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\documentclass[12pt]{article} |
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\usepackage{graphicx} |
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\usepackage{color} |
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\usepackage[utf8]{inputenc} |
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\usepackage[T2A]{fontenc} |
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\usepackage[russian]{babel} |
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\usepackage{amsmath} |
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\usepackage{amssymb} |
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\topmargin=-1.8cm |
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\oddsidemargin=-15mm |
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\evensidemargin=-15mm |
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\textheight=24.5cm |
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\textwidth=19cm |
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\tolerance=1000 |
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\parskip=5pt plus 4pt minus 2pt |
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\tolerance=9000 |
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% |
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\title{Двухкомпонентные Бозе-конденсаты} |
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\begin{document} |
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\maketitle |
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\section{Нелинейные конденсаты разной природы} |
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Система уравнений: |
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% |
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\begin{equation} |
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\begin{aligned} |
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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,\\ |
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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. |
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\end{aligned} |
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\label{nlinsys} |
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\end{equation} |
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% |
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Пространственная и временная сетки: |
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% |
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\begin{equation} |
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\begin{gathered} |
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\{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,\\ |
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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. |
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\end{gathered} |
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\label{nlingrid} |
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\end{equation} |
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% |
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Дискретизация Крэнка\,--\,Николсона: |
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\begin{equation} |
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\begin{aligned} |
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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)-\\ |
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&-\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,\\ |
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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)-\\ |
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&-\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. |
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\end{aligned} |
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\end{equation} |
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В форме, подходящей для решения |
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\begin{equation} |
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\begin{aligned} |
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&\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}=\\ |
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=&-\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+\\ |
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&+\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),\\ |
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&\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}=\\ |
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=&-\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+\\ |
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&+\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). |
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\end{aligned} |
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\end{equation} |
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\section{Нелинейные конденсаты с взаимодействием Раби} |
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Система уравнений содержит дополнительный член по сравнению с \eqref{nlinsys} |
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% |
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\begin{equation} |
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\begin{aligned} |
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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,\\ |
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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. |
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\end{aligned} |
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\label{nlinrsys} |
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\end{equation} |
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% |
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Введём дополнительные обозначения |
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% |
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\begin{equation} |
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\begin{aligned} |
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&\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. |
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\end{aligned} |
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\label{nlinrnl} |
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\end{equation} |
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% |
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Сетка та же, что и в предыдущем случае \eqref{nlingrid} с дополнением для $\phi$ и $\psi$ |
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% |
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\begin{equation} |
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\begin{gathered} |
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\{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,\\ |
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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,\\ |
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\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). |
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\end{gathered} |
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\label{nlinrgrid} |
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\end{equation} |
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% |
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Дискретизация Крэнка\,--\,Николсона: |
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\begin{equation} |
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\begin{gathered} |
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\begin{aligned} |
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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)-\\ |
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&-\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,\\ |
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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)-\\ |
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&-\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, |
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\end{aligned}\\ |
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\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 |
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\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. |
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\end{gathered} |
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\end{equation} |
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В форме, удобной для решения |
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\begin{equation} |
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\begin{aligned} |
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&\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}=\\ |
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=&-\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,\\ |
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&\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}=\\ |
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=&-\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. |
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\end{aligned} |
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\end{equation} |
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В матричном виде |
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\begin{equation} |
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Ay^{n+1}=My^n,\qquad A=-\Re{M}+i\Im{M}, |
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\end{equation} |
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где $A$ и $M$~--- пятидиагональные матрицы. |
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\end{document} |
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