Understanding the development of wave instabilities is a fundamental problem in mathematics and physics. Not only it is a theoretically challenging question per se, but it has relevant implications from the applied point of view, for many natural phenomena occur as a result of some instabilities. For instance, in nonlinear optics and fluid dynamics, a type of instability known as modulational instability has been proposed as a mechanism for the generation of rogue waves.
Recently, Degasperis, Lombardo & Sommacal (2018) introduced a new technique for investigating the stability of continuous wave solutions of 1+1, multicomponent, nonlinear partial differential equations of integrable type, such as the vector nonlinear Schrödinger equation or the three-wave resonant interaction system. This approach employs only the associated Lax pair, with no reference to boundary conditions, and allows to build the so-called stability spectra, providing a necessary condition in the parameters space for the onset of instability. The derivation of the spectra is completely algorithmic and relies on algebraic-geometry. It turns out that, for a Lax Pair that is polynomial in the spectral parameter, the problem of classifying the spectra is transformed into the problem of classifying certain algebraic varieties.
This project will utilise this new technique to extend the classes of integrable systems, as well as the families of solutions and perturbations, for which the stability spectrum can be constructed, allowing predictions and shining a new light on the mechanisms behind wave instabilities.
Depending on the successful candidate’s profile, the emphasis can be either on the algebraic-geometric study of the varieties associated to the stability spectra, or on the physical applications, or, ideally, on both aspects.
Working knowledge of Mathematica and Matlab is desirable. Knowledge in one or more of the following areas is desirable: (computational) algebraic geometry; integrable systems; nonlinear waves; numerical methods for PDEs.
The principal supervisor for this project is Matteo Sommacal.
Eligibility and How to Apply:
Please note eligibility requirement:
• Academic excellence of the proposed student i.e. 2:1 (or equivalent GPA from non-UK universities [preference for 1st class honours]); or a Masters (preference for Merit or above); or APEL evidence of substantial practitioner achievement.
• Appropriate IELTS score, if required.
• Applicants cannot apply for this funding if currently engaged in Doctoral study at Northumbria or elsewhere.
For further details of how to apply, entry requirements and the application form, see https://www.northumbria.ac.uk/research/postgraduate-research-degrees/how-to-apply/
Please note: Applications that do not include a research proposal of approximately 1,000 words (not a copy of the advert), or that do not include the advert reference (e.g. RDF19/EE/MPEE/SOMMACAL) will not be considered.
Deadline for applications: Friday 25 January 2019
Start Date: 1 October 2019
Northumbria University is an equal opportunities provider and in welcoming applications for studentships from all sectors of the community we strongly encourage applications from women and under-represented groups.
A. Degasperis, S. Lombardo, and M. Sommacal (2018) Integrability and linear stability of nonlinear waves. Journal of Nonlinear Science, 28 (4), 1251-1291.
B. R. Randoux, P. Suret, A. Chabchoub, B. Kibler, and G. El (2018) Nonlinear spectral analysis of Peregrine solitons observed in optics and in hydrodynamic experiments. Physical Review E, 98 (2), 022219.
C. M. Maiden, D. Anderson, N. Franco, G. El, and M. Hoefer (2018) Solitonic Dispersive Hydrodynamics: Theory and Observation. Physical Review Letters, 120 (14).
D. F. Demontis, G. Ortenzi, and M. Sommacal (2018) Heisenberg ferromagnetism as an evolution of a spherical indicatrix: localized solutions and elliptic dispersionless reduction. Electronic Journal of Differential Equations, 2018 (106), 1-34.
E. P. Suret, G. El, M. Onorato, and S. Randoux (2017) Rogue Waves in Integrable Turbulence: Semi-Classical Theory and Fast Measurements, in: Guided wave optics: A testbed for extreme events, S. Wabnitz (ed.), IOP e-book.
F. G. El, and M. Hoefer (2016) Dispersive shock waves and modulation theory, Physica D 333, 11–65.