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Nonlinear Waves & Integrable Systems

Nonlinear phenomena appear everywhere in nature, from water waves to magnetic materials, from optics to weather forecasts, hence their description and understanding is of fundamental importance both from the theoretical and the applicative point of view. Nonlinear phenomena are generally described by differential equations whose solution often stands out as a challenging problem. Nevertheless, there is a special class of differential equations which are solvable (in some sense) - they are called integrable systems.

Many concepts of modern mathematical physics such as solitons, instantons and quantum groups have their origin in the theory of integrable systems. When a physical phenomenon is described by an integrable system, its behaviour can be understood globally and can be often predicted. The beauty of this theory lies in its universality: many fundamental nonlinear equations turn out to be both widely applicable and integrable. Moreover, in several cases of applicative relevance, nonintegrable nonlinear equations can be approximated, under certain assumptions, by nonlinear integrable equations, allowing a better understanding of the phenomena modelled by them.

Current research focuses on: exact and asymptotic methods for nonlinear PDEs; classification problems in the theory of nonlinear integrable PDEs; applications to classical electrodynamics - nonlinear optics, magnetic droplets, ferromagnetic nanostructures; biological systems; critical phenomena and shocks; Automorphic Lie Algebras (ALiAs) and algebraic structures underlying integrability theory.

The research on Automorphic Lie Algebras and on Dynamics of domain walls in ferromagnetic nanostructures has been supported by EPSRC grants. 

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