One Hat Cyber Team

Your IP : 216.73.216.216
Server IP : 194.44.31.54
Server : Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software : Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version : 5.6.40
Buat File | Buat Folder
Dir : ~/home/congress/3new/
View File Name : sidorenko.tex

\documentclass[12pt]{article}
\usepackage[koi8-u]{inputenc}
\usepackage[english,ukrainian]{babel}
\textheight 255mm
\textwidth 170mm
\pagestyle{headings}
\hoffset=-25mm
\voffset=-45mm
\begin{document}
\title {Dressing Method and Generalized Darboux Transformations}
\author {Sidorenko Yu.M. (Lviv National University)}
\date{}
\maketitle
Let us consider  a system of linear evolution equations (1): 
$$ 
\left\{  
\begin{array}{rcl}
\alpha f_{y} & = & \sum \limits_{i=0}^{n_{1}} u_{1i}(x,y,t) f^{(i)} :=L(f) \\
\beta  f_{t} & = & \sum \limits_{i=0}^{n_{2}} u_{2i}(x,y,t) f^{(i)} :=M(f)
\end{array} \ \  f^{(i)}:=\partial^{i}f(x,y,t) / \partial x^{i}; 
\alpha , \beta \in {\bf C}, \right.
\eqno(1)
$$
and formally transponed system: 
$$ 
\left\{  
\begin{array}{rcl}
\alpha g_{y} & = & - \sum \limits_{i=0}^{n_{1}} (-1)^{i} 
(u_{1i}^{T}(x,y,t) g)^{(i)} :=L^{\tau }(g) \\
\beta  g_{t} & = & - \sum \limits_{i=0}^{n_{2}} (-1)^{i} 
(u_{2i}^{T}(x,y,t) g)^{(i)} :=M^{\tau }(g)
\end{array} 
\right.
\eqno(2)
$$
($ " T " $ is a symbol of transposition in matrix ring) 
with smooth $(N\times N)$--matrix coeficients 
$ u_{ij} \in {\bf C}^{(\infty )}({\bf R}^{3} \to {\rm Mat}_{N\times N} 
({\bf C});  j=\overline{0,n_{i}}, i=1,2. $ 

Let $\varphi, \psi \in {\bf C}^{(\infty)}({\bf R}^3 \to {\rm Mat}_{N \times K}({\bf C}))$ be smooth
($N\times K$)-matrix solutions of the system (1), (2) correspondigly,
$C=const \in {\rm Mat}_{K\times K}({\bf C})$ and 
$\Omega(x,y,t):=C+\int_{-\infty}^x\psi^{T}\varphi ds$ is nondegenerate
in $(x,y,t)\in \sigma \subset {\bf R}^2\times{\bf R}_+$.
Define the functions $\Phi:=\Phi(x,y,t),\Psi:=\Psi(x,y,t)$
and integral operator $W$ by the following way: 
$\Phi:=\varphi\Omega^{-1}, \Psi^{T}=\Omega^{-1}\psi^{T},
W(f)=f-\int_{-\infty}^x\Phi(x,y,t)\psi^{T}(s,y,t)f(s,y,t)ds.$

{\bf Definition 1.} Let $f$ be an arbitrary solution of the
system (1). We define generalized binary
Darboux-type trasformation $BD_{(\varphi, \psi)}$
of the solution of the system (1) by the formula:
$f\to F:=W(f)$

{\bf Theorem.} \\ 1. Operator $W$ has an inverse $W^{-1}$ in explicit 
form:
$$
W^{-1}=I+\int\limits_{-\infty}^x\varphi (x,y,t)\Psi^{T}(s,y,t)\cdot ds.
$$ 
\\ 2. The function $F$ is a solution of system:
$$
\left\{
\begin{array}{l}
\alpha F_{y} = \sum \limits_{i=0}^{n_{1}} {{\tilde u}_{1i}}(x,y,t) F^{(i)}: =
{\tilde L}(F), \\[5mm]
\beta  F_{t} = \sum \limits_{i=0}^{n_{2}} {{\tilde u}_{2i}}(x,y,t) F^{(i)}: =
{\tilde M}(F), 
\end{array}
\right.  \eqno(1^*)
$$
where coefficients $ {\tilde u}_{ji}, j=1,2, i=\overline{0,n_{j}}, $
are differential polynomials of functions $ u_{ji}$, $\varphi$ ,  
$\varphi^{*}$, $ \Omega^{-1}[\varphi^{*}, \varphi ]. $  

{\bf Definition 2. } The operators $ {\tilde L}, {\tilde M} $ of the form $(1^*)$ are 
called dressed by the help of generalized binary Darboux transformation: 
$$
\alpha\frac{\partial}{\partial y}-{\tilde L}:=W(\alpha\frac{\partial}
{\partial y}-L)W^{-1},
\beta\frac{\partial}{\partial t}-{\tilde M}:=W(\beta\frac{\partial}
{\partial t}-M)W^{-1}.
$$
\end{document}