Supersymmetric solvable and integrable systems in quantum and classical physics

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In this research program, I am proposing two work themes which are all related to the study of solvable and integrable supersymmetric systems appearing in quantum and classical physics using group theoretical methods. In particle physics, they are describing in a unified way bosons and fermions. Supersymmetry is also used to include gravitation in the unified scheme involving the different forces of nature. In the first theme, I intend to work on generating new solutions for supersymmetric non-linear differential equations, such as the Korteweg de Vries equation. These solutions are called multi-solitons (or multi solitary waves) as they are well-localised waves that maintain their shape even after interactions with other such waves. In the supersymmetric context, there is a large body of equations to be solved. I hope to extend symmetry reduction and bilinearisation methods to an extensive range of new systems of equations. I will also work with supersymmetric extensions of models used in gauge theories that have been very successful in elementary particle physics. In this work, important properties of supersymmetric sigma models will be exhibited with the aim to better understand the geometry of these systems. In the second theme, I plan to relate known multidimensional integrable quantum systems through supersymmetry and thus determine explicitly the solutions. I also plan the establish relations between supercharges and accidental degeneracy in the spectrum. Finally, from my work on coherent and squeezed states for the one-dimensional Morse potential, I plan to establish rigorously connections between quantum and classical states. Such states are important because they are, in particular, related to the quantum treatment of optical coherence and behave in a similar fashion to classical states for oscillating systems. Nowadays, they are seen as an important instrument for studying quantum information processes and the question of how our different constructions will fit within this scheme is relevant for future developments in the field.