非線性薛丁格方程組的極限問題與泊松-能斯特-普朗克方程組的穩態解

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.