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Stationary Solutions For The 1+1 Nonlinear Schrodinger Equation Modeling Repulsive Bose-Einstein Condensates In Small Potentials, Kristina Mallory, Robert A. Van Gorder
Stationary Solutions For The 1+1 Nonlinear Schrodinger Equation Modeling Repulsive Bose-Einstein Condensates In Small Potentials, Kristina Mallory, Robert A. Van Gorder
Faculty Bibliography 2010s
Stationary solutions for the 1 + 1 cubic nonlinear Schrodinger equation modeling repulsive Bose-Einstein condensates (BEC) in a small potential are obtained through a form of nonlinear perturbation. In particular, for sufficiently small potentials, we determine the perturbation theory of stationary solutions, by use of an expansion in Jacobi elliptic functions. This idea was explored before in order to obtain exact solutions [Bronski, Carr, Deconinck, and Kutz, Phys. Rev. Lett. 86, 1402 (2001)], where the potential itself was fixed to be a Jacobi elliptic function, thereby reducing the nonlinear ODE into an algebraic equation, (which could be easily solved). However, …
Scaling Laws And Accurate Small-Amplitude Stationary Solution For The Motion Of A Planar Vortex Filament In The Cartesian Form Of The Local Induction Approximation, Robert A. Van Gorder
Scaling Laws And Accurate Small-Amplitude Stationary Solution For The Motion Of A Planar Vortex Filament In The Cartesian Form Of The Local Induction Approximation, Robert A. Van Gorder
Faculty Bibliography 2010s
We provide a formulation of the local induction approximation (LIA) for the motion of a vortex filament in the Cartesian reference frame (the extrinsic coordinate system) which allows for scaling of the reference coordinate. For general monotone scalings of the reference coordinate, we derive an equation for the planar solution to the derivative nonlinear Schrodinger equation governing the LIA. We proceed to solve this equation perturbatively in small amplitude through an application of multiple-scales analysis, which allows for accurate computation of the period of the planar vortex filament. The perturbation result is shown to agree strongly with numerical simulations, and …