Limit Problems of Solutions for the Coupled Nonlinear Schr?dinger Equations and Steady-state Solutions of the Poisson-Nernst-Planck Systems
Date Issued
2010
Date
2010
Author(s)
Lee, Chiun-Chang
Abstract
In this thesis, we investigate two different types partial differential equations, one is the coupled nonlinear Schrödinger equations and the other is the renormalize Poisson-Boltzmann equations (the steady-state solutions of the Poisson-Nernst-Planck systems).
Recently, a rich variety of dynamical phenomena and a turbulent relaxation have been observed in rotating Bose-Einstein condensates depicted by Gross-Pitaevskii equations coupled with rotating fields and trap potentials. The dynamical phenomena range from shock-wave formation to anisotropic sound propagation. The turbulent relaxation leads to the crystallization of vortex lattices. To see the dynamical phenomena and the turbulent relaxation of two-component rotating Bose-Einstein condensates, we study the incompressible and the compressible limits of two-component systems of Gross-Pitaevskii equations. Our arguments generalize the idea of [22] and define "H-function" a modulated energy functional which may control the propagation of densities and linear momentums under the effect of rotating fields and trap potentials.
The Poisson-Boltzmann (PB) equation is conventionally used to model the equilibrium of bulk ionic species in different media and solvents. We study a renormalized Poisson-Boltzmann (RPB) equation with a small dielectric parameter ∈2 and nonlocal nonlinearity which takes into consideration of the preservation of the total amounts of each individual ion. This equation can be derived from the original Poisson-Nernst-Planck (PNP) system. Under Robin type boundary conditions with various coefficient scales, we demonstrate the asymptotic behaviors of one dimensional solutions of RPB equations as the parameter $epsilon$ approaches to zero. In particular, we show that in case of electro-neutrality, i.e., (∑N1 κ=1 ακακ=∑N2 l=1 blβl), we prove that φ∈''s solutions of 1-D RPB equations may tend to a nonzero constant c at every interior point as $epsilon$ goes to zero. The value c can be uniquely determined by ακ, bl''s valences of ions, ακ, βl''s total concentrations of ions and the limit of φ∈''s at the boundary x=±1. In particular, when N1=1, N2=2, a1=b1=1 and b2=2, a precise formula of the value c and the ratio β1/β2 is given in (4.1.3). Such a result can not be found in conventional 1-D Poisson-Boltzmann (PB) equations. On the other hand, as (∑N1 κ=1 ακακ≠∑N2 l=1 blβl) (non-electroneutrality), solutions of 1-D RPB equations have blow-up behavior which also may not be obtained in 1-D PB equations.
Subjects
Bose-Einstein condensates
Schrodinger equation
compressible limit
incompressible limit
Poisson-Nernst-Planck Equation
Poisson-Boltzmann Equation
Type
thesis
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