From 15f85d1826f2eb3893932ae7a731180c97e903c2 Mon Sep 17 00:00:00 2001 From: Michael Uleysky Date: Fri, 18 Jan 2019 13:41:40 +1000 Subject: [PATCH] Text about Bose condensates --- shred.tex | 126 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 126 insertions(+) create mode 100644 shred.tex diff --git a/shred.tex b/shred.tex new file mode 100644 index 0000000..eff8253 --- /dev/null +++ b/shred.tex @@ -0,0 +1,126 @@ +\documentclass[12pt]{article} +\usepackage{graphicx} +\usepackage{color} +\usepackage[utf8]{inputenc} +\usepackage[T2A]{fontenc} +\usepackage[russian]{babel} +\usepackage{amsmath} +\usepackage{amssymb} + +\topmargin=-1.8cm +\oddsidemargin=-15mm +\evensidemargin=-15mm +\textheight=24.5cm +\textwidth=19cm +\tolerance=1000 + +\parskip=5pt plus 4pt minus 2pt +\tolerance=9000 +% +\title{Двухкомпонентные Бозе-конденсаты} +\begin{document} +\maketitle +\section{Нелинейные конденсаты разной природы} +Система уравнений: +% +\begin{equation} +\begin{aligned} +i\hbar\frac{\partial\Psi_1}{\partial t}&=\left(-\frac{\hbar^2}{2m}\nabla^2+V(x)+g_1\lvert\Psi_1\rvert^2+k\lvert\Psi_2\rvert^2\right)\Psi_1,\\ +i\hbar\frac{\partial\Psi_2}{\partial t}&=\left(-\frac{\hbar^2}{2m}\nabla^2+V(x)+g_2\lvert\Psi_2\rvert^2+k\lvert\Psi_1\rvert^2\right)\Psi_2. +\end{aligned} +\label{nlinsys} +\end{equation} +% +Пространственная и временная сетки: +% +\begin{equation} +\begin{gathered} +\{x_j\}: x_{j+1}-x_j=\Delta x,\quad 0\leqslant j\leqslant N,\qquad \{t_n\}: t_{n+1}-t_n=\Delta t,\quad 0\leqslant n\leqslant M,\\ +u_j^n=\Psi_1(x_j,t_n),\quad v_j^n=\Psi_2(x_j,t_n),\quad u_0^n=u_N^n=v_0^n=v_N^n=0,\quad V(x_j)=V_j. +\end{gathered} +\label{nlingrid} +\end{equation} +% +Дискретизация Крэнка\,--\,Николсона: +\begin{equation} +\begin{aligned} +i\hbar\frac{1}{\Delta t}\left(u_j^{n+1}-u_j^n\right)&+\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}\left(u_{j+1}^n-2u_j^n+u_{j-1}^n+u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}\right)-\\ +&-\left(V_j+\frac{g_1}{2}\left(\lvert u_j^{n+1}\rvert^2+\lvert u_j^n\rvert^2\right)+\frac{k}{2}\left(\lvert v_j^{n+1}\rvert^2+\lvert v_j^n\rvert^2\right)\right)\frac{u_j^n+u_j^{n+1}}{2}=0,\\ +i\hbar\frac{1}{\Delta t}\left(v_j^{n+1}-v_j^n\right)&+\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}\left(v_{j+1}^n-2v_j^n+v_{j-1}^n+v_{j+1}^{n+1}-2v_j^{n+1}+v_{j-1}^{n+1}\right)-\\ +&-\left(V_j+\frac{g_2}{2}\left(\lvert v_j^{n+1}\rvert^2+\lvert v_j^n\rvert^2\right)+\frac{k}{2}\left(\lvert u_j^{n+1}\rvert^2+\lvert u_j^n\rvert^2\right)\right)\frac{v_j^n+v_j^{n+1}}{2}=0. +\end{aligned} +\end{equation} +В форме, подходящей для решения +\begin{equation} +\begin{aligned} +&\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}u_{j-1}^{n+1}+\left(i\hbar\frac{1}{\Delta t}-\frac{\hbar^2}{2m}\frac{1}{\Delta x^2}-\frac{V_j}{2}\right)u_j^{n+1}+\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}u_{j+1}^{n+1}=\\ +=&-\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}u_{j-1}^n+\left(i\hbar\frac{1}{\Delta t}+\frac{\hbar^2}{2m}\frac{1}{\Delta x^2}+\frac{V_j}{2}\right)u_j^n-\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}u_{j+1}^n+\\ +&+\frac{g_1\left(\lvert u_j^{n+1}\rvert^2+\lvert u_j^n\rvert^2\right)+k\left(\lvert v_j^{n+1}\rvert^2+\lvert v_j^n\rvert^2\right)}{4}\left(u_j^n+u_j^{n+1}\right),\\ +&\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}v_{j-1}^{n+1}+\left(i\hbar\frac{1}{\Delta t}-\frac{\hbar^2}{2m}\frac{1}{\Delta x^2}-\frac{V_j}{2}\right)v_j^{n+1}+\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}v_{j+1}^{n+1}=\\ +=&-\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}v_{j-1}^n+\left(i\hbar\frac{1}{\Delta t}+\frac{\hbar^2}{2m}\frac{1}{\Delta x^2}+\frac{V_j}{2}\right)v_j^n-\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}v_{j+1}^n+\\ +&+\frac{g_2\left(\lvert v_j^{n+1}\rvert^2+\lvert v_j^n\rvert^2\right)+k\left(\lvert u_j^{n+1}\rvert^2+\lvert u_j^n\rvert^2\right)}{4}\left(v_j^n+v_j^{n+1}\right). +\end{aligned} +\end{equation} + + +\section{Нелинейные конденсаты с взаимодействием Раби} +Система уравнений содержит дополнительный член по сравнению с \eqref{nlinsys} +% +\begin{equation} +\begin{aligned} +i\hbar\frac{\partial\Psi_1}{\partial t}&=\left(-\frac{\hbar^2}{2m}\nabla^2+V(x)+g_1\lvert\Psi_1\rvert^2+k\lvert\Psi_2\rvert^2\right)\Psi_1-\frac{\Omega}{2}\Psi_2,\\ +i\hbar\frac{\partial\Psi_2}{\partial t}&=\left(-\frac{\hbar^2}{2m}\nabla^2+V(x)+g_2\lvert\Psi_2\rvert^2+k\lvert\Psi_1\rvert^2\right)\Psi_2-\frac{\Omega}{2}\Psi_1. +\end{aligned} +\label{nlinrsys} +\end{equation} +% +Введём дополнительные обозначения +% +\begin{equation} +\begin{aligned} +&\phi=g_1\lvert\Psi_1\rvert^2+k\lvert\Psi_2\rvert^2, &\psi=g_2\lvert\Psi_2\rvert^2+k\lvert\Psi_1\rvert^2. +\end{aligned} +\label{nlinrnl} +\end{equation} +% +Сетка та же, что и в предыдущем случае \eqref{nlingrid} с дополнением для $\phi$ и $\psi$ +% +\begin{equation} +\begin{gathered} +\{x_j\}: x_{j+1}-x_j=\Delta x,\quad 0\leqslant j\leqslant N,\qquad \{t_n\}: t_{n+1}-t_n=\Delta t,\quad 0\leqslant n\leqslant M,\\ +u_j^n=\Psi_1(x_j,t_n),\quad v_j^n=\Psi_2(x_j,t_n),\quad u_0^n=u_N^n=v_0^n=v_N^n=0,\quad V(x_j)=V_j,\\ +\phi_j^{n+\frac12}=\phi(x_j,t_n+\Delta t/2),\qquad \psi_j^{n+\frac12}=\psi(x_j,t_n+\Delta t/2). +\end{gathered} +\label{nlinrgrid} +\end{equation} +% +Дискретизация Крэнка\,--\,Николсона: +\begin{equation} +\begin{gathered} +\begin{aligned} +i\hbar\frac{1}{\Delta t}\left(u_j^{n+1}-u_j^n\right)&+\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}\left(u_{j+1}^n-2u_j^n+u_{j-1}^n+u_{j+1}^{n+1}-2u_j^{n+1}+u_{j-1}^{n+1}\right)-\\ +&-\left(V_j+\phi_j^{n+\frac12}\right)\frac{u_j^n+u_j^{n+1}}{2}+\frac{\Omega}{2}\frac{v_j^n+v_j^{n+1}}{2}=0,\\ +i\hbar\frac{1}{\Delta t}\left(v_j^{n+1}-v_j^n\right)&+\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}\left(v_{j+1}^n-2v_j^n+v_{j-1}^n+v_{j+1}^{n+1}-2v_j^{n+1}+v_{j-1}^{n+1}\right)-\\ +&-\left(V_j+\psi_j^{n+\frac12}\right)\frac{v_j^n+v_j^{n+1}}{2}+\frac{\Omega}{2}\frac{u_j^n+u_j^{n+1}}{2}=0, +\end{aligned}\\ +\frac{\phi_j^{n+\frac12}+\phi_j^{n-\frac12}}{2}=g_1\lvert u_j^n\rvert^2+k\lvert v_j^n\rvert^2,\qquad +\frac{\psi_j^{n+\frac12}+\psi_j^{n-\frac12}}{2}=g_2\lvert v_j^n\rvert^2+k\lvert u_j^n\rvert^2. +\end{gathered} +\end{equation} + +В форме, удобной для решения +\begin{equation} +\begin{aligned} +&\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}u_{j-1}^{n+1}+\left(i\hbar\frac{1}{\Delta t}-\frac{\hbar^2}{2m}\frac{1}{\Delta x^2}-\frac{V_j+\phi_j^{n+\frac12}}{2}\right)u_j^{n+1}+\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}u_{j+1}^{n+1}+\frac\Omega4 v_j^{n+1}=\\ +=&-\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}u_{j-1}^n+\left(i\hbar\frac{1}{\Delta t}+\frac{\hbar^2}{2m}\frac{1}{\Delta x^2}+\frac{V_j+\phi_j^{n+\frac12}}{2}\right)u_j^n-\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}u_{j+1}^n-\frac\Omega4 v_j^n,\\ +&\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}v_{j-1}^{n+1}+\left(i\hbar\frac{1}{\Delta t}-\frac{\hbar^2}{2m}\frac{1}{\Delta x^2}-\frac{V_j+\psi_j^{n+\frac12}}{2}\right)v_j^{n+1}+\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}v_{j+1}^{n+1}+\frac\Omega4 u_j^{n+1}=\\ +=&-\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}v_{j-1}^n+\left(i\hbar\frac{1}{\Delta t}+\frac{\hbar^2}{2m}\frac{1}{\Delta x^2}+\frac{V_j+\psi_j^{n+\frac12}}{2}\right)v_j^n-\frac{\hbar^2}{4m}\frac{1}{\Delta x^2}v_{j+1}^n-\frac\Omega4 u_j^n. +\end{aligned} +\end{equation} + +В матричном виде +\begin{equation} +Ay^{n+1}=My^n,\qquad A=-\Re{M}+i\Im{M}, +\end{equation} +где $A$ и $M$~--- пятидиагональные матрицы. +\end{document} \ No newline at end of file