Gross-Pitaevskii方程简化模型的几类精确解

Gross-Pitaevskii方程的精确解对理解玻色-爱因斯坦凝聚动力学演化具有重要作用.应用sine-cosine方法对Gross-Pitaevskii方程的简化模型进行了求解.获得了孤波解、三角函数周期波解等一些不同形式的精确解.     

A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N-1Va(xi-xj) where Va(x) = a-2V (x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schrdinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices {k ut, k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrdinger equation. Under the assumption that a = N-ε for 0 ε 3/4, we prove that as N →∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V (x) dx.     

Gross-Pitaevskii方程爆破解的极限图景

考虑Gross-Pitaevskii方程的爆破解. 利用集中紧原理和对应基态变分特征, 在对应能量空间中得到具临界质量爆破解的极限图景. 进一步利用变分法和尺度变换技术将上述结果推广到小超临界质量情形.     

几何规划问题的一种分枝定界方法

本文通过指数函数变换，把解几何规划GP（Ω）等价地转化为另外一个非线优化问题NLP（－↑Ω），根据问题（－↑Ω）的结构特征，构造它的一个线性规划松驰上确定它的最优值的一个下界，由此给出问题GP（Ω）的一个新的分枝定界算法。最后证明了这个算法是收敛的。     

SKYRMIONS IN GROSS-PITAEVSKII FUNCTIONALS

In Bose-Einstein condensates （BECs）, skyrmions can be characterized by pairs of linking vortex rings coming from two-component wave functions. Here we construct skyrmions by studying critical points of Gross-Pitaevskii functionals with two-component wave functions. Using localized energy method, we rigorously prove the existence, and describe the configurations of skyrmions in such BECs.     

数形结合解复合函数的零点个数的常见解法

本文通过利用函数图像的方法研究复合函数y=g（f（x））的零点问题,即复合函数方程g（f（x））=0的根,令u=f（x）（内层方程）,这样g（f（x））=0就转化成g（u）=0.当外层方程g（u）=0容易求解时,可以先解方程g（u）=0,再解内层方程u=f（x）,这样方程的总个数即为复合函数y=g（f（x））的零点个数.     

方程 λ(1 ce~(-τλ) a be~(-τλ)= 0所有根具有负实部的充要条件

考虑方程λ（１＋ｃｅ~（－τλ）＋ａ＋ｂｅ~（－τλ）＝０,（２）其中ａ,ｂ和ｃ为任意常数,τ为正常数,ｃ≠０．方程（１）为中立型方程　　　　　　　　ｘ（ｔ）＋ｃｘ（ｔ－τ）＋ａｘ（ｔ）＋ｂｘ（ｔ－τ）＝０　（２）的特征方程．方程（１）为一常见的拟多项式方程．关于拟多项式函数, Ｐｏｎｔｒｙａｇｉｎ在 １９４２年给出了判断这类函数所有零点位于左半复平面的充要条件．但对中立型方程来说,由于这些条件往往难以验证,使得人们长期以来无法用Ｐｏｎｔｒｙａｐｉｎ定理找出方程（１）所有根具有负实部的充要条件．本文在克服了上述困难后,用Ｐｏｎｔｒｇｉｎ定理找出方程（１）所有根具有负实部的充要条件．     

几类一阶微分方程的统一方程和统一变换

在高等数学的微分方程一章中,一阶变量可分离方程是最基本的方程,分离变量后积分,即得通解（通积分）：式中x（s）羊0（否则y一C）,C为积分常数（下同）．对于一阶非齐次线性方程由于其对应的齐次方程将常数U变易为函数U什）,即作变量代换方程（3）就变为关于函数X（X）和变量X的变量可分离方程积分后代入（5）式,即得非齐次线性方程（3）的通解对,右端为零次齐次。数。（于卜齐次方程当机y缸）学VX（否则,变量可分离）,作变换可得变量可分离方程UU‘＋U一J（U）,于是（7）的通解对于伯努利（Bernoulli）方程（n羊0,豆,否…     

Vlasov方程的精确解

该文将等熵磁流体力学（ＭＨＤ）或等熵电磁流体力学（ＥＭＨＤ）的基本方程组以及（非相对论的或相对论的）Ｖｌａｓｏｖ方程，分别化为等熵流体力学（ＨＤ）表象，建立了上述三类等熵方程之间的对应关系．从而使非相对论Ｖｌａｓｏｖ方程的精确解（它与等熵ＭＨＤ方程的精确解相对应）和相对论Ｖｌａｓｏｖ方程的精确解（它与等熵ＥＭＨＤ方程的精确解相对应）都可以用（非相对论的和相对论的）等熵ＨＤ方程的精确解来表示．     

从一道高考题看求解的转化策略

求解超越方程（指数、对数、三角、反三角方程），特别是合参数的方程，一般用等价转化的思想和方法，转化为代数方程求解．下面拟通过一道例题来探讨有关转化策略．例已知关于x的方程lg（ax）=2lg（x—1），（1）求a为何值时方程有解；（2）求出方程的解．（1986年广东省高考题改编）分析（1）即求方程有解的充分条件；（2）即求在（1）中条件下的解．原方程等价于即其中①、②两式成立，则ax＞0必成立．故③式可舍．这样原问题等价于：（1）求a的范围使厂～、T＿“”（。）”IxlM0成立；（2）求出（。）式中方程的解．说明上面通过把原…     

On the Marchenko System and the Long-time Behavior of Multi-soliton Solutions of the One-dimensional Gross-Pitaevskii Equation

We establish a rigorous well-posedness results for the Marchenko system associated to the scattering theory of the one dimensional Gross-Pitaevskii equation (GP). Under some assumptions on the scattering data, these well-posedness results provide regular solutions for (GP). We also construct particular solutions, called Nsoliton solutions as an approximate superposition of traveling waves. A study for the asymptotic behaviors of such solutions when t → ± ∞ is also made.     

The quintic NLS as the mean field limit of a boson gas with three-body interactions

We investigate the dynamics of a boson gas with three-body interactions in dimensions d=1,2. We prove that in the limit of infinite particle number, the BBGKY hierarchy of k-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. Factorized solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schrödinger equation. Our proof is based on, and extends, methods of Erdös-Schlein-Yau, Klainerman-Machedon, and Kirkpatrick-Schlein-Staffilani.     

On the linear wave regime of the Gross-Pitaevskii equation

We study long-wavelength asymptotics for the Gross-Pitaevskii equation corresponding to perturbations of a constant state of modulus one. We exhibit lower bounds on the first occurrence of possible zeros (vortices) and compare the solutions with the corresponding solutions to the linear wave equation or variants. The results rely on the use of the Madelung transform, which yields the hydrodynamical form of the Gross-Pitaevskii equation, as well as of an augmented system.     

Asymptotic limit of the Gross-Pitaevskii equation with general initial data

This paper mainly concerns the mathematical justification of the asymptotic limit of the Gross-Pitaevskii equation with general initial data in the natural energy space over the whole space. We give a rigorous proof of the convergence of the velocity fields defined through the solutions of the Gross-Pitaevskii equation to the strong solution of the incompressible Euler equations. Furthermore, we also obtain the rates of the convergence.     

Bounds on the tight-binding approximation for the Gross-Pitaevskii equation with a periodic potential

We justify the validity of the discrete nonlinear Schrödinger equation for the tight-binding approximation in the context of the Gross-Pitaevskii equation with a periodic potential. Both piecewise-constant and smooth potentials are considered in the semi-classical limit. While justification of stationary equations is developed in our previous work (Pelinovsky et al. (2008) [11]), this work deals with time-dependent space-decaying solutions on large but finite time intervals.     

Upper bound of weak-limitation for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of Gross-Pitaevskii equation. We obtain the upper bound of weak-limitation for the blow-up solutions by using the method of Cazenave (2003) [3] as well as the concentration compact principle.     

Eigenvalues of a nonlinear ground state in the Thomas-Fermi approximation

We study a nonlinear ground state of the Gross-Pitaevskii equation with a parabolic potential in the hydrodynamics limit often referred to as the Thomas-Fermi approximation. Existence of the energy minimizer has been known in literature for some time but it was only recently when the Thomas-Fermi approximation was rigorously justified. The spectrum of linearization of the Gross-Pitaevskii equation at the ground state consists of an unbounded sequence of positive eigenvalues. We analyze convergence of eigenvalues in the hydrodynamics limit. Convergence in norm of the resolvent operator is proved and the convergence rate is estimated. We also study asymptotic and numerical approximations of eigenfunctions and eigenvalues using Airy functions.     

Unconditional convergence of linearized implicit finite difference method for the 2D/3D Gross-Pitaevskii equation with angular momentum rotation

Science China Mathematics - This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum...     

Symplectic structure-preserving integrators for the two-dimensional Gross-Pitaevskii equation for BEC

Symplectic integrators have been developed for solving the two-dimensional Gross-Pitaevskii equation. The equation is transformed into a Hamiltonian form with symplectic structure. Then, symplectic integrators, including the midpoint rule, and a splitting symplectic scheme are developed for treating this equation. It is shown that the proposed codes fulfill the discrete charge conservation law. Furthermore, the global error of the numerical solution is theoretically estimated. The theoretical analysis is supported by some numerical simulations.