量子局域感应近似下超流涡丝上的量子Hasimoto变换和非线性波
Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation
摘要：
经典LIA（局域诱导近似，一种近似薄涡丝运动的模型）和非线性薛定谔方程（NLS）之间的Hasimoto变换在过去被证明是非常有用的，因为一旦已知NLS的解，它允许构造LIA的新解。在本文中，量子形式的LIA（其中包括相互摩擦效应）对应于一种复杂的非线性色散偏微分方程（PDE）与立方非线性（类似的形式Ginsburg-Landau方程，与额外的非线性项）。以这种方式转换量子LIA，使人们能够获得量子涡丝解决方案，一旦解决方案，这个色散PDE是已知的。从我们的量子Hasimoto变换，我们确定的形式和行为的斯托克斯波和一个常设的1-孤子解正常和副正常摩擦的影响。量子涡丝上的孤子解是经典1-孤子解的自然推广，该解由Hasimoto数学构造（激发了随后的真实世界实验）。当正常流体速度相对较弱时，量子Hasimoto变换是有用的，因此对于正常流体速度占主导地位的情况，我们采用其他方法。我们认为直接从量子LIA的涡丝的切向矢量的动力学，这种方法，而不是优雅的量子桥本变换，使我们能够研究波主要由正常的流体速度驱动。
英文摘要：
The Hasimoto transformation between the classical LIA (local induction approximation, a model approximating the motion of a thin vortex filament) and the nonlinear Schrödinger equation (NLS) has proven very useful in the past, since it allows one to construct new solutions to the LIA once a solution to the NLS is known. In the present paper, the quantum form of the LIA (which includes mutual friction effects) is put into correspondence with a type of complex nonlinear dispersive partial differential equation (PDE) with cubic nonlinearity (similar in form to a Ginsburg-Landau equation, with additional nonlinear terms). Transforming the quantum LIA in such a way enables one to obtain quantum vortex filament solutions once solutions to this dispersive PDE are known. From our quantum Hasimoto transformation, we determine the form and behavior of Stokes waves and a standing 1-soliton solution under normal and binormal friction effects. The soliton solution on a quantum vortex filament is a natural generalization of the classical 1-soliton solution constructed mathematically by Hasimoto (which motivated subsequent real-world experiments). The quantum Hasimoto transformation is useful when normal fluid velocity is relatively weak, so for the case where the normal fluid velocity is dominant we resort to other approaches. We consider the dynamics of the tangent vector to the vortex filament directly from the quantum LIA, and this approach, while less elegant than the quantum Hasimoto transformation, enables us to study waves primarily driven by the normal fluid velocity.
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