You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
126 lines
6.9 KiB
126 lines
6.9 KiB
\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} |