Generalized Nonlinear Schrodinger Equations and Applications to Superfluid Turbulence

Paul Roberts， Natalia Berloff广义非线性薛定谔方程及其在超流体湍流中的应用提出的研究有三个密切相关的目标。首先是开发和研究更符合超流氦的非线性薛定谔方程（NLS）的新变体。在真实的氦中，即使在低温范围内，也存在与超流体耦合的正常流体，并且通过其粘度提供高波数能量汇。因此，现有的模型需要修改，以包括与正常流体的相互摩擦。为了计算与正常流体相互作用的能量损失，需要使用朗道双流体模型中的动力学系数。这些系数的计算通常采用简单的色散关系，即声波频率与其波数之间的关系。例如，色散关系假定是反常的，并且没有旋转最小值。我们将研究相互作用势是非局域的并且包含正确色散曲线的模型。第二个目标是从动力系统的角度分析这种耗散NLS模型。我们将研究并分类吸引子、分岔序列和通往混沌的路径。第三个目标是利用这些结果来阐明低温状态下的超流体湍流，在低温状态下，正常流体组分的密度小于超流体组分的密度，并且人们可以期望超流体中的湍流在很大程度上决定正常流体中的湍流。我们的研究将为低温范围内的实验工作提供目标。俄勒冈大学的唐纳利小组目前正处于计划阶段。这可能有重要的意义，远远超出了液氦研究领域：类似的想法可能适用于新发现的高温超导体，磁自旋系统，晶体的熔化转变，以及一些低温工程应用。超流氦被用作超导磁体和红外探测器中的冷却剂，超流湍流会影响这些设备中的热量传递。对超流体湍流的理解可能使设计更有效的超导装置制冷方法成为可能。普通流体中的湍流是物理学和工程学中最难研究的课题之一。超流体湍流是类似的，但在某些方面理论研究更简单，相信我们将开发的一些模型将有助于了解普通流体中的湍流行为。

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.