Copy of Generation of spatial dispersive shocks in the supersonic flow of Bose-Einstein condensate past an obstacle

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Bose-Einstein condensate (BEC) represents a unique state of matter demonstrating quantum properties on macroscopic spatial scales. After the first experimental realisation of the BEC of dilute alkali gases in 1995 (followed by the Nobel Prize for Physics in 2001) this new physical object has attracted a great deal of attention of specialists in nonlinear wave dynamics. The reason of this interest is, in particular, connected with a possibility to conduct very subtle experiments where so-called matter waves are realised. These waves represent a manifestation of quantum behavior at macroscopic scales and, unlike most quantum mechanics effects, are essentially nonlinear. Indeed, in the experiments, a number of effects have been observed which cannot be explained using conventional linear wave theory. One of the series of recent experiments of JILA Cornell group (Boulder, Colorado) on BEC flow past macroscopic obstacles revealed tthe existence of specific shock waves in the BEC similar to the so-called collisionless shocks in rarefied plasmas and undular bores on shallow water. Such dispersive shock waves consist of a large number of interacting solitons (nonlinear solitary waves with the unique particle-like behaviour). The aim of the present project is, using the Gross-Pitaevskii equation, which is a general mathematical model describing BEC dynamics, to construct an analytical theory of the dispersive shock waves in BEC. This theory will enable one to understand and quantitatively explain results of the mentioned recent experiments on the BEC flows past obstacles and, possibly, to predict new nonlinear dispersive wave effects in BEC. The project will require a substantial development of the mathematical theory of the Gross-Pitaevskii equation, which could be used by applied mathematicians and physicists in other physical contexts.