRA), 92 (053630). ISSN 1050-2947

Huard, Benoit, Easton, Jonathan and Angelova, Maia (2015) Investigation of stability in a two-delay model of the ultradian oscillations in glucose-insulin regulation. Communications in Nonlinear Science and Numerical Simulation, 26 (1-3). pp. 211-222. ISSN 1007-5704

Barra, Adriano and Moro, Antonio (2015) Exact solution of the van der Waals model in the critical region. Annals of Physics, 359. pp. 290-299. ISSN 0003-4916

Arsie, Alessandro, Lorenzoni, Paolo and Moro, Antonio (2015) Integrable viscous conservation laws. Nonlinearity, 28 (6). pp. 1859-1895. ISSN 0951-7715

Dubrovin, Boris, Grava, Tamara, Klein, Christian and Moro, Antonio (2015) On critical behaviour in systems of Hamiltonian partial differential equations. Journal of Nonlinear Science, 25 (3). pp. 631-707. ISSN 0938-8974

Goussev, Arseni (2015) Manipulating quantum wave packets via time-dependent absorption. Physical Review A (PRA), 91 (4). 043638. ISSN 1050-2947

2014

Barra, Adriano, Guerra, Francesco, Di Lorenzo, Andrea and Moro, Antonio (2014) On quantum and relativistic mechanical analogues in mean field spin models. Proceedings of the Royal Society A, 470 (2172). ISSN 1471-2946

Arsie, Alessandro, Lorenzoni, Paolo and Moro, Antonio (2014) On integrable conservation laws. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (201401). ISSN 1471-2946

Moro, Antonio (2014) Shock dynamics of phase diagrams. Annals of Physics, 343. pp. 49-60. ISSN 0003-4916

Goussev, Arseni, Robbins, Jonathan and Slastikov, Valeriy (2014) Domain wall motion in thin ferromagnetic nanotubes: Analytic results. Europhysics Letters, 105 (6). p. 67006. ISSN 0295-5075

Dubertrand, Remy and Goussev, Arseni (2014) Origin of the exponential decay of the Loschmidt echo in integrable systems. Physical Review E, 89. 022915. ISSN 1539-3755

Moro, Antonio and Trillo, Stefano (2014) Mechanism of wave breaking from a vacuum point in the defocusing nonlinear Schrödinger equation. Physical Review E (PRE), 89 (2). ISSN 1550-2376

de Nittis, Giuseppe, Lorenzoni, Paolo and Moro, Antonio (2014) Integrable multi-phase thermodynamic systems and Tsallis' composition rule. Journal of Physics: Conference Series, 482 (1). 012009. ISSN 1742-6588

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