Text about Bose condensates

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+\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}