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

玻色-爱因斯坦凝聚（BEC）是一种独特的物质状态，在宏观空间尺度上表现出量子特性。1995年稀碱气体的BEC首次实验实现后（2001年获得诺贝尔物理学奖），这一新的物理对象引起了非线性波动动力学专家的极大关注。这种兴趣的原因，特别地，与进行所谓物质波的非常微妙的实验的可能性有关。这些波代表了宏观尺度上量子行为的表现，与大多数量子力学效应不同，它们本质上是非线性的。事实上，在实验中，已经观察到一些不能用传统线性波动理论解释的效应。JILA Cornell小组（Boulder, Colorado）最近对BEC流过宏观障碍物的一系列实验之一揭示了BEC中特定激波的存在，类似于稀薄等离子体和浅水波纹孔中所谓的无碰撞激波。这种色散激波由大量相互作用的孤子（具有独特的类粒子行为的非线性孤子波）组成。本课题的目的是利用描述BEC动力学的一般数学模型Gross-Pitaevskii方程，建立BEC中色散激波的解析理论。这一理论将使人们能够理解和定量解释上述最近关于BEC流过障碍物的实验结果，并可能预测BEC中新的非线性色散波效应。该项目将需要对Gross-Pitaevskii方程的数学理论进行实质性的发展，这些理论可以被应用数学家和物理学家在其他物理环境中使用。

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.