DMS 9803480 Paul Roberts, Natalia Berloff GENERALIZED NONLINEAR SCHRODINGER EQUATIONS AND APPLICATIONS TO SUPERFLUID TURBULENCE The proposed research has three closely related objectives. The first is to develop and study new variants of the nonlinear Schroedinger equation (NLS) that are more faithful to superfluid helium. In real helium, even in the low temperature range, normal fluid is present which is coupled to the superfluid and which, through its viscosity, provides a high wavenumber energy sink. So the existing models need to be modified to include the mutual friction with the normal fluid. To evaluate the energy loss of interactions with the normal fluid one needs the kinetic coefficients in the Landau two-fluid model. Calculations of these coefficients are usually performed with a simplistic dispersion relation, i.e., the relation between the frequency of sound waves and their wavenumber. For example, the dispersion relation is supposed anomalous and with no roton minimum. We shall study models in which the interaction potential is nonlocal and which imply the correct dispersion curve. The second objective is to analyze such dissipative NLS models from the point of view of dynamical systems. We shall study and classify attractors, bifurcation sequences, and routes to chaos. The third objective is to make use of the results to elucidate superfluid turbulence in the low temperature regime where the density of the normal fluid component is smaller than the density of the superfluid component and where one may expect turbulence in the superfluid largely to determine turbulence in the normal fluid. Our investigations will provide targets for experimental work in the low temperature range. Such experiments are currently in the planning stage by the Donnelly group at the University of Oregon. There may be important implications well beyond the field of liquid helium research: similar ideas may apply to the newly discovered, high temperature super conductors, to systems of magnetic spins, to the melting transition of crystals, and to some cryogenic engineering applications. Superfluid Helium is used as a coolant in superconducting magnets and infrared detectors, and superfluid turbulence affects the transfer of heat in such devices. An understanding of superfluid turbulence may make it possible to design more efficient methods of refrigeration for superconducting devices. Turbulence in ordinary fluids is one of the most formidably difficult subjects in physics and engineering. Superfluid turbulence is similar but in some respects simpler for theoretical study, and it is believed that some of the models we shall develop will throw light onto turbulent behaviors in ordinary fluids.
