# Integration of the nonlinear Schrödinger equation with a self-consistent
source and nonzero boundary conditions

Anvar Reyimberganov  Urgench State University, Uzbekistan<EMAIL_ADDRESS>http://www.urdu.uz

###### Abstract.

This paper is devoted to the study of the defocusing nonlinear Schrödinger
equation with a self-consistent source and nonzero boundary conditions by the
method of the inverse scattering problem. In cases where the source consists
of a combination of eigenfunctions of the corresponding spectral problem for
the Zakharov-Shabat system, the complete integrability of the nonlinear
Schrödinger equation is investigated. Namely, the evolutions of the scattering
data of the self-adjoint Zakharov-Shabat system, whose potential is a solution
of the defocusing nonlinear Schrödinger equation with a self-consistent
source, are obtained.

###### Key words and phrases:

Defocusing nonlinear Schrödinger equation; Zakharov-Shabat system; inverse
scattering theory; nonzero boundary condition; self-consistent source.

##  1\. Introduction

Nonlinear Schrödinger (NLS) equation

(1) $iu_{t}-2\chi\left|u\right|^{2}u+u_{xx}=0$

with various boundary conditions models a wide class of nonlinear phenomena in
physics. In the work [16], V. Zakharov and A. Shabat showed that NLS equation
can be applied in the study of optical self-focusing and splitting of optical
beams. The sign of the coupling constant corresponds to the attraction
($\chi<0$) and repulsion ($\chi>0$) of particles. In the case of attraction,
the problem of a finite number of particles and their bound states has a
physical meaning. In the classical limit, this is modeled by rapidly
decreasing boundary conditions. In the case of repulsion, the problem
corresponding to a gas of particles with a finite density is of interest.

It is well known that inverse scattering method for integration of the NLS
equation is based on the so-called Zakharov-Shabat and Ablowitz-Kaup-Newell-
Segur scattering problem (see: [1, 16, 8]).

In 1971, V. Zakharov and A. Shabat [16] showed that the NLS equation can be
solved by means of the inverse scattering transform (IST) technique. The IST
as a method to solve the initial-value problem for the NLS equation has been
extensively studied in the literature, both in the focusing ($\chi=-1$) and in
the defocusing ($\chi=1$) dispersion regimes. The IST for the defocusing NLS
equation with nonzero boundary conditions was first studied in 1973 by V.
Zakharov and A. Shabat [17] and a detailed study can be found in the monograph
[8].

In the work [4] a rigorous theory of the IST for the defocusing nonlinear
Schrödinger equation with nonvanishing boundary values
$u_{\pm}=u_{0}e^{i\theta_{\pm}}$ as $x\to\pm\infty$ is presented. The IST
theory for the defocusing NLS equation with nonzero boundary conditions was
studied by B. Gino, F. Emily and B. Prinari [2] and the focusing case has been
studied by G. Biondini, G. Kovačić [3], F. Demontis, B. Prinari, C. van der
Mee, F. Vitale [5].

V.K. Melnikov [10, 11] showed that the NLS equation remains its integrability
by the inverse scattering method, if a source is added to them in the form of
a combination of eigenfunctions of the corresponding spectral problem. Namely,
the term “self-consistent source” was introduced in the works of V.K.
Melnikov.

The NLS equation with the self-consistent sources in various classes of
functions were considered by A.B. Khasanov [9], I.D. Rakhimov [13], A.B.
Yakhshimuratov [15].

In the matrix case, the inverse scattering theory for the matrix Zakharov-
Shabat system was investigated by P. Barbara, F. Demontis and C. Van der Mee
[12, 6, 7] and applied for the integration of the matrix NLS equation. In
[14], the matrix NLS equation with the self-consistent source was considered
in the class of rapidly decreasing matrix functions.

##  2\. Formulation of the problem

We consider the following system of equations

(2) $\displaystyle
iu_{t}-2u\left|u\right|^{2}+u_{xx}=-2i\sum_{n=1}^{N}(f_{1,n}^{*}g_{2,n}^{*}+f_{2,n}g_{1,n})$
(3) $\displaystyle\frac{\partial f_{1,n}}{\partial
x}-u^{*}f_{2,n}+i\xi_{n}f_{1,n}=\frac{\partial f_{2,n}}{\partial
x}-uf_{1,n}-i\xi_{n}f_{2,n}=0,\,\,\,n=1,2,...,N,$ (4)
$\displaystyle\frac{\partial g_{1,n}}{\partial
x}-ug_{2,n}-i\xi_{n}g_{1,n}=\frac{\partial g_{2,n}}{\partial
x}-u^{*}g_{1,n}+i\xi_{n}g_{2,n}=0,\,\,\,n=1,2,...,N$

under the initial condition

(5) $u(x,0)=u_{0}(x),\quad x\in R.$

Here the initial function $u_{0}(x)$, $x\in R$ satisfies the following
properties:

1)

(6) $\int_{-\infty}^{0}(1-x)\left|u_{0}(x)-\rho
e^{i\alpha_{-}}\right|dx\,+\int_{0}^{\infty}(1+x)\left|u_{0}(x)-\rho
e^{i\alpha_{+}}\right|dx<\infty,\,$

where $\rho>0$ and $0\leq\alpha_{\pm}<2\pi$ are arbitrary constants.

2) The system of equations (3) with coefficient $u_{0}(x)$ possesses exactly
$N$ eigenvalues $\xi_{1}(0)$, $\xi_{2}(0)$, … , $\xi_{N}(0)$.

We assume that the solution $u(x,t)$ of the equation (2) is sufficiently
smooth and tends to its limits sufficiently rapidly as $x\to\pm\infty$, i.e.,
for all $t\geq 0$ satisfies the condition

$\displaystyle\int_{-\infty}^{0}(1-x)\left|u(x,t)-\rho
e^{i\alpha_{-}-2i\rho^{2}t}\right|dx+$
$\displaystyle\int_{0}^{\infty}(1+x)\left|u(x,t)-\rho
e^{i\alpha_{+}-2i\rho^{2}t}\right|dx$ (7) $\displaystyle+$
$\displaystyle\int_{-\infty}^{\infty}\sum_{k=1}^{2}\left|\frac{\partial^{k}u(x,t)}{\partial
x^{k}}\right|dx<\infty.$

It follows from this condition that the left-hand side of equation (2) for all
$t\geq 0$ tends to zero as $x\to\pm\infty$. Taking this into account, the
solutions $\,F_{n}(x,t)=(f_{1,n},f_{2,n})^{T}$ and
$G_{n}(x,t)=(g_{1,n},g_{2,n})^{T}$ of equations (3) and (4), respectively, are
to be chosen so that the expression in the right-hand side of equation (2) for
all $t\geq 0$ should tend to zero rapidly enough as $x\to\pm\infty$. This can
be done in the following two ways:

(A) Let the functions $F_{n}(x,t)$ and $G_{n}(x,t)$ be the eigenfunctions of
equations (3) and (4) respectively, corresponding to the eigenvalues
$\xi_{n}(t)$. In this case, the functions $F_{n}(x,t)$ and $G_{n}(x,t)$ belong
to the $L^{2}(R)$ for all $t\geq 0$. We assume that

(8) $\int_{-\infty}^{\infty}G_{n}^{T}(s,t)F_{n}(s,t)ds=A_{n}(t),\quad t\geq
0,\quad n=1,2,...,N.$

Where $A_{n}(t)$ are given and the continuous functions of $t$.

(B) Let the function $F_{n}(x,t)$ be the eigenfunctions of equations (3)
corresponding to the eigenvalues $\xi_{n}(t)$ and let the function
$G_{n}(x,t)$ be unbounded solution of the equation (4), satisfying equalities

(9) $f_{1,n}g_{1,n}-f_{2,n}g_{2,n}=B_{n}(t),\quad t\geq 0,\quad n=1,2,...,N.$

Where the functions $B_{n}(t)$ are given continuous real valued functions of
$t$.

Our main goal is to obtain representations for solutions $u(x,t)$,
$F_{n}(x,t)$, $G_{n}(x,t)$, $n=1,2,...,N$ of problem (2)-(7), within the
framework of the inverse scattering method.

##  3\. Preliminaries

Let a function $u(x,t)$ belong to the class of functions (7). In this section,
we give well known [8], necessary information concerning the theory of direct
and inverse scattering problems for the operator

$L(t)=i\left(\begin{array}[]{cc}{\frac{\partial}{\partial x}}&{-u^{*}(x,t)}\\\
{u(x,t)}&{-\frac{\partial}{\partial x}}\end{array}\right),\quad t\geq 0.$

Consider the equation

(10) $(L(t)-\xi I)f=0$

with respect to an unknown $2\times 2$ square matrix function $f(x,\xi,t)$.
Here $\xi$ is a complex spectral parameter.

We introduce a new spectral parameter $p$ as follows
$p=\sqrt{\xi^{2}-\rho^{2}}$. The variable $p$ is then thought of as belonging
to a Riemann surface $\Gamma$ consisting of a sheet $\Gamma_{+}$ and a sheet
$\Gamma_{-}$ which both coincide with the complex plane cut along the semi
lines $\Sigma=(-\infty,-\rho]\cup[\rho,\infty)$ with its edges glued in such a
way that $p(\xi)$ is continuous through the cut. The variable $p$ is thought
of as belonging to the complex plane consisting of the upper half complex
plane $\Gamma_{+}$ and the lower half complex plane $\Gamma_{-}$ glued
together along the whole real line. For all $\xi\in\Sigma$, the branch of the
square root is fixed by the condition $sign\,p(\xi)=sign\,\xi$.

For $\xi\in\Sigma$, we define matrix Jost solutions $f^{-}(x,\xi,t)$ and
$f^{+}(x,\xi,t)$ from the right and the left, respectively as those square
matrix solutions to (10) satisfying asymptotics

(11) $f^{\pm}\sim E^{\pm}(x,\xi,t)\quad\text{as}\quad x\to\pm\infty$

where

$E^{\pm}(x,\xi,t)=\left(\begin{array}[]{cc}{1}&{-\frac{i(\xi-p)}{\rho}e^{-i\alpha_{\pm}+2i\rho^{2}t}}\\\
{\frac{i(\xi-p)}{\rho}e^{i\alpha_{\pm}-2i\rho^{2}t}}&{1}\end{array}\right)e^{-ip\sigma_{3}x}.$

Here and everywhere below we will use the standard Pauli matrices

$\sigma_{1}=\left(\begin{array}[]{cc}{0}&{1}\\\
{1}&{0}\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cc}{0}&{-i}\\\
{i}&{0}\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cc}{1}&{0}\\\
{0}&{-1}\end{array}\right).$

If a function $u(x,t)$ belongs to the class of functions (7), then such a
solution to equations (10) exists and is unique.

It can be shown that

(12) $\frac{d}{dx}\det f^{\pm}(x,\xi,t)=0.$

From (12) and (11) it follows that

(13) $\det f^{\pm}(x,\xi,t)=\frac{2p(\xi-p)}{\rho^{2}}.$

The system (10) is invariant with respect to the involution

(14)
$\bar{f}^{\pm}(x,\xi,t)=\sigma_{1}f^{\pm}(x,\xi,t)\sigma_{1},\quad\xi\in\Sigma.$

We now call the columns of

$f^{+}(x,\xi,t)=\left(\bar{\psi}(x,\xi,t)\,\,\,\,\psi(x,\xi,t)\right),\,\,\,f^{-}(x,\xi,t)=\left(\varphi(x,\xi,t)\,\,\,\,\bar{\varphi}(x,\xi,t)\right)$

the Jost solutions from the right and the left, respectively. For the Jost
solutions we get the following asymptotic estimates

(15)
$\psi(x,\xi,t)\sim\left(\begin{array}[]{c}{-\frac{i(\xi-p)}{\rho}e^{-i\alpha_{+}+2i\rho^{2}t}}\\\
{1}\end{array}\right)e^{ipx},\quad x\to\infty,$
$\bar{\psi}(x,\xi,t)\sim\left(\begin{array}[]{c}{1}\\\
{\frac{i(\xi-p)}{\rho}e^{i\alpha_{+}-2i\rho^{2}t}}\end{array}\right)e^{-ipx},\quad
x\to\infty,$ (16) $\varphi(x,\xi,t)\sim\left(\begin{array}[]{c}{1}\\\
{\frac{i(\xi-p)}{\rho}e^{i\alpha_{-}-2i\rho^{2}t}}\end{array}\right)e^{-ipx},\quad
x\to-\infty,$
$\bar{\varphi}(x,\xi,t)\sim\left(\begin{array}[]{c}{-\frac{i(\xi-p)}{\rho}e^{-i\alpha_{-}+2i\rho^{2}t}}\\\
{1}\end{array}\right)e^{ipx},\quad x\to-\infty,$

Since $f^{-}(x,\xi,t)$ and $f^{+}(x,\xi,t)$ are square matrix solutions of the
homogeneous first order equation (10), we necessarily have for $\xi\in\Sigma$

(17) $f^{-}(x,\xi,t)=f^{+}(x,\xi,t)S(\xi,t)$

where $S(\xi,t)$ is the transition coefficient matrix.

From the involution property (14) for $\xi\in\Sigma$, it follows that

$\bar{S}(\xi,t)=\sigma_{1}S(\xi,t)\sigma_{1}.$

Hence, we have

(18) $S(\xi,t)=\left(\begin{array}[]{cc}{a(\xi,t)}&{\bar{b}(\xi,t)}\\\
{b(\xi,t)}&{\bar{a}(\xi,t)}\end{array}\right).$

Coefficients $a(\xi,t)$ and $b(\xi,t)$ are called scattering coefficients.
From relations (13) and (18) we obtain

$a(\xi,t)\bar{a}(\xi,t)-b(\xi,t)\bar{b}(\xi,t)=1.$

By using (17) we can represent the scattering coefficients as

(19)
$a(\xi,t)=\frac{\rho^{2}}{2p(\xi-p)}\det(\varphi(x,\xi,t),\,\psi(x,\xi,t))$

and

$b(\xi,t)=\frac{\rho^{2}}{2p(\xi-p)}\det(\bar{\psi}(x,\xi,t),\varphi(x,\xi,t)).$

If a function $u(x,t)$ belongs to the class of functions (7), then for each
$x\in R$ the Jost solutions $\psi(x,\xi,t)e^{-ipx}$ and
$\varphi(x,\xi,t)e^{ipx}$ are analytic for $\xi\in\Gamma_{+}$ excluding branch
points $\xi=\pm\rho$, there are asymptotes for $\left|\xi\right|\to\infty$

(20) $\varphi(x,\xi,t)e^{ipx}=\left(\begin{array}[]{c}{1}\\\
{\frac{i(\xi-p)}{\rho}e^{i\alpha_{-}-2ip^{2}t}}\end{array}\right)+O\left(\frac{\left|1+\xi-p\right|}{\left|\xi\right|}\right),$
(21)
$\psi(x,\xi,t)e^{-ipx}=\left(\begin{array}[]{c}{-\frac{i(\xi-p)}{\rho}e^{-i\alpha_{+}+2ip^{2}t}}\\\
{1}\end{array}\right)+O\left(\frac{\left|1+\xi-p\right|}{\left|\xi\right|}\right).$

It follows from the analyticity properties of the Jost solutions and equality
(19) that the function $a(\xi,t)$ can be analytically continued to the sheet
$\Gamma_{+}$ excluding branch points $\xi=\pm\rho$.

From (20) and (21) we obtain that for $\left|\xi\right|\to\infty$, the
function $a(\xi,t)$ has the asymptotics

(22) $a(\xi,t)=1+O\left(\frac{1}{\left|\xi\right|}\right)\,\,\,{\rm
as}\,\,Im\,\xi>0$

and

(23) $a(\xi,t)=e^{-i\theta}+O\left(\frac{1}{\left|\xi\right|}\right)\,\,{\rm
as}\,\,Im\,\xi<0$

where we recall $\theta=\alpha_{+}-\alpha_{-}$.

Similarly, the function $\bar{a}(\xi,t)$ can be analytically continued to the
sheet $\Gamma_{-}$, excluding branch points $\xi=\pm\rho$.

It follows from the analyticity with respect to the function $a(\xi,t)$ on
$\Gamma_{+}$ and from the asymptotics (22), (23) that the function $a(\xi,t)$
can have only a finite number of zeros on the sheet $\Gamma_{+}$. These zeros
will be denoted by $\xi_{1},\xi_{2},...,\xi_{N}$. In [4] it is shown that all
zeros are simple and all belong to the $(-\rho,\rho)$.

It is seen from representation (19) that $\xi=\xi_{n}$ the functions
$\varphi(x,\xi,t)$ and $\psi(x,\xi,t)$ are proportional to each other

(24)
$\varphi_{n}(x,t)=c_{n}(t)\psi_{n}(x,t),\,\,\,\,\,\,\,\bar{\varphi}_{n}(x,t)=c_{n}^{*}(t)\bar{\psi}_{n}(x,t),\,\,\,\,\,n=1,2,...,N.$

Where
$\varphi_{n}(x,t)=\varphi(x,\xi_{n},t),\,\,\,\psi_{n}(x,t)=\psi(x,\xi_{n},t)$.

The zeros of $a(\xi,t)$ correspond to the eigenvalues of the equation (10).
The equation (10) is self-adjoint, so its eigenvalues and thus the zeros of
the function $a(\xi,t)$ are real.

Note, the vector functions

(25)
$h_{n}(x,t)=\frac{\frac{\partial}{\partial\xi}\left.\left(\varphi(x,\xi,t)-c_{n}\psi(x,\xi,t)\right)\right|_{\xi=\xi_{n}}}{\dot{a}(\xi_{n},t)},\,\,n=1,2,...,N$

are a solution to the equations $(L(t)-\xi_{n}I)h_{n}=0$. Where
$\dot{a}(\xi_{n},t)=\frac{\partial}{\partial\xi}\left.a(\xi,t)\right|_{\xi=\xi_{n}}$.

From equality (25) it follows that

$h_{n}(x,t)\sim-
c_{n}(t)\left(\begin{array}[]{c}{-\frac{i(\xi_{n}-p_{n})}{\rho}e^{-i\alpha_{-}+2ip^{2}t}}\\\
{1}\end{array}\right)e^{ip_{n}x}\,\,as\,\,x\to-\infty,$ (28) $\displaystyle
h_{n}(x,t)\sim\left(\begin{array}[]{c}{1}\\\
{\frac{i(\xi_{n}-p_{n})}{\rho}e^{i\alpha_{+}-2ip^{2}t}}\end{array}\right)e^{-ip_{n}x}\,\,as\,\,x\to\infty,$

where $p_{n}=i\sqrt{\rho^{2}-\xi_{n}^{2}}$. In particular, we have

(29)
$\det(\varphi_{n},h_{n})=-\frac{2p_{n}(\xi_{n}-p_{n})}{\rho^{2}}c_{n},\,\,n=1,2,...,N.$

######  Definition 1.

The set
$\\{a(\xi,t),\,\,b(\xi,t),\,\,\xi_{n}(t),\,\,c_{n}(t),\,\,n=1,2,...,N\\}$ is
called the scattering data for equation (10). The direct scattering problem is
to find the scattering data via the given potentials $u(x,t)$ and the inverse
scattering problem is to find the potentials $u(x,t)$ of the equation (10) via
the given scattering data.

Before we proceed further solving the inverse problem, it is convenient to
introduce a uniformization variable $z$ (see [4, 8]) defined by the conformal
mapping: $z=z(\xi)=\xi+p(\xi)$. Inverse mapping given by

$\xi=\frac{1}{2}\left(z+\frac{\rho^{2}}{z}\right),\,\,\,p=z-\xi=\frac{1}{2}\left(z-\frac{\rho^{2}}{z}\right).$

With this mapping the sheets $\Gamma_{+}$ and $\Gamma_{-}$ of the Riemann
surface $\Gamma$ are, respectively, mapped onto the upper and lower complex
half-planes $Im\,z>0$ and $Im\,z<0$ of the complex $z$-plane. The cut $\Sigma$
on the Riemann surface is mapped onto the real $z$ axis. The segments
$[-\rho,\rho]$ on $\Gamma_{+}$ and $\Gamma_{-}$ are mapped onto the upper and
lower semicircles of radius $\rho$ and center at the origin of the $z$-plane.
The neighborhood of the point $\xi=\infty$ on $\Gamma_{\pm}$ with the
condition $\pm Im\xi>0$ is mapped into the neighborhood of the point
$z=\infty$, and the neighborhood of the point $\xi=\infty$ on $\Gamma_{\pm}$
with the condition $\pm Im\xi<0$ is mapped into the neighborhood of the point
$z=0$.

In terms of variable $z$, relation (17) can be written when $Imz=0$ following
form

(30) $f^{-}(x,z,t)=f^{+}(x,z,t)S(z,t)$

here $f^{\pm}(x,z,t)=f^{\pm}(x,\xi(z),t)$, $S(z,t)=S(\xi(z),t)$ and one can
obtain the symmetries of the scattering coefficients:

(31) $a(z,t)=\bar{a}\left(\frac{\rho^{2}}{z},t\right),\,\,\,Imz\geq 0,$
$b(z,t)=-\bar{b}\left(\frac{\rho^{2}}{z},t\right),\,\,\,Imz=0.$

Equality (31), together with the self-adjointness of the equation (10), ensure
that the scattering coefficient $a(z,t)$ ($\bar{a}(z,t)$) can only have zeros
at $z_{n}=\xi_{n}+iv_{n}$ ($\bar{z}_{n}=\xi_{n}-iv_{n}$), with
$-\rho<\xi_{n}<\rho$ and $v_{n}=\sqrt{\rho^{2}-\xi_{n}^{2}}>0$.

Taking into account the analyticity properties of $a(z,t)$ in the upper half
plane $Imz>0$ we can obtain the following representation

$a(z,t)=\prod_{n=1}^{N}\frac{z-z_{n}}{z-z_{n}^{*}}\exp\left[-\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{\log(1-|r(\zeta,t)|^{2})}{\zeta-z}d\zeta\right].$

Where $r(z,t)\equiv\frac{b(z,t)}{a(z,t)}$ is called reflection coefficient.
According to (23), for $z\to 0$ we obtain that

$e^{-i\theta}=\prod_{n=1}^{N}\frac{z_{n}}{z_{n}^{*}}\exp\left[-\frac{1}{2\pi
i}\int_{-\infty}^{\infty}\frac{\log(1-|r(\zeta,t)|^{2})}{\zeta}d\zeta\right].$

If $\varphi_{n}=\left(\begin{array}[]{l}{\varphi_{1,n}}\\\
{\varphi_{2,n}}\end{array}\right)$ is an eigenfunction of the equation (10)
corresponding to $z_{n}$, then we define
$\bar{\varphi}_{n}=\left(\begin{array}[]{l}{\varphi_{2,n}^{*}}\\\
{\varphi_{1,n}^{*}}\end{array}\right)$ to be the eigenfunction of the equation
(10) corresponding to $\bar{z}_{n}$.

It is well known that the inverse scattering theory of (10) can be formulated
in terms of the Gelfand-Levitan-Marchenko equations. The Jost solution
$\psi(x,z,t)$ of the equation (10) can be represented in the following form

(32)
${\psi(x,z,t)=\left(\begin{array}[]{c}{-\frac{i\rho}{z}e^{-i\alpha_{+}+2i\rho^{2}t}}\\\
{1}\end{array}\right)e(x,z)+\int_{x}^{\infty}\mathcal{K}(x,y,t)\left(\begin{array}[]{c}{-\frac{i\rho}{z}e^{-i\alpha_{+}+2i\rho^{2}t}}\\\
{1}\end{array}\right)e(y,z)dy,}$

here $e(x,z)=e^{\frac{i}{2}\left(z-\frac{\rho^{2}}{z}\right)x}$ and
$\mathcal{K}(x,y)$ is a 2 $\mathrm{\times}$ 2 matrix function which has to
satisfy the following Gelfand-Levitan-Marchenko equation:

$\mathcal{K}(x,y,t)+\mathcal{F}(x+y,t)+\int_{x}^{\infty}\mathcal{K}(x,s,t)\mathcal{F}(s+y,t)ds=0,\,\,\,y\geq
x,$

where $\mathcal{K}(x,y,t)$ and $\mathcal{F}(x+y,t)$ are defined as

$\mathcal{K}(x,y,t)=\left(\begin{array}[]{cc}{K_{11}(x,y,t)}&{K_{12}(x,y,t)}\\\
{K_{21}(x,y,t)}&{K_{22}(x,y,t)}\end{array}\right),\,\,\mathcal{F}(x+y,t)=\left(\begin{array}[]{cc}{F_{1}(x+y,t)}&{F_{2}^{*}(x+y,t)}\\\
{F_{2}(x+y,t)}&{F_{1}(x+y,t)}\end{array}\right)$

with

$F_{1}(x,t)=\frac{\rho e^{i\alpha_{+}-2i\rho^{2}t}}{4\pi
i}\int_{-\infty}^{\infty}\frac{b(z,t)}{za(z,t)}\cdot
e(x,z)dz-\frac{1}{2}\sum_{n=1}^{N}\frac{c_{n}(t)\rho
e^{-i\alpha_{+}+2i\rho^{2}t}}{\dot{a}(z_{n},t)z_{n}}\cdot e(x,z_{n}),$
$F_{2}(x,t)=\frac{1}{4\pi}\int_{-\infty}^{\infty}\frac{b(z,t)}{a(z,t)}\cdot
e(x,z)dz-\frac{1}{2}\sum_{n=1}^{N}\frac{ic_{n}(t)}{\dot{a}(z_{n},t)}\cdot
e(x,z_{n}).$

In representations (32), the component $K_{21}(x,x,t)$ of the matrix
$\mathcal{K}(x,y,t)$ have relations with the potential

$2K_{21}(x,x,t)=\rho e^{i\alpha_{+}-2i\rho^{2}t}-u(x,t).$

In the work [8], it was proven the uniquely determining of the potential
$u(x,t)$ by the scattering data.

##  4\. Time evolution

The use of the inverse scattering method for integration of the problem
(2)-(7) is based on the following. Let the function $u(x,t)$ be a solution of
equation (2), from the class of functions (6). Consider equation (10) with a
potential $u(x,t)$ and find the evolution from the scattering data.

Assuming that

(33) $F_{N+n}=\left(\begin{array}[]{l}{f_{2,n}^{*}}\\\
{f_{1,n}^{*}}\end{array}\right),\,\,\,G_{N+n}=\left(\begin{array}[]{l}{g_{2,n}^{*}}\\\
{g_{1,n}^{*}}\end{array}\right),\,\,\,\,n=1,2,...,N,$

equation (2) can be represented as an equality of operators in the class of
smooth functions $f(x,\xi,t)$ satisfying the equation (10):

$\frac{\partial L}{\partial
t}+[L,A]=i\sum_{n=1}^{2N}[\sigma_{3},\,F_{n}G_{n}^{T}].$

Where $[L,A]=LA-AL$ and

$A=\left(\begin{array}[]{cc}{i\left|u\right|^{2}+2i\xi^{2}}&{-iu_{x}^{*}-2\xi
u^{*}}\\\ {iu_{x}-2\xi u}&{-i\left|u\right|^{2}-2i\xi^{2}}\end{array}\right).$

######  Lemma 1.

Let $f(x,\xi,t)$ be solution of the equation (10) and let
$\phi_{n}(x,\xi,t),\,\,\,n=1,2,...,2N$ be any functions, which satisfy the
conditions

(34) $\frac{\partial\phi_{n}}{\partial
x}=G_{n}^{T}f,\,\,\,\,\,n=1,\;2,\;...,\;2N.$

Then, the function $G_{n}(x,t)$ satisfy the equalities

(35)
$G_{n}^{T}\sigma_{3}f+i(\xi-\xi_{n})\phi_{n}=0\,,\,\,\,n=1,\;2,\;...,\;2N$

and the function

(36) $H=\frac{\partial f}{\partial t}-Af+\sum_{n=1}^{2N}F_{n}\phi_{n}$

satisfies the equation (10) for any $\xi\in\Sigma$.

###  4.1. Evolution equation for the scattering data in the case of a source
satisfying the conditions (A)

Let us take matrix Jost solutions $f^{-}(x,\xi,t)$ and $f^{+}(x,\xi,t)$ for
$\xi\in\Sigma$ as the solution $f(x,\xi,t)$ and $\xi=\xi_{n},\,\,n=1,2,...,N$
are eigenvalues of the equation (10). According to the definition of
eigenfunctions, there are $\alpha_{n}(t)$ and $\beta_{n}(t)$ such that the
relations hold

(37)
$F_{n}(x,t)=\alpha_{n}(t)\psi_{n}(x,t),\,\,\,\,G_{n}(x,t)=\beta_{n}(t)\sigma_{1}\varphi_{n}(x,t),\,\,n=1,2,...,N$

According to these relations, due to the assumptions (33), we obtain

(38)
$F_{N+n}(x,t)=\alpha_{n}^{*}(t)\bar{\psi}_{n}(x,t),\,\,G_{N+n}(x,t)=\beta_{n}^{*}(t)\sigma_{1}\bar{\varphi}_{n}(x,t),\,\,\,n=1,2,...,N.$

By definition functions $G_{n}(x,t)$, belong to the $L^{2}(R)$ for all $t\geq
0$ and matrix Jost solutions $f^{-}(x,\xi,t)$, $f^{+}(x,\xi,t)$ are bounded
for all $\xi\in\Sigma$. Therefore $\phi_{n}^{-}\in L^{2}(R)$ and
$\phi_{n}^{+}\in L^{2}(R)$ for all $t\geq 0$ and $\xi\in\Sigma$. Hence, by
virtue of (35) it follows that at any $\xi\in\Sigma$ and $n=1,2,...,2N$ the
asymptotics

(39) $\displaystyle\phi_{n}^{-}(x,\xi,t)\to 0\,\,{\rm as}\,\,x\to-\infty,$
$\displaystyle\phi_{n}^{+}(x,\xi,t)\to 0\,\,{\rm as}\,\,x\to\infty$

are valid. So, from (34) for $n=1,2,...,2N$ we obtain the following
expressions

(40)
$\phi_{n}^{-}=\int_{-\infty}^{x}G_{n}^{T}(s{\rm,}t)f^{-}(s,\xi,t)ds,\,\,\phi_{n}^{+}=-\int_{x}^{\infty}G_{n}^{T}(s,t)f^{+}(s,\xi,t)ds.$

Using the matrix Jost solutions $f^{+}$ and $f^{-}$ of equation (10), we
rewrite equality (36) in the form

(41) $H^{-}=\frac{\partial f^{-}}{\partial
t}-Af^{-}+\sum_{n=1}^{2N}F_{n}\phi_{n}^{-}$

and

(42) $H^{+}=\frac{\partial f^{+}}{\partial
t}-Af^{+}+\sum_{n=1}^{2N}F_{n}\phi_{n}^{+}.$

These functions satisfy the equation (10) for any $\xi\in\Sigma$. Therefore,
$H^{+}$ and $H^{-}$ are linearly dependent on $f^{+}$ and $f^{-}$,
respectively, i.e., there exist such $C_{0}^{-}(\xi,t)$ and $C_{0}^{+}(\xi,t)$
that the following identities hold

$H^{-}(x,\xi,t)=f^{-}(x,\xi,t)C_{0}^{-}(\xi,t),\,\,\,H^{+}(x,\xi,t)=f^{+}(x,\xi,t)C_{0}^{+}(\xi,t).$

By virtue of the definition of the matrix $A$, from relations (41), (42) and
from asymptotics (11), (39) we obtain

(43) $\displaystyle{H^{-}(x,\xi,t)\to-(i\rho^{2}+2i\xi
p)E^{-}(x,\xi,t)\sigma_{3},\,\,\,x\to-\infty,}$ (44)
$\displaystyle{H^{+}(x,\xi,t)\to-(i\rho^{2}+2i\xi
p)E^{+}(x,\xi,t)\sigma_{3},\,\,\,x\to\infty.}$

By the uniqueness of the Jost solutions we get

(45) $\displaystyle H^{-}(x,\xi,t)=-(i\rho^{2}+2i\xi
p)f^{-}(x,\xi,t)\sigma_{3},$ $\displaystyle H^{+}(x,\xi,t)=-(i\rho^{2}+2i\xi
p)f^{+}(x,\xi,t)\sigma_{3}.$

We introduce the function $\mathcal{H}$ in the following form

$\mathcal{H}=H^{-}(x,\xi,t)-H^{+}(x,\xi,t)S(\xi,t).$

Based on equalities (17) and (LABEL:ZEqnNum408091), the function $\mathcal{H}$
can be rewritten in the form

(46) $\mathcal{H}=(i\rho^{2}+2i\xi p)f^{+}(x,\xi,t)[\sigma_{3},S(\xi,t)].$

On the other hand, by virtue of (17), (41) and (42) the equality

$\displaystyle\mathcal{H}=$ $\displaystyle
H^{-}(x,\xi,t)-H^{+}(x,\xi,t)S(\xi,t)=f^{+}(x,\xi,t)S_{t}(x,\xi,t)+$ (47)
$\displaystyle+$
$\displaystyle\sum_{n=1}^{2N}[F_{n}(x,t)\phi_{n}^{-}(x,\xi,t)-F_{n}(x,t)\phi_{n}^{+}(x,\xi,t)S(\xi,t)]$

holds.

Based on equality (35) this relation becomes

$\mathcal{H}=f^{+}(x,\xi,t)S_{t}(\xi,t)+$
$+\sum_{n=1}^{2N}\frac{i}{\xi-\xi_{n}}[F_{n}(x,t)G_{n}^{T}(x,t)\sigma_{3}f^{-}(x,\xi,t)-F_{n}(x,t)G_{n}^{T}(x,t)\sigma_{3}f^{+}(x,\xi,t)S(\xi,t)].$

Finally, based on (17), we obtain

(48) $\mathcal{H}=f^{+}(x,\xi,t)S_{t}(\xi,t).$

Comparing equalities (46) and (48) we have

$(i\rho^{2}+2i\xi
p)f^{+}(x,\xi,t)[\sigma_{3},S(\xi,t)]=f^{+}(x,\xi,t)S_{t}(\xi,t).$

Therefore, for all $\xi\in\Sigma$ we have the relation

$S_{t}(\xi,t)-(i\rho^{2}+2i\xi p)[\sigma_{3},S(\xi,t)]=0,$

i.e.

$\frac{d}{dt}a(\xi,t)=0,\,\,\,\frac{d}{dt}b(\xi,t)=-2(i\rho^{2}+2i\xi
p)b(\xi,t).$

Since, the function $a(\xi,t)$ does not depend on $t$, hence we conclude that
its zeros $\xi_{n}$ also do not depend on $t$.

Based on identities (41) and (42), we write the following equalities

(49) $\displaystyle
H_{1}^{-}(x,\xi_{n},t)=\frac{\partial\varphi_{m}(x,t)}{\partial
t}-A(x,\xi_{n},t)\varphi_{m}(x,t)+\sum_{n=1}^{2N}F_{n}(x,t)\phi_{1,n}^{-}(x,\xi_{n},t)$
(50) $\displaystyle
H_{2}^{+}(x,\xi_{n},t)=\frac{\partial\psi_{m}(x,t)}{\partial
t}-A(x,\xi_{n},t)\psi_{m}(x,t)+\sum_{n=1}^{2N}F_{n}(x,t)\phi_{2,n}^{+}(x,\xi_{n},t)$

By virtue of the definition of the matrix $A$, from relations (49), (50) and
from asymptotics (15), (16) we obtain

(51) $\displaystyle
H_{1}^{-}(x,\xi_{m},t)=(-i\rho^{2}-2i\xi_{m}p_{m})\varphi_{m}(x,t),$
$\displaystyle
H_{2}^{+}(x,\xi_{m},t)=(i\rho^{2}+2i\xi_{m}p_{m})\psi_{m}(x,t).$

We now introduce the following functions

$\mathcal{H}_{m}=H_{1}^{-}(x,\xi_{m},t)-c_{m}(t)H_{2}^{+}(x,\xi_{m},t),\,\,\,m=1,2,...,2N.$

Using equalities (24), (51) the function $\mathcal{H}_{m}$ can be rewritten in
the form

(52) $\mathcal{H}_{m}=(-2i\rho^{2}-4i\xi_{m}p_{m})\varphi_{m}(x,t).$

Substituting instead of $\phi_{1,n}^{-}(x,\xi,t)$ and
$\phi_{2,n}^{+}(x,\xi,t)$ the expressions from (40) into equalities (49), (50)
and using (24), we obtain

(53)
$H_{1}^{-}(x,\xi_{m},t)-c_{m}(t)H_{2}^{+}(x,\xi_{m},t)=\frac{dc_{m}(t)}{dt}\psi_{m}(x,t)+\sum_{n=1}^{2N}F_{n}(x,t)\int_{-\infty}^{\infty}G_{n}^{T}(s,t)\varphi_{m}(s,t)ds$

If $\xi_{m}\neq\xi_{n}$, according to equation (35) we get

$\int_{-\infty}^{\infty}G_{n}^{T}(s,t)\varphi_{m}(s,t)ds=0.$

According to (37) and (38), equality (53) can be rewritten in the form

(54)
$\begin{gathered}{H_{1}^{-}(x,\xi_{m},t)-c_{m}(t)H_{2}^{+}(x,\xi_{m},t)=\frac{dc_{m}(t)}{dt}\psi_{m}(x,t)+}\\\
{+\left(\int_{-\infty}^{\infty}G_{m}^{T}(s,t)F_{m}(s,t)ds+\int_{-\infty}^{\infty}G_{N+m}^{T}(s,t)F_{N+m}(s,t)ds\right)\varphi_{m}(x,t).}\end{gathered}$

Comparing equalities (52) and (54) we obtain

$(-2i\rho^{2}-4i\xi_{m}p_{m})\varphi_{m}(x,t)=$
$=\frac{dc_{m}(t)}{dt}\psi_{m}(x,t)+\left(\int_{-\infty}^{\infty}G_{m}^{T}(s,t)F_{m}(s,t)ds+\int_{-\infty}^{\infty}G_{N+m}^{T}(s,t)F_{N+m}(s,t)ds\right)\varphi_{m}(x,t).$

Finally, using these equalities and taking into account (8) and (24) we
determine

$\frac{dc_{m}(t)}{dt}=(-2i\rho^{2}-4i\xi_{m}p_{m}-A_{m}(t)-A_{m}^{*}(t))c_{m}(t).$

Thus, we have proved the following theorem.

######  Theorem 1.

If functions $u(x,t),$ $F_{k}(x,t)$, $G_{k}(x,t)$, $k=1,\;2,\;...,N$ are the
solutions of the problem (2)-(7) in the case of a source satisfying the
conditions (A), then the scattering data for the equation (10) satisfy the
following relations

$a(\xi,t)=a(\xi,0),$ $b(\xi,t)=b(\xi,0)\exp(-2i\rho^{2}t-4i\xi
pt)\quad\text{for}\quad\xi\in\Sigma,$ $\xi_{k}(t)=\xi_{k}(0),\quad
k=1,\;2,\;...,N.$
$c_{k}(t)=c_{k}(0)\exp(-2i\rho^{2}t-4i\xi_{k}p_{k}t-\int_{0}^{t}(A_{k}(\tau)+A_{k}^{*}(\tau))d\tau),\quad
k=1,\;2,\;...,N.$

###  4.2. Evolution equation for the scattering data in the case of a source
satisfying the conditions (B)

Let us take matrix Jost solutions $f^{-}(x,\xi,t)$ and $f^{+}(x,\xi,t)$ for
$\xi\in\Sigma$ as the solution $f(x,\xi,t)$ and $\xi=\xi_{n},\,\,n=1,2,...,N$
are eigenvalues of the equation (10). According to the definition of
eigenfunctions of equation (10), there are $\alpha_{n}(t)$ such that the
relations

(55)
$F_{n}(x,t)=\alpha_{n}(t)\psi_{n}(x,t),\,\,\,\,\,F_{N+n}(x,t)=\alpha_{n}^{*}(t)\bar{\psi}_{n}(x,t),\,\,\,\,n=1,2,...,N.$

Due to the assumptions (B) the functions $G_{n}(x,t)$ are unbounded functions.
So, there are $\beta_{n}(t)$ such that which follow the equalities

(56) $\displaystyle
G_{n}(x,t)=\frac{\beta_{n}(t)}{\dot{a}(\xi_{n},t)}\sigma_{1}\varphi_{n}(x,t)+\sigma_{1}h_{n}(x,t),$
$\displaystyle
G_{N+n}(x,t)=\frac{\beta_{n}^{*}(t)}{\dot{\bar{a}}(\xi_{n},t)}\sigma_{1}\bar{\varphi}_{n}(x,t)+\sigma_{1}\bar{h}_{n}(x,t),\,\,n=1,2,...,N.$

One can easily see from (9) and (29), that the quantities $\alpha_{n}(t)$
satisfy the following equalities

(57)
$\alpha_{n}(t)=-\frac{\rho^{2}}{2p_{n}(\xi_{n}-p_{n})}B_{n}(t),\,\,\,\,\alpha_{n}^{*}(t)=\frac{\rho^{2}}{2p_{n}(\xi_{n}+p_{n})}B_{n}(t),\,\,n=1,2,...,N.$

Using equalities (24), (35), (55), (56) and asymptotics (15), (28) we can
verify that at any $\xi\in\Sigma$ and when $x\to\infty$ the following
asymptotics are valid:

$F_{n}\phi_{n}^{+}\sim\frac{i\alpha_{n}(t)}{\xi-\xi_{n}}\left(\begin{array}[]{cc}{\frac{(\xi_{n}-p_{n})^{2}}{\rho^{2}}}&{-\frac{i(\xi_{n}-p_{n})}{\rho}e^{-i\alpha_{+}+2i\rho^{2}t}}\\\
{\frac{i(\xi_{n}-p_{n})}{\rho}e^{i\alpha_{+}-2i\rho^{2}t}}&{1}\end{array}\right)\sigma_{3}E^{+}(x,\xi,t),$
$F_{N+n}\phi_{N+n}^{+}\sim\frac{i\alpha_{n}^{*}(t)}{\xi-\xi_{n}}\left(\begin{array}[]{cc}{1}&{-\frac{i(\xi_{n}+p_{n})}{\rho}e^{-i\alpha_{+}+2i\rho^{2}t}}\\\
{\frac{i(\xi_{n}+p_{n})}{\rho}e^{i\alpha_{+}-2i\rho^{2}t}}&{\frac{(\xi_{n}+p_{n})^{2}}{\rho^{2}}}\end{array}\right)\sigma_{3}E^{+}(x,\xi,t).$

Taking into account of equalities (24), (35), (55), (56) and asymptotics (15),
(28) we are convinced that at any $\xi\in\Sigma$ and $x\to-\infty$ there hold
the asymptotics

$F_{n}\phi_{n}^{-}\sim-\frac{i\alpha_{n}(t)}{\xi-\xi_{n}}\left(\begin{array}[]{cc}{1}&{-\frac{i(\xi_{n}-p_{n})}{\rho}e^{-i\alpha_{-}+2i\rho^{2}t}}\\\
{\frac{i(\xi_{n}-p_{n})}{\rho}e^{i\alpha_{-}-2i\rho^{2}t}}&{\frac{(\xi_{n}-p_{n})^{2}}{\rho^{2}}}\end{array}\right)\sigma_{3}E^{-}(x,\xi,t),$
$F_{N+n}\phi_{N+n}^{-}\sim-\frac{i\alpha_{n}^{*}(t)}{\xi-\xi_{n}}\left(\begin{array}[]{cc}{\frac{(\xi_{n}+p_{n})^{2}}{\rho^{2}}}&{-\frac{i(\xi_{n}+p_{n})}{\rho}e^{-i\alpha_{-}+2i\rho^{2}t}}\\\
{\frac{i(\xi_{n}+p_{n})}{\rho}e^{i\alpha_{-}-2i\rho^{2}t}}&{1}\end{array}\right)\sigma_{3}E^{-}(x,\xi,t).$

Using equalities (57) one can easily verify that at any $\xi\in\Sigma$ the
asymptotic

$F_{n}\phi_{n}^{+}+F_{N+n}\phi_{N+n}^{+}\to 0\quad\mathrm{for}\quad
x\to\infty,$

and

$F_{n}\phi_{n}^{-}+F_{N+n}\phi_{N+n}^{-}\to 0\quad\mathrm{for}\quad
x\to-\infty$

are valid.

Hence, it follows that the quantities $H^{-}(x,\xi,t)$ and $H^{+}(x,\xi,t)$
determined by (41) and (42) satisfy equalities

(58) $\begin{gathered}{H^{-}(x,\xi,t)=f^{-}(x,\xi,t)(-i\rho^{2}-2i\xi
p)\sigma_{3},\,\,\,}\\\ {H^{+}(x,\xi,t)=f^{+}(x,\xi,t)(-i\rho^{2}-2i\xi
p)\sigma_{3}.}\end{gathered}$

Now, consider the function $\mathcal{H}_{m}$ of the form

$\mathcal{H}_{m}=H^{-}(x,\xi,t)-H^{+}(x,\xi,t)S(\xi,t).$

Taking into account (58) we find that

(59) $\mathcal{H}_{m}=(i\rho^{2}+2i\xi p)f^{+}(x,\xi,t)[\sigma_{3},S(\xi,t)].$

From equalities (41), (42) and (17) it is easy to get that

(60) $\displaystyle
H^{-}(x,\xi,t)-H^{+}(x,\xi,t)S(\xi,t)=f^{+}(x,\xi,t)S_{t}(x,\xi,t)+$
$\displaystyle+\sum_{n=1}^{2N}[F_{n}(x,t)\phi_{n}^{-}(x,\xi,t)-F_{n}(x,t)\phi_{n}^{+}(x,\xi,t)S(\xi,t)].$

Using equalities (35), we obtain

$H^{-}(x,\xi,t)-H^{+}(x,\xi,t)S(\xi,t)=f^{+}(x,\xi,t)S_{t}(\xi,t)+$
$+\sum_{n=1}^{2N}\frac{i}{\xi-\xi_{n}}[F_{n}(x,t)G_{n}^{T}(x,t)\sigma_{3}f^{-}(x,\xi,t)-F_{n}(x,t)G_{n}^{T}(x,t)\sigma_{3}f^{+}(x,\xi,t)S(\xi,t)].$

By virtue of (17), it follows that

(61) $H^{-}(x,\xi,t)-H^{+}(x,\xi,t)S(\xi,t)=f^{+}(x,\xi,t)S_{t}(\xi,t)$

Comparing equalities (59) and (61) we have

$(i\rho^{2}+2i\xi
p)f^{+}(x,\xi,t)[\sigma_{3},S(\xi,t)]=f^{+}(x,\xi,t)S_{t}(\xi,t).$

Therefore, for all $\xi\in\Sigma$ we have

$S_{t}(\xi,t)-(i\rho^{2}+2i\xi p)[\sigma_{3},S(\xi,t)]=0,$

i.e.

$\frac{d}{dt}a(\xi,t)=0,\,\,\,\frac{d}{dt}b(\xi,t)=-2(i\rho^{2}+2i\xi
p)b(\xi,t).$

Thus, we conclude that the function $a(\xi,t)$ does not depend on $t$, so the
zeros $\xi_{n}(t)$ of function $a(\xi,t)$ do not depend on $t$.

Let us now find the evolution of the normalizing constants
$c_{m}(t),\,\,m=1,2,...,N$.

We now introduce the following functions

(62)
$\mathcal{H}_{m}=H_{1}^{-}(x,\xi_{m},t)-c_{m}(t)H_{2}^{+}(x,\xi_{m},t),\,\,m=1,2,...,N,$

where

(63) $\displaystyle
H_{1}^{-}(x,\xi_{m},t)=\frac{\partial\varphi_{m}(x,t)}{\partial
t}-A(x,\xi_{m},t)\varphi_{m}(x,t)+\sum_{n=1}^{2N}\Phi_{n}(x,t)\varphi_{1,n}^{-}(x,\xi_{m},t),$
(64) $\displaystyle
H_{2}^{+}(x,\xi_{m},t)=\frac{\partial\psi_{m}(x,t)}{\partial
t}-A(x,\xi_{m},t)\psi_{m}(x,t)+\sum_{n=1}^{2N}\Phi_{n}(x,t)\varphi_{2,n}^{+}(x,\xi_{m},t).$

It is easy to show that

(65)
$H_{1}^{-}(x,\xi_{m},t)=(-i\rho^{2}-2i\xi_{m}p_{m})\varphi_{m}(x,t),\,\,H_{2}^{+}(x,\xi_{m},t)=(i\rho^{2}+2i\xi_{m}p_{m})\psi_{m}(x,t).$

Substituting (65) into (62) and using equalities (24), we get for
$m=1,2,...,N$

(66) $\mathcal{H}_{m}=(-2i\rho^{2}-4i\xi_{m}p_{m})\varphi_{m}(x,t).$

On the other hand, using equalities (63), (64) and (24) we obtain

$H_{1}^{-}(x,\xi_{m},t)-c_{m}(t)H_{2}^{+}(x,\xi_{m},t)=\frac{dc_{m}(t)}{dt}\psi_{m}(x,t)+$
$+\sum_{{}^{n=1}_{n\neq
m\,}}^{2N}\frac{i}{\xi_{m}-\xi_{n}}[F_{n}(x,t)G_{n}^{T}(x,t)\sigma_{3}\varphi_{m}(x,t)-c_{m}(t)F_{n}^{+}(x,t)G_{n}^{T}(x,t)\sigma_{3}\psi_{m}(x,t)]+$
$+iF_{m}(x,t)G_{m}^{T}(x,t)\sigma_{3}\frac{\partial}{\partial\xi}\left.\left(\varphi(x,\xi,t)-c_{m}(t)\psi(x,\xi,t)\right)\right|_{\xi=\xi_{m}}+$
$+iF_{N+m}(x,t)G_{N+m}^{T}(x,t)\sigma_{3}\frac{\partial}{\partial\xi}\left.\left(\varphi(x,\xi,t)-c_{m}(t)\psi(x,\xi,t)\right)\right|_{\xi=\xi_{m}}.$

According to (24) and (25), this equation can be rewritten in the following
form

$H_{1}^{-}(x,\xi_{m},t)-c_{m}(t)H_{2}^{+}(x,\xi_{m},t)=\frac{dc_{m}(t)}{dt}\psi_{m}(x,t)+$
$+i\dot{a}(\xi_{n},t)F_{m}(x,t)G_{m}^{T}(x,t)\sigma_{3}h_{m}(x,t)+i\dot{a}(\xi_{n},t)F_{N+m}(x,t)G_{N+m}^{T}(x,t)\sigma_{3}h_{m}(x,t).$

Further, by virtue (9), (55) and (56), we obtain the equality

$\displaystyle
H_{1}^{-}(x,\xi_{m},t)-c_{m}(t)H_{2}^{+}(x,\xi_{m},t)=\frac{dc_{m}(t)}{dt}\psi_{m}(x,t)-$
(67)
$\displaystyle-i\beta_{m}(t)B_{m}(t)\varphi_{m}(x,t)+i\beta_{m}^{*}(t)B_{m}(t){\varphi}_{m}(x,t).$

Comparing equalities (66) and (67), we obtain

$(-2i\rho^{2}-4i\xi_{m}p_{m})\varphi_{m}(x,t)=\frac{dc_{m}(t)}{dt}\psi_{m}(x,t)-i\beta_{m}(t)B_{m}(t)\varphi_{m}(x,t)+i\beta_{m}^{*}(t)B_{m}(t){\varphi}_{m}(x,t)$

hence, taking into account equalities (24), we find

$\frac{dc_{m}(t)}{dt}=(-2i\rho^{2}-4i\xi_{m}p_{m}+i(\beta_{m}(t)-\beta_{m}^{*}(t))B_{m}(t))c_{m}(t).$

Thus, we have proved the following theorem.

######  Theorem 2.

If functions $u(x,t)$, $F_{k}(x,t)$, $G_{k}(x,t)$, $k=1,\;2,\;...,N$ are the
solutions of the problem (2)-(7) in the case of a source satisfying the
conditions (B), then the scattering data for the equation (10) satisfy the
following relations

$a(\xi,t)=a(\xi,0),$ $b(\xi,t)=b(\xi,0)\exp(-2i\rho^{2}t-4i\xi
pt)\quad\text{for}\quad\xi\in\Sigma,$ $\xi_{k}(t)=\xi_{k}(0),\quad
k=1,\;2,\;...,N.$
$c_{k}(t)=c_{k}(0)\exp(-2i\rho^{2}-4i\xi_{k}p_{k}+i\int_{0}^{t}\left(\beta_{k}(\tau)-\beta_{k}^{*}(\tau)\right)B_{k}(\tau)d\tau,k=1,\;2,\;...,N.$

We will illustrate inverse scattering method of constructing exact solutions
to the NLS equation with concrete example.

######  Example 1.

Let the initial function $u_{0}(x)$ have the form

$u_{0}(x)=\rho\cdot\frac{e^{i\alpha_{+}}e^{\nu x}+e^{i\alpha_{-}}e^{-\nu
x}}{e^{\nu x}+ce^{-\nu x}}.$

Where $\alpha_{+},\,\alpha_{-},\,\rho,\,\nu$, $c$ are positive real numbers
and $\rho>\nu$.

In this case, solving the direct scattering problem for the equation (10), we
obtain

$a(\xi,0)=\frac{\xi+p-\zeta-i\nu}{\xi+p-\zeta+i\nu},\,\,\zeta=\sqrt{\rho^{2}-\nu^{2}},$
$b(\xi,0)=0,\quad\xi_{1}(0)=\zeta,\quad
c_{1}(0)=\frac{i(\zeta-i\nu)}{\rho}ce^{i\alpha_{-}}.$

Based on Theorem 1, we can show the evolution of the scattering data in the
following form

$a(\xi,t)=\frac{\xi+p-\zeta-i\nu}{\xi+p-\zeta+i\nu},\,\,\zeta=\sqrt{\rho^{2}-\nu^{2}},$
$b(\xi,t)=0,\quad\xi_{1}(t)=\zeta,$ $c_{1}(t)=\frac{i(\zeta-i\nu)}{\rho}\cdot
c\cdot\exp(i\alpha_{-}-2i\rho^{2}t+4\zeta\nu
t-\int_{0}^{t}(A_{k}(\tau)+A_{k}^{*}(\tau))d\tau).$

Applying the procedure of the inverse scattering problem, we find

$u(x,t)=\rho e^{-2i\rho^{2}t}\cdot\frac{e^{i\alpha_{+}}e^{\nu
x}+e^{i\alpha_{-}}ce^{-\nu x+4\zeta\nu t-g(t)}}{e^{\nu x}+ce^{-\nu x+4\zeta\nu
t-g(t)}},$

where $g(t)=\int_{0}^{t}(A_{1}(\tau)+A_{1}^{*}(\tau))\,d\tau$.

Using representation (32) and conditions (8), we obtain

${F_{1}=\alpha_{1}(t)\cdot\left(\begin{array}[]{c}{-\frac{i(\zeta-i\nu)}{\rho}\cdot
e^{-i\alpha_{+}+2i\rho^{2}t}}\\\ {1}\end{array}\right)\cdot\frac{1}{e^{\nu
x}+ce^{-\nu x+4\zeta\nu t-g(t)}},}$ ${G_{1}=\frac{\nu
A_{1}(t)}{\alpha_{1}(t)}\cdot\left(\begin{array}[]{c}{\frac{i(\zeta-i\nu)}{\rho}\cdot
e^{i\alpha_{-}-2i\rho^{2}t}}\\\ {1}\end{array}\right)\cdot\frac{ce^{4\zeta\nu
t-g(t)}}{e^{\nu x}+ce^{-\nu x+4\zeta\nu t-g(t)}}.}$

Analogously, in the case (B), using results of Theorem 2, we obtain

$u(x,t)=\rho e^{-2i\rho^{2}t}\cdot\frac{e^{i\alpha_{+}}e^{\nu
x}+e^{i\alpha_{-}}ce^{-\nu x+4\zeta\nu t-g(t)}}{e^{\nu x}+ce^{-\nu x+4\zeta\nu
t-g(t)}},$
$F_{1}=-\frac{\rho^{2}B_{1}(t)}{2i\nu(\zeta-i\nu)}\cdot\left(\begin{array}[]{c}{-\frac{i(\zeta-i\nu)}{\rho}\cdot
e^{-i\alpha_{+}+2i\rho^{2}t}}\\\ {1}\end{array}\right)\cdot\frac{1}{e^{\nu
x}+ce^{-\nu x+4\zeta\nu t-g(t)}},$
${G_{1}=\frac{-2\nu}{\zeta+i\nu}\cdot\left(\nu\beta_{1}(t)-2x\zeta+i\sigma_{3}\right)\cdot\left(\begin{array}[]{c}{\frac{i(\zeta-i\nu)}{\rho}\cdot
e^{i\alpha_{-}-2i\rho^{2}t}}\\\ {1}\end{array}\right)\cdot\frac{ce^{4\zeta\nu
t-g(t)}}{e^{\nu x}+ce^{-\nu x+4\zeta\nu t-g(t)}}+}$
$+\left(\begin{array}[]{c}{\frac{i(\zeta-i\nu)}{\rho}\cdot
e^{i\alpha_{+}-2i\rho^{2}t}}\\\ {1}\end{array}\right)\cdot\frac{e^{2\nu
x}}{e^{\nu x}+ce^{-\nu x+4\zeta\nu t-g(t)}}+$
${+\left(\begin{array}[]{c}{-\frac{i(\zeta-i\nu)}{\rho}\cdot
e^{i\alpha_{-}-2i\rho^{2}t}}\\\
{e^{i\theta}}\end{array}\right)\cdot\frac{c^{2}e^{-2\nu x+8\zeta\nu
t-2g(t)}}{e^{\nu x}+ce^{-\nu x+4\zeta\nu t-g(t)}}},$

where
$g(t)=-i\int_{0}^{t}\left(\beta_{1}(\tau)-\beta_{1}^{*}(\tau)\right)B_{1}(\tau)d\tau$.

## Acknowledgments

This research was supported by program “Short-term research internships of
young scientists in leading foreign scientific organizations” of the Ministry
of Innovative Development of the Republic of Uzbekistan. Endless gratitude to
Professor Rogrigo Lopez for the great support of my research visit to
University of Santiago de Compostela.

## References

  * [1] Ablowitz M.J., Kaup D.J., Newell A.C., Segur H., The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), 249–315.
  * [2] Biondini G., Fagerstrom E., Prinari B., Inverse scattering transform for the defocusing nonlinear Schrödinger equation with fully asymmetric non-zero boundary conditions, Phys. D 333 (2016), 117–136.
  * [3] Biondini G., Kovačič G., Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions, J. Math. Phys. 55 (2014), 031506, 22.
  * [4] Demontis F., Prinari B., van der Mee C., Vitale F., The inverse scattering transform for the defocusing nonlinear Schrödinger equations with nonzero boundary conditions, Stud. Appl. Math. 131 (2013), 1–40.
  * [5] Demontis F., Prinari B., van der Mee C., Vitale F., The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions, J. Math. Phys. 55 (2014), 101505, 40.
  * [6] Demontis F., van der Mee C., Characterization of scattering data for the AKNS system, Acta Appl. Math. 131 (2014), 29–47.
  * [7] Demontis F., van der Mee C., A matrix Schrödinger approach to focusing nonlinear Schrödinger equations with nonvanishing boundary conditions, J. Nonlinear Sci. 32 (2022), Paper No. 57, 29.
  * [8] Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987.
  * [9] Khasanov A.B., Reĭimberganov A.A., On the finite-density solution of the nonlinear Schrödinger equation with a self-consistent source, Uzbek. Mat. Zh. (2009), 123–130.
  * [10] Melnikov V.K., Integration of the nonlinear Schroedinger equation with a self-consistent source, Comm. Math. Phys. 137 (1991), 359–381.
  * [11] Melnikov V.K., Integration of the nonlinear Schrödinger equation with a source, Inverse Problems 8 (1992), 133–147.
  * [12] Prinari B., Demontis F., Li S., Horikis T.P., Inverse scattering transform and soliton solutions for square matrix nonlinear Schrödinger equations with non-zero boundary conditions, Phys. D 368 (2018), 22–49.
  * [13] Reyimberganov A.A., Rakhimov I.D., The soliton solutions for the nonlinear Schrödinger equation with self-consistent source, Izv. Irkutsk. Gos. Univ. Ser. Mat. 36 (2021), 84–94.
  * [14] Urazboev G.U., Reyimberganov A.A., Babadjanova A.K., Integration of the matrix nonlinear Schrödinger equation with a source, Izv. Irkutsk. Gos. Univ. Ser. Mat. 37 (2021), 63–76.
  * [15] Yakhshimuratov A., The nonlinear Schrödinger equation with a self-consistent source in the class of periodic functions, Math. Phys. Anal. Geom. 14 (2011), 153–169.
  * [16] Zakharov V.E., Shabat A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Journal of Experimental and Theoretical Physics 61 (1971), 118–134.
  * [17] Zakharov V.E., Shabat A.B., Interaction between solitons in a stable medium, Soviet Journal of Experimental and Theoretical Physics 37 (1973), 823–828.


Despite their simple intuition, convolutions are more tedious to analyze than
dense layers, which complicates the generalization of theoretical and
algorithmic ideas. We provide a new perspective onto convolutions through
tensor networks (TNs) which allow reasoning about the underlying tensor
multiplications by drawing diagrams, and manipulating them to perform function
transformations, sub-tensor access, and fusion. We demonstrate this expressive
power by deriving the diagrams of various autodiff operations and popular
approximations of second-order information with full hyper-parameter support,
batching, channel groups, and generalization to arbitrary convolution
dimensions. Further, we provide convolution-specific transformations based on
the connectivity pattern which allow to re-wire and simplify diagrams before
evaluation. Finally, we probe computational performance, relying on
established machinery for efficient TN contraction. Our TN implementation
speeds up a recently-proposed KFAC variant up to 4.5 x and enables new
hardware-efficient tensor dropout for approximate backpropagation.

§ INTRODUCTION

§ PRELIMINARIES

§ TENSOR NETWORKS FOR CONVOLUTION OPERATIONS

§ TENSOR NETWORK TRANSFORMATIONS & IMPLEMENTATION

§ EXPERIMENTS

§ RELATED WORK

Structured convolutions: We use the TN formulation of convolution
from Hayashi et al., 2019 whose focus lies on connecting factorizations
of the kernel to existing structured convolutions (depth-wise
separable [Howard et al., 2017, Sandler et al., 2018],
factored [Szegedy et al., 2016], bottleneck [He et al., 2016],
flattened/CP decomposed, low-rank
filter [Smith, 1997, Rigamonti et al., 2013, Tai et al., 2015]) and
explore new factorizations. Our work focuses on operations related to
convolutions, diagram manipulations, the index pattern structure, and the
performance/flexibility of TNs. Structured convolutions integrate seamlessly
with our derivations by swapping the kernel with its factorized diagram.

Higher-order autodiff: ML frameworks prioritize differentiating
scalar-valued objectives once. A recent line of works
[Laue et al., 2018, Laue et al., 2020, Ma et al., 2020] developed a tensor
calculus framework for computing (higher-order) derivatives of
matrix/tensor-valued functions, along with compilation techniques based on
linear algebra and common sub-expression elimination (CSE). By phrasing
convolution as tensor multiplication, we allow it to be integrated into such
frameworks and make it amenable to their optimizations. Our convolution-specific
simplifications complement existing techniques.

§ CONCLUSION

We proposed using tensor networks (TNs), a diagrammatic representation of tensor
multiplications, to simplify the analysis and provide white-box implementations
of convolutions, as well as related routines for autodiff and approximation of
second-order information. We illustrated their elegance and expressiveness to
apply function transformations (differentiation, batching) and partial operand
access (diagonal extraction) by simple graphical manipulations. Concretely, we
derived the diagrams of the above operations, supporting all hyper-parameters of
convolution, features like channel groups and batching, and generalization to
arbitrary dimensions. TNs enable automated performance optimization via
contraction tree search, symbolic simplification, and distribution through a
single interface (). We extended those techniques by
convolution-specific simplifications based on structure in the connectivity
pattern and demonstrated their effectiveness to speed-up computations.

We believe that TNs are a versatile tool to simplify the analysis of
convolutions and open up new algorithmic possibilities due to their simplicity
and flexibility, while offering abstractions for automated performance
optimization. Our results motivate their application to efficient approximation
of higher-order information (up to 4.5 x speedup on a recently proposed KFAC
variant) and randomized autodiff with structured dropout to reduce cost. We look
forward to exploring those, among others, in more detail in future work.

§.§.§ Acknowledgments

The author would like to thank Luca Thiede for providing feedback to the

[Adelman et al., 2021]
Menachem Adelman, Kfir Levy, Ido Hakimi, and Mark Silberstein.
Faster neural network training with approximate tensor operations.
In Advances in Neural Information Processing Systems
(NeurIPS), 2021.

[Arora et al., 2019]
Sanjeev Arora, Simon S Du, Wei Hu, Zhiyuan Li, Russ R Salakhutdinov, and
Ruosong Wang.
On exact computation with an infinitely wide neural net.
Advances in neural information processing systems (NeurIPS),

[Bahamou et al., 2023]
Achraf Bahamou, Donald Goldfarb, and Yi Ren.
A mini-block fisher method for deep neural networks.
In International Conference on Artificial Intelligence and
Statistics (AISTATS), 2023.

[Becker & Lecun, 1989]
Suzanna Becker and Yann Lecun.
Improving the convergence of back-propagation learning with
second-order methods.

[Biamonte & Bergholm, 2017]
Jacob Biamonte and Ville Bergholm.
Tensor networks in a nutshell, 2017.

[Botev et al., 2017]
Aleksandar Botev, Hippolyt Ritter, and David Barber.
Practical Gauss-Newton optimisation for deep learning.
In International Conference on Machine Learning (ICML), 2017.

[Bradbury et al., 2018]
James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary,
Dougal Maclaurin, and Skye Wanderman-Milne.
JAX: composable transformations of Python+NumPy programs,

[Bridgeman & Chubb, 2017]
Jacob C Bridgeman and Christopher T Chubb.
Hand-waving and interpretive dance: an introductory course on tensor
Journal of Physics A: Mathematical and theoretical, 2017.

[Chellapilla et al., 2006]
Kumar Chellapilla, Sidd Puri, and Patrice Simard.
High performance convolutional neural networks for document
In International Workshop on Frontiers in Handwriting
Recognition, 2006.

[Chen et al., 2023]
Joya Chen, Kai Xu, Yuhui Wang, Yifei Cheng, and Angela Yao.
DropIT: Dropping intermediate tensors for memory-efficient DNN
In International Conference on Learning Representations
(ICLR), 2023.

[Chen et al., 2020]
Yinpeng Chen, Xiyang Dai, Mengchen Liu, Dongdong Chen, Lu Yuan, and Zicheng
Dynamic convolution: Attention over convolution kernels.
In IEEE/CVF conference on computer vision and pattern
recognition (CVPR), 2020.

[cuQuantum development team, 2023]
The cuQuantum development team.
cuQuantum SDK: A high-performance library for accelerating
quantum information science, 2023.

[Damm et al., 2002]
Carsten Damm, Markus Holzer, and Pierre McKenzie.
The complexity of tensor calculus.
computational complexity, 2002.

[Dangel, 2021]
Felix Dangel.
unfoldNd: (n=1,2,3)-dimensional unfold (im2col) and fold (col2im)
in pytorch, 2021.

[Dangel et al., 2020]
Felix Dangel, Stefan Harmeling, and Philipp Hennig.
Modular block-diagonal curvature approximations for feedforward
In International Conference on Artificial Intelligence and
Statistics (AISTATS), 2020a.

[Dangel et al., 2020]
Felix Dangel, Frederik Kunstner, and Philipp Hennig.
BackPACK: Packing more into backprop.
In International Conference on Learning Representations
(ICLR), 2020b.

[Dangel et al., 2022]
Felix Dangel, Lukas Tatzel, and Philipp Hennig.
ViViT: Curvature access through the generalized
gauss-newton’s low-rank structure.
Transactions on Machine Learning Research (TMLR), 2022.

[Dangel, 2023]
Felix Julius Dangel.
Backpropagation beyond the gradient.

[Daxberger et al., 2021]
Erik Daxberger, Agustinus Kristiadi, Alexander Immer, Runa Eschenhagen,
Matthias Bauer, and Philipp Hennig.
Laplace redux - effortless bayesian deep learning.
In Advances in Neural Information Processing Systems
(NeurIPS), 2021.

[Dumoulin & Visin, 2016]
Vincent Dumoulin and Francesco Visin.
A guide to convolution arithmetic for deep learning.

[Elsayed & Mahmood, 2023]
Mohamed Elsayed and A. Rupam Mahmood.
HesScale: Scalable computation of hessian diagonals.

[Eschenhagen, 2022]
Runa Eschenhagen.
Kronecker-factored approximate curvature for linear weight-sharing

[Goldfarb et al., 2021]
Donald Goldfarb, Yi Ren, and Achraf Bahamou.
Practical quasi-newton methods for training deep neural networks,

[Gray & Kourtis, 2021]
Johnnie Gray and Stefanos Kourtis.
Hyper-optimized tensor network contraction.
Quantum, 2021.

[Griewank & Walther, 2008]
Andreas Griewank and Andrea Walther.
Evaluating derivatives: principles and techniques of
algorithmic differentiation.
SIAM, 2008.

[Grosse & Martens, 2016]
Roger Grosse and James Martens.
A kronecker-factored approximate Fisher matrix for convolution
In International Conference on Machine Learning (ICML), 2016.

[Hayashi et al., 2019]
Kohei Hayashi, Taiki Yamaguchi, Yohei Sugawara, and Shin-ichi Maeda.
Exploring unexplored tensor network decompositions for convolutional
neural networks.
In Advances in Neural Information Processing Systems
(NeurIPS), 2019.

[He et al., 2016]
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
Deep residual learning for image recognition.
In IEEE conference on computer vision and pattern recognition
(CVPR), 2016.

[Howard et al., 2017]
Andrew G Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang,
Tobias Weyand, Marco Andreetto, and Hartwig Adam.
Mobilenets: Efficient convolutional neural networks for mobile vision

[Hu et al., 2018]
Jie Hu, Li Shen, and Gang Sun.
Squeeze-and-excitation networks.
In IEEE/CVF conference on computer vision and pattern
recognition (CVPR), 2018.

[Huang et al., 2021]
Cupjin Huang, Fang Zhang, Michael Newman, Xiaotong Ni, Dawei Ding, Junjie Cai,
Xun Gao, Tenghui Wang, Feng Wu, Gengyan Zhang, et al.
Efficient parallelization of tensor network contraction for
simulating quantum computation.
Nature Computational Science, 2021.

[Huang et al., 2017]
Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger.
Densely connected convolutional networks.
In IEEE conference on computer vision and pattern recognition
(CVPR), 2017.

[Jacot et al., 2020]
Arthur Jacot, Franck Gabriel, and Clément Hongler.
Neural tangent kernel: Convergence and generalization in neural
networks, 2020.

[Jastrzebski et al., 2020]
Stanislaw Jastrzebski, Maciej Szymczak, Stanislav Fort, Devansh Arpit, Jacek
Tabor, Kyunghyun Cho*, and Krzysztof Geras*.
The break-even point on optimization trajectories of deep neural
In International Conference on Learning Representations
(ICLR), 2020.

[Krizhevsky et al., 2012]
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton.
Imagenet classification with deep convolutional neural networks.
In Advances in Neural Information Processing Systems
(NeurIPS), 2012.

[Laue et al., 2018]
Sören Laue, Matthias Mitterreiter, and Joachim Giesen.
Computing higher order derivatives of matrix and tensor expressions.
Advances in Neural Information Processing Systems (NeurIPS),

[Laue et al., 2020]
Sören Laue, Matthias Mitterreiter, and Joachim Giesen.
A simple and efficient tensor calculus.
In AAAI Conference on Artificial Intelligence, 2020.

[LeCun et al., 1989]
Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and
L. D. Jackel.
Backpropagation applied to handwritten zip code recognition.
Neural Computation, 1989.

[Li et al., 2022]
Chao Li, Aojun Zhou, and Anbang Yao.
Omni-dimensional dynamic convolution.
In International Conference on Learning Representations
(ICLR), 2022.

[Liu et al., 2023]
Shiwei Liu, Tianlong Chen, Xiaohan Chen, Xuxi Chen, Qiao Xiao, Boqian Wu, Tommi
Kärkkäinen, Mykola Pechenizkiy, Decebal Constantin Mocanu, and
Zhangyang Wang.
More convnets in the 2020s: Scaling up kernels beyond 51x51 using
In International Conference on Learning Representations
(ICLR), 2023.

[Liu et al., 2022]
Zhuang Liu, Hanzi Mao, Chao-Yuan Wu, Christoph Feichtenhofer, Trevor Darrell,
and Saining Xie.
A convnet for the 2020s.
IEEE/CVF Conference on Computer Vision and Pattern Recognition
(CVPR), 2022.

[Ma et al., 2020]
Linjian Ma, Jiayu Ye, and Edgar Solomonik.
Autohoot: Automatic high-order optimization for tensors.
In International Conference on Parallel Architectures and
Compilation Techniques (PACT), 2020.

[Magnus & Neudecker, 1999]
J. R. Magnus and H. Neudecker.
Matrix Differential Calculus with Applications in
Statistics and Econometrics.
Probabilistics and Statistics. 1999.

[Martens, 2010]
James Martens.
Deep learning via Hessian-free optimization.
In International Conference on Machine Learning (ICML), 2010.

[Martens, 2020]
James Martens.
New insights and perspectives on the natural gradient method, 2020.

[Martens & Grosse, 2015]
James Martens and Roger Grosse.
Optimizing neural networks with Kronecker-factored approximate
In International Conference on Machine Learning (ICML), 2015.

[Martens et al., 2018]
James Martens, Jimmy Ba, and Matt Johnson.
Kronecker-factored curvature approximations for recurrent neural
In International Conference on Learning Representations
(ICLR), 2018.

[Novak et al., 2022]
Roman Novak, Jascha Sohl-Dickstein, and Samuel S. Schoenholz.
Fast finite width neural tangent kernel.
In International Conference on Machine Learning (ICML), 2022.

[Oktay et al., 2021]
Deniz Oktay, Nick McGreivy, Joshua Aduol, Alex Beatson, and Ryan P Adams.
Randomized automatic differentiation.
In International Conference on Learning Representations
(ICLR), 2021.

[Osawa et al., 2023]
Kazuki Osawa, Satoki Ishikawa, Rio Yokota, Shigang Li, and Torsten Hoefler.
Asdl: A unified interface for gradient preconditioning in pytorch,

[Paszke et al., 2019]
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory
Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban
Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan
Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith
PyTorch: An imperative style, high-performance deep learning
In Advances in Neural Information Processing Systems
(NeurIPS). 2019.

[Penrose, 1971]
Roger Penrose.
Applications of negative dimensional tensors.
Combinatorial Mathematics and its Applications, 1971.

[Petersen et al., 2023]
Felix Petersen, Tobias Sutter, Christian Borgelt, Dongsung Huh, Hilde Kuehne,
Yuekai Sun, and Oliver Deussen.
ISAAC newton: Input-based approximate curvature for newton's
In International Conference on Learning Representations
(ICLR), 2023.

[Ren et al., 2022]
Yi Ren, Achraf Bahamou, and Donald Goldfarb.
Kronecker-factored quasi-newton methods for deep learning, 2022.

[Rigamonti et al., 2013]
Roberto Rigamonti, Amos Sironi, Vincent Lepetit, and Pascal Fua.
Learning separable filters.
In IEEE Conference on Computer Vision and Pattern Recognition
(CVPR), 2013.

[Rochette et al., 2019]
Gaspar Rochette, Andre Manoel, and Eric W. Tramel.
Efficient per-example gradient computations in convolutional neural
networks, 2019.

[Rogozhnikov, 2022]
Alex Rogozhnikov.
Einops: Clear and reliable tensor manipulations with einstein-like
In International Conference on Learning Representations
(ICLR), 2022.

[Sandler et al., 2018]
Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh
Mobilenetv2: Inverted residuals and linear bottlenecks.
In IEEE conference on computer vision and pattern recognition
(CVPR), 2018.

[Schneider et al., 2019]
Frank Schneider, Lukas Balles, and Philipp Hennig.
DeepOBS: A deep learning optimizer benchmark suite.
In International Conference on Learning Representations
(ICLR), 2019.

[Schraudolph, 2002]
Nicol N Schraudolph.
Fast curvature matrix-vector products for second-order gradient
Neural computation, 2002.

[Singh et al., 2021]
Sidak Pal Singh, Gregor Bachmann, and Thomas Hofmann.
Analytic insights into structure and rank of neural network hessian
Advances in Neural Information Processing Systems (NeurIPS),

[Singh et al., 2023]
Sidak Pal Singh, Thomas Hofmann, and Bernhard Schölkopf.
The hessian perspective into the nature of convolutional neural

[Smith & Gray, 2018]
Daniel G. A. Smith and Johnnie Gray.
opt_einsum - A python package for optimizing contraction order
for einsum-like expressions.
Journal of Open Source Software (JOSS), 2018.

[Smith, 1997]
Steven W. Smith.
The scientist and engineer's guide to digital signal processing.

[Springenberg et al., 2015]
Jost Tobias Springenberg, Alexey Dosovitskiy, Thomas Brox, and Martin
Striving for simplicity: The all convolutional net, 2015.

[Szegedy et al., 2016]
Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew
Rethinking the inception architecture for computer vision.
In IEEE conference on computer vision and pattern recognition
(CVPR), 2016.

[Tai et al., 2015]
Cheng Tai, Tong Xiao, Yi Zhang, Xiaogang Wang, et al.
Convolutional neural networks with low-rank regularization.

[Vaswani et al., 2017]
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones,
Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin.
Attention is all you need.
In Advances in Neural Information Processing Systems
(NeurIPS), 2017.

[Xie et al., 2017]
Saining Xie, Ross Girshick, Piotr Dollár, Zhuowen Tu, and Kaiming He.
Aggregated residual transformations for deep neural networks.
In IEEE conference on computer vision and pattern recognition
(CVPR), 2017.

[Yang et al., 2019]
Brandon Yang, Gabriel Bender, Quoc V Le, and Jiquan Ngiam.
Condconv: Conditionally parameterized convolutions for efficient
Advances in Neural Information Processing Systems (NeurIPS),

[Zagoruyko & Komodakis, 2016]
Sergey Zagoruyko and Nikos Komodakis.
Wide residual networks.

[Zhang, 2020]
Fang Zhang.
A parallel tensor network contraction algorithm and its applications
in quantum computation.

(Supplementary Material)
0.3in minus 0.1in

In this supplementary material we provide additional details of the main text. Overview:

§ LIMITATIONS

§ VISUAL TOUR OF TENSOR NETWORK OPERATIONS FOR CONVOLUTIONS

§ IMPLEMENTATION DETAILS

§ CONVOLUTION LAYER HYPER-PARAMETER ANALYSIS

§ PERFORMANCE EVALUATION DETAILS (GPU)

Here we provide all details on the run time evaluation from the main text. We
consider the convolutions from the CNNs from <Ref>.
Experiments were carried out on an Nvidia Tesla T4 (16 GB memory). We use a
batch size of 32 for the ImageNet architectures, and 128 for the others.

§.§ Protocol & Overview

§.§ Forward Pass

§.§ Input VJP

§.§ Weight VJP

§.§ KFC Factor (KFAC-expand)

§.§ KFAC-reduce Factor

§ MISCELLANEOUS

§.§ Example: Associativity of Tensor


# A new stability result for the 2D Boussinesq equations with horizontal
dissipation

Hong Sung Jin  Minkyu Kwak  Bataa Lkhagvasuren  Department Of
Mathematics,Chonnam National University, Gwangju, South Korea

###### Abstract

For $\mathbb{R}^{2}$, the stability of smooth solutions of 2D anisotropic
Boussinesq equations with horizontal dissipation is an open problem. In this
work, we present a partial answer to this problem in a rougher function space
$H^{0,s}(\mathbb{R}^{2})$. Moreover, the previous stability results with the
regularity index $\frac{1}{2}<s<1$ is extended to an integer $1\leq s$ and the
elementary techniques are used.

###### keywords:

2D Boussinesq equations , anisotropic viscosity , stability

††journal: Journal Name

##  1 Introduction

In this paper, we study the following Boussinesq equation with partial
dissipations:

$\begin{split}\partial_{t}u+u\cdot\nabla u=&-\nabla P+\nu\partial_{11}u+\Theta
e_{2},\\\
\partial_{t}\Theta+u\cdot\nabla\Theta=&~{}\kappa\partial_{11}\Theta,\\\
\nabla\cdot u=&0,\end{split}$ (1)

where $u=(u_{1},u_{2})$ is the velocity field, $P$ is the pressure, $\Theta$
is the temperature, $e_{2}=(0,1)$ is the vertical unit vector and
$\nu,\kappa>0$ are the viscosity and the thermal diffusivity, respectively.
The system (1) is the anisotropic version of the standard 2D Boussinesq
equations:

$\begin{split}\partial_{t}u+u\cdot\nabla u=&-\nabla P+\nu\Delta u+\Theta
e_{2},\\\ \partial_{t}\Theta+u\cdot\nabla\Theta=&~{}\kappa\Delta\Theta,\\\
\nabla\cdot u=&0.\end{split}$ (2)

Recently, the Boussinesq equations (2) have attracted great interests among
the researchers because of their applications in geophysical fluid mechanics
(oceanic, atmospheric flows) and mathematical importance. The Boussinesq
equations are derived from the fluid flow equations coupled with the state
equations (temperature, salinity, humidity, etc.) by the Boussinesq
approximation, which assumes that the fluid density varies not much around a
mean value. In the result, the Boussinesq equations are the coupled system of
Navier-Stokes equations with the convection-diffusion equation of scalars such
as temperature or density variation ([40, 15]). The Boussinesq equations are
the fundamental model for buoyancy-driven fluids such as large-scale
geophysical flows, stratification, and the Rayleigh-Bénard convection ([14,
36]).

Mathematically, 2D Boussinesq equations share some key similarities with 3D
Navier-Stokes and Euler equations such as the vortex stretching mechanism and
it is expected that the qualitative behavior of solutions for those systems
are identical for the axisymmetric swirling flows [37]. A lot of results have
been carried out to understand the fundamental questions of global existence
and regularity problems for various Boussinesq systems. We refer to [1, 2, 3,
4, 7, 6, 8, 10, 5, 9, 11, 13, 16, 17, 18, 21, 22, 23, 24, 25, 26, 32, 33, 34,
38, 39, 42, 43, 46, 47, 48, 19, 45] for an inexhaustible list.

More recently, the stability problem of a hydrostatic equilibrium for
anisotropic 2D Boussinesq equations (1) have attracted a considerable
interests among mathematical fluid dynamics community. The hydrostatic
equilibrium refers to the stationary solution $(u_{e},\Theta_{e},P_{e})$ with

$u_{e}=0,~{}\Theta_{e}=x_{2},~{}P_{e}=\frac{1}{2}x^{2}_{2}.$

The equations for the perturbation $(u,p,\theta)$ near the hydrostatic
equilibrium:

$p=P-\frac{1}{2}x^{2}_{2},~{}\theta=\Theta-x_{2}$

can be written as

$\begin{split}\partial_{t}u+u\cdot\nabla u=&-\nabla p+\nu\partial_{11}u+\theta
e_{2},\\\
\partial_{t}\theta+u\cdot\nabla\theta+u_{2}=&~{}\kappa\partial_{11}\theta,\\\
\nabla\cdot u=&0\end{split}$ (3)

with the initial data $u(x,0)=u_{0}(x),~{}\theta(x,0)=\theta_{0}(x)$.

For the system (1), the global existence and uniqueness results in Sobolev
spaces even with $\nu=0$ or $\kappa=0$ were established in [17] for the domain
$\mathbb{R}^{2}$ and in [34] for the periodic domain. However, the stability
problem for the system (1) or (3) remains open when the spatial domain is
$\mathbb{R}^{2}$. Interestingly, for the spatial domain
$\Omega=\mathbb{T}\times\mathbb{R}$ with $\mathbb{T}$ is 1D torus, the
stability problem was tackled in [19]. The key point in their work is the
decomposition of horizontal average and oscillation part and it allows one to
use the Poincaré type inequality for the oscillation part. The stability
problems concerned with the same spatial domain are established in the works
[44, 35, 27, 30] for the different systems and mixed dissipations.

In this paper, we address the stability problem in $H^{0,s}$ for the domain
$\mathbb{R}^{2}$. The functions in $H^{0,s}(\mathbb{R}^{2})$ are square-
integrable in the $x_{1}$ direction and $H^{s}(\mathbb{R})$ in $x_{2}$
direction (see the definition below). The same type of function spaces are
used in the works [41, 12, 28, 20, 31]. The following theorem is our main
result:

######  Theorem 1.

There exists a universal constant $C$ such that for all the initial data
$u_{0},\theta_{0}\in H^{0,s}$ with a integer $s>\frac{1}{2}$ satisfying

$\|u_{0}\|_{0,s}^{2}+\|\theta_{0}\|_{0,s}^{2}\leq\epsilon$ (4)

with
$\delta=\min\\{\frac{1}{4C},\frac{1}{16C^{2}}\\},~{}0<\epsilon\leq\frac{\delta}{4}$,
the system (3) has a global solution $u(t,x),\theta(t,x)$ in
$L^{\infty}([0;\infty);H^{0,s})\cap L^{2}([0;\infty);H^{1,s})$ and it is
uniformly bounded as

$\begin{split}\|u(t)\|^{2}_{0,s}+\|\theta(t)\|^{2}_{0,s}+\mu\int_{0}^{t}(\|\partial_{1}u(\tau)\|^{2}_{0,s}+\|\partial_{1}\theta(\tau)\|^{2}_{0,s})d\tau\leq\delta\end{split}$
(5)

for all $t>0$. In particular, the trivial solution is stable.

######  Remark 1.

For $\frac{1}{2}<s<1$, the same result was established for the 3D Navier-
Stokes equation [12] and quasi-geostropic equations [31]. Our result extends
the range of regularity index $s$ to some integer $s>\frac{1}{2}$. Moreover,
we use elementary methods to estimate nonlinear terms.

######  Remark 2.

It is very desirable to obtain the same result in more regular spaces
$H^{t,s}$ for $t>0$ and $s>\frac{1}{2}$. To compensate the lack of smoothing
effect from anisotropic dissipation, one usually exploits the divergence
condition $\partial_{2}u_{2}=-\partial_{1}u_{1}$ to obtain regularization
effect. We find the divergence condition can not be applied to the sort of
integrals
$\int_{\mathbb{R}^{2}}u_{2}\partial_{2}f\cdot\Lambda_{1}^{2t}\Lambda_{2}^{2s}fdx$
unless $t=0$ or $f=u_{2}$.

##  2 Notation and Preliminary

In this work, $C$ denotes a generic constant which may change from line to
line and depends on the other parameters. We denote by $\langle\xi\rangle$ the
quantity $\langle\xi\rangle=(1+|\xi|^{2})^{\frac{1}{2}}$.

######  Definition 1.

We define the anisotropic Sobolev space $H^{s,s^{\prime}}$ by the completion
of $\mathcal{S}(\mathbb{R}^{2})$ for the norm

$\|f\|_{s,s^{\prime}}\stackrel{{\scriptstyle\text{def}}}{{=}}\|\langle\xi_{1}\rangle^{s}\langle\xi_{2}\rangle^{s^{\prime}}\widehat{f}(\xi)\|_{L^{2}(\mathbb{R}^{2})}.$

Note that the above norm can be equivalently written in the form

$\|f\|_{s,s^{\prime}}=\|\|f\|_{{H}^{s}(\mathbb{R}_{x_{1}})}\|_{H^{s^{\prime}}(\mathbb{R}_{x_{2}})}=\|2^{js^{\prime}}\|\Delta_{j}f\|_{L^{2}(\mathbb{R}_{x_{2}};{H}^{s}(\mathbb{R}_{x_{1}}))}\|_{l^{2}(\mathbb{Z})},$

where ${H}^{s}(\mathbb{R})$ is the usual nonhomogeneous Sobolev space in
dimension one. We denote by $\Lambda_{k}$ the operator
$\Lambda_{k}=(1-\partial^{2}_{k})^{\frac{1}{2}}$, that is, the multiplicative
operator with the symbol $\langle\xi_{k}\rangle$ in the frequency space. The
simple application of Hölder inequality yields the interpolation result for
the anisotropic Sobolev spaces.

######  Proposition 1.

For $s,s^{\prime},t,t^{\prime}\in\mathbb{R}$ and $\alpha\in[0,1]$, if $f\in
H^{s,s^{\prime}}\cap H^{t,t^{\prime}}$, then $f\in H^{\alpha
s+(1-\alpha)t,\alpha s^{\prime}+(1-\alpha)t^{\prime}}$ and

$\|f\|_{\alpha s+(1-\alpha)t,\alpha
s^{\prime}+(1-\alpha)t^{\prime}}\leq\|f\|_{s,s^{\prime}}^{\alpha}\|f\|_{t,t^{\prime}}^{1-\alpha}.$

As pointed out in [28], the product estimation

$\|fg\|_{s+t-\frac{1}{2},t^{\prime}}\leq
C\|f\|_{s,s^{\prime}}\|g\|_{t,t^{\prime}},~{}\text{for}~{}t^{\prime}\leq
s^{\prime},~{}s^{\prime}+t^{\prime}>0$

can be proved, however, the following simple product rule can be applied in
our case.

######  Proposition 2.

For $s,t<\frac{1}{2},s+t>0,s^{\prime}>\frac{1}{2}$, there exists a constant
$C$ such that

$\|fg\|_{s+t-\frac{1}{2},0}\leq C\|f\|_{s,s^{\prime}}\|g\|_{t,0},$

or

$\|fg\|_{s+t-\frac{1}{2},0}\leq C\|f\|_{s,0}\|g\|_{t,s^{\prime}}.$

###### Proof.

Assuming $x_{2}$ as a parameter, we can write 1D product rule as

$\|fg\|_{{H}^{s+t-\frac{1}{2}}(\mathbb{R}_{x_{1}})}\leq
C\|f\|_{{H}^{s}(\mathbb{R}_{x_{1}})}\|g\|_{{H}^{t}(\mathbb{R}_{x_{1}})}.$

Then, it gives

$\begin{split}\|\|fg\|_{{H}^{s+t-\frac{1}{2}}(\mathbb{R}_{x_{1}})}\|_{L^{2}(\mathbb{R}_{x_{2}})}\leq
C\|\|f\|_{{H}^{s}(\mathbb{R}_{x_{1}})}\|g\|_{{H}^{t}(\mathbb{R}_{x_{1}})}\|_{L^{2}(\mathbb{R}_{x_{2}})}\\\
\leq
C\|f\|_{L^{\infty}(\mathbb{R}_{x_{2}};{H}^{s}(\mathbb{R}_{x_{1}}))}\|g\|_{t,0}.\end{split}$

Next, by Littlewood-Paley decomposition and Bernstein’s lemma, we have

$\begin{split}\|f\|_{L^{\infty}(\mathbb{R}_{x_{2}};{H}^{s}(\mathbb{R}_{x_{1}}))}\leq&\sum_{j}\|\Delta_{j}\|f\|_{{H}^{s}(\mathbb{R}_{x_{1}})}\|_{L^{\infty}(\mathbb{R}_{x_{2}})}\\\
\leq&\sum_{j}2^{-j(s^{\prime}-\frac{1}{2})}2^{js^{\prime}}\|\Delta_{j}\|f\|_{{H}^{s}(\mathbb{R}_{x_{1}})}\|_{L^{2}(\mathbb{R}_{x_{2}})}\\\
\leq&C\|f\|_{H^{s,s^{\prime}}}.\end{split}$

This proves the first inequality. The second inequality is proved analogously.
∎

##  3 The proof of Theorem 1

The existence of local smooth solution can be obtained easily. Thus, if we
obtain global a priori bound for the smooth solution, the standard compactness
argument give the global existence result. Multiplying the equations of $u$
and $\theta$ in (3) by $\Lambda_{2}^{2s}u$ and $\Lambda_{2}^{2s}\theta$
respectively, integrating by space variable and combining the resulting
equations, we obtain

$\begin{split}\frac{d}{dt}(\|u(t)\|^{2}_{0,s}+\|\theta(t)\|^{2}_{0,s})+2\nu\|\partial_{1}u(t)\|^{2}_{0,s}+2\kappa\|\partial_{1}\theta(t)\|^{2}_{0,s}\\\
=-\int_{\mathbb{R}^{2}}u\cdot\nabla
u\cdot\Lambda_{2}^{2s}udx-\int_{\mathbb{R}^{2}}u\cdot\nabla\theta\cdot\Lambda_{2}^{2s}\theta
dx.\end{split}$ (6)

We decompose the first integral in (6) and estimate each term.

$\begin{split}\int_{\mathbb{R}^{2}}u\cdot\nabla
u\cdot\Lambda_{2}^{2s}udx=\int_{\mathbb{R}^{2}}u_{1}\partial_{1}u_{1}\cdot\Lambda_{2}^{2s}u_{1}dx+\int_{\mathbb{R}^{2}}u_{2}\partial_{2}u_{1}\cdot\Lambda_{2}^{2s}u_{1}dx\\\
+\int_{\mathbb{R}^{2}}u_{1}\partial_{1}u_{2}\cdot\Lambda_{2}^{2s}u_{2}dx+\int_{\mathbb{R}^{2}}u_{2}\partial_{2}u_{2}\cdot\Lambda_{2}^{2s}u_{2}dx\\\
=I_{1}+I_{2}+I_{3}+I_{4}.\end{split}$ (7)

To estimate $I_{1}$, we first apply Plancherel’s formula, Hölder’s inequality
and obtain

$\begin{split}I_{1}=&\int_{\mathbb{R}^{2}}u_{1}\partial_{1}u_{1}\cdot\Lambda_{2}^{2s}u_{1}dx\\\
=&\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{4}}\langle\xi_{2}\rangle^{s}\widehat{u_{1}}(\xi-\eta)\widehat{\partial_{1}u_{1}}(\eta)\langle\xi_{2}\rangle^{s}\overline{\widehat{u_{1}}}(\xi)d\eta
d\xi.\end{split}$ (8)

One can easily check the inequality

$\begin{split}\langle\xi_{2}\rangle^{s}\leq
2^{\frac{s}{2}}\sum_{k=0}^{s}\binom{s}{k}\langle\xi_{2}-\eta_{2}\rangle^{s-k}\langle\eta_{2}\rangle^{k}.\end{split}$
(9)

Thus, by Proposition 2, we get

$\begin{split}I_{1}&\leq
C\sum_{k=0}^{s}\binom{s}{k}\int_{\mathbb{R}^{4}}\widehat{\Lambda_{2}^{s-k}u_{1}}(\xi-\eta)\widehat{\partial_{1}\Lambda_{2}^{k}u_{1}}(\eta)\langle\xi_{2}\rangle^{s}\overline{\widehat{u_{1}}}(\xi)d\eta
d\xi\\\ &\leq
C\sum_{k=0}^{s}\binom{s}{k}\int_{\mathbb{R}^{4}}\langle\xi_{1}\rangle^{-\frac{1}{4}}\widehat{\Lambda_{2}^{s-k}u_{1}}(\xi-\eta)\widehat{\partial_{1}\Lambda_{2}^{k}u_{1}}(\eta)\langle\xi_{1}\rangle^{\frac{1}{4}}\langle\xi_{2}\rangle^{s}\overline{\widehat{u_{1}}}(\xi)d\eta
d\xi\\\ &\leq
C\sum_{k=0}^{s}\binom{s}{k}\|\Lambda_{2}^{s-k}u_{1}\partial_{1}\Lambda_{2}^{k}u_{1}\|_{-\frac{1}{4},0}\|u_{1}\|_{\frac{1}{4},s}\\\
&\leq C\|\partial_{1}u_{1}\|_{0,s}\|u_{1}\|_{\frac{1}{4},s}^{2}.\end{split}$
(10)

Similarly, the integral $I_{3}$ can be written as

$\begin{split}I_{3}\leq
C\|\partial_{1}u_{2}\|_{0,s}\|u_{1}\|_{\frac{1}{4},s}\|u_{2}\|_{\frac{1}{4},s}.\end{split}$
(11)

Thanks to the divergence condition $\partial_{2}u_{2}=-\partial_{1}u_{1}$, the
similar computation gives

$\begin{split}I_{4}\leq
C\|\partial_{1}u_{1}\|_{0,s}\|u_{2}\|_{\frac{1}{4},s}^{2}.\end{split}$ (12)

The estimation of $I_{2}$ involves more delicacy. Applying Plancherel’s
formula yields

$\begin{split}I_{2}=&\int_{\mathbb{R}^{2}}u_{2}\partial_{2}u_{1}\cdot\Lambda_{2}^{2s}u_{1}dx\\\
=&\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\widehat{u_{2}\partial_{2}u_{1}}(\xi)\langle\xi_{2}\rangle^{2s}\overline{\widehat{u_{1}}}(\xi)d\xi\\\
=&\frac{1}{(2\pi)^{4}}\int_{\mathbb{R}^{2}}\widehat{u_{2}}*\widehat{\partial_{2}u_{1}}(\xi)\langle\xi_{2}\rangle^{2s}\widehat{u_{1}}(-\xi)d\xi\\\
=&\frac{i}{(2\pi)^{4}}\int_{\mathbb{R}^{4}}\eta_{2}\widehat{u_{2}}(\xi-\eta)\widehat{u_{1}}(\eta)\langle\xi_{2}\rangle^{2s}\widehat{u_{1}}(-\xi)d\eta
d\xi.\end{split}$ (13)

By the change of variables $(\xi,\eta)\rightarrow(-\eta,-\xi)$, one obtains

$\begin{split}I_{2}=\frac{i}{2(2\pi)^{4}}\int_{\mathbb{R}^{4}}\left(\eta_{2}\langle\xi_{2}\rangle^{2s}-\xi_{2}\langle\eta_{2}\rangle^{2s}\right)\widehat{u_{2}}(\xi-\eta)\widehat{u_{1}}(\eta)\widehat{u_{1}}(-\xi)d\eta
d\xi.\end{split}$ (14)

Furthermore, we have

$\begin{split}\eta_{2}\langle\xi_{2}\rangle^{2s}-&\xi_{2}\langle\eta_{2}\rangle^{2s}=(\eta_{2}-\xi_{2})\langle\xi_{2}\rangle^{2s}+\xi_{2}(\langle\xi_{2}\rangle^{2s}-\langle\eta_{2}\rangle^{2s})\\\
=&(\eta_{2}-\xi_{2})\left(\langle\xi_{2}\rangle^{2s}-\xi_{2}(\eta_{2}+\xi_{2})\sum_{k=0}^{s-1}\langle\eta_{2}\rangle^{2k}\langle\xi_{2}\rangle^{2(s-k-1)}\right)\\\
\leq&|\eta_{2}-\xi_{2}|\underbrace{\left(\langle\xi_{2}\rangle^{2s}+\langle\xi_{2}\rangle(\langle\eta_{2}\rangle+\langle\xi_{2}\rangle)\sum_{k=0}^{s-1}\langle\eta_{2}\rangle^{2k}\langle\xi_{2}\rangle^{2(s-k-1)}\right)}_{P(\xi_{2},\eta_{2})}.\end{split}$
(15)

From (14) and (15), one can write as

$\begin{split}I_{2}\leq\frac{i}{2(2\pi)^{4}}\int_{\mathbb{R}^{4}}|\xi_{2}-\eta_{2}||\widehat{u_{2}}(\xi-\eta)||\widehat{u_{1}}(\eta)|P(\xi_{2},\eta_{2})|\widehat{u_{1}}(-\xi)|d\eta
d\xi,\end{split}$ (16)

where $P(\xi_{2},\eta_{2})$ is a polynomial with terms
$\langle\eta_{2}\rangle^{m}\langle\xi_{2}\rangle^{n}$ for $m+n\leq 2s$. In the
sum of integrals of (16), if we make change of variables
$(\xi,\eta)\rightarrow(-\eta,-\xi)$ whenever $m>n$, (16) takes the form

$\begin{split}I_{2}\leq\frac{i}{2(2\pi)^{4}}\int_{\mathbb{R}^{4}}|\xi_{2}-\eta_{2}||\widehat{u_{2}}(\xi-\eta)||\widehat{u_{1}}(\eta)|\widetilde{P}(\xi_{2},\eta_{2})|\widehat{u_{1}}(-\xi)|d\eta
d\xi,\end{split}$ (17)

where $\widetilde{P}(\xi_{2},\eta_{2})$ consists of the sum of elements
$\langle\eta_{2}\rangle^{m}\langle\xi_{2}\rangle^{n}$ for $m\leq n$. It is
clear that $m\leq s$ and $n\geq s$. Thus, for $n>s$, we have

$\begin{split}\langle\eta_{2}\rangle^{m}\langle\xi_{2}\rangle^{n}=&\langle\eta_{2}\rangle^{m}\langle\xi_{2}\rangle^{s}\langle\xi_{2}\rangle^{n-s}\\\
\leq&\langle\eta_{2}\rangle^{m}\langle\xi_{2}\rangle^{s}2^{\frac{n-s}{2}}\sum_{k=0}^{n-s}\binom{n-s}{k}\langle\xi_{2}-\eta_{2}\rangle^{n-s-k}\langle\eta_{2}\rangle^{k}.\end{split}$
(18)

Applying (18) to $\widetilde{P}(\xi_{2},\eta_{2})$, we get
$\widehat{P}(\xi_{2},\eta_{2})$ that satisfies

$\begin{split}\widetilde{P}(\xi_{2},\eta_{2})\leq\widehat{P}(\xi_{2},\eta_{2})\end{split}$

and whose term has the form
$\langle\eta_{2}\rangle^{m}\langle\xi_{2}-\eta_{2}\rangle^{n}\langle\xi_{2}\rangle^{k}$
with $m,n,k\leq s,~{}m+n+k\leq 2s$. Therefore, (17) becomes

$\begin{split}I_{2}\leq\frac{i}{2(2\pi)^{4}}\int_{\mathbb{R}^{4}}|\xi_{2}-\eta_{2}||\widehat{u_{2}}(\xi-\eta)||\widehat{u_{1}}(\eta)|\widehat{P}(\xi_{2},\eta_{2})|\widehat{u_{1}}(-\xi)|d\eta
d\xi.\end{split}$ (19)

Now, using the divergence condition $\partial_{2}u_{2}=-\partial_{1}u_{1}$,
for each term in (19), one can estimate as

$\begin{split}&\int_{\mathbb{R}^{4}}\langle\xi_{2}-\eta_{2}\rangle^{n}|\widehat{\partial_{1}u_{1}}(\xi-\eta)|\langle\eta_{2}\rangle^{m}|\widehat{u_{1}}(\eta)|\langle\xi_{2}\rangle^{k}|\widehat{u_{1}}(-\xi)|d\eta
d\xi\\\
&\leq\int_{\mathbb{R}^{4}}\langle\xi_{1}\rangle^{-\frac{1}{4}}|\widehat{\Lambda_{2}^{n}\partial_{1}u_{1}}(\xi-\eta)||\widehat{\Lambda_{2}^{m}u_{1}}(\eta)|\langle\xi_{1}\rangle^{\frac{1}{4}}|\widehat{\Lambda_{2}^{k}u_{1}}(-\xi)|d\eta
d\xi\\\
&\leq\|\Lambda_{2}^{n}\partial_{1}u_{1}\Lambda_{2}^{m}u_{1}\|_{-\frac{1}{4},0}\|u_{1}\|_{\frac{1}{4},k}\\\
&\leq\left\\{\begin{array}[]{l}C\|\Lambda_{2}^{n}\partial_{1}u_{1}\|_{0,s-n}\|\Lambda_{2}^{m}u_{1}\|_{\frac{1}{4},0}\|u_{1}\|_{\frac{1}{4},k},~{}\text{if}~{}n<s\\\
C\|\Lambda_{2}^{n}\partial_{1}u_{1}\|_{0,0}\|\Lambda_{2}^{m}u_{1}\|_{\frac{1}{4},s-m}\|u_{1}\|_{\frac{1}{4},k},~{}\text{if}~{}n=s\end{array}\right.\\\
&\leq
C\|\partial_{1}u_{1}\|_{0,s}\|u_{1}\|_{\frac{1}{4},s}\|u_{1}\|_{\frac{1}{4},k}\leq
C\|\partial_{1}u_{1}\|_{0,s}\|u_{1}\|_{\frac{1}{4},s}^{2},\end{split}$ (20)

since $m+n\leq s$. Applying the above calculation to all the integrals in
(19), we obtain

$\begin{split}I_{2}\leq
C\|\partial_{1}u_{1}\|_{0,s}\|u_{1}\|_{\frac{1}{4},s}^{2}.\end{split}$ (21)

Collecting (8),(11),(12) and (21), we get

$\begin{split}\int_{\mathbb{R}^{2}}u\cdot\nabla u\cdot\Lambda_{2}^{2s}udx\leq
C\|\partial_{1}u\|_{0,s}\|u\|_{\frac{1}{4},s}^{2}.\end{split}$ (22)

Following the estimations for $I_{1}$ and $I_{2}$, we also have

$\begin{split}\int_{\mathbb{R}^{2}}u\cdot\nabla\theta\cdot\Lambda_{2}^{2s}\theta
dx\leq
C(\|\partial_{1}\theta\|_{0,s}\|u_{1}\|_{\frac{1}{4},s}\|\theta\|_{\frac{1}{4},s}+\|\partial_{1}u_{1}\|_{0,s}\|\theta\|_{\frac{1}{4},s}^{2})\\\
\leq
C(\|\partial_{1}\theta\|_{0,s}\|u\|_{\frac{1}{4},s}\|\theta\|_{\frac{1}{4},s}+\|\partial_{1}u\|_{0,s}\|\theta\|_{\frac{1}{4},s}^{2}).\end{split}$
(23)

Set $X(t,x)=(u(t,x),\theta(t,x))^{T}$. From (6),(22) and (23), one obtains

$\begin{split}\frac{d}{dt}\|X(t)\|^{2}_{0,s}+2\mu\|\partial_{1}X(t)\|^{2}_{0,s}\leq
C\|\partial_{1}X(t)\|_{0,s}\|X(t)\|_{\frac{1}{4},s}^{2}\end{split}$ (24)

with $\mu=\min\\{\nu,\kappa\\}$. By invoking the embedding
$H^{\frac{1}{2},s}\hookrightarrow H^{\frac{1}{4},s}$ and using the
interpolation inequality

$\|f\|_{\frac{1}{2},s}^{2}\leq\|f\|_{0,s}\|f\|_{1,s}$

with the norm equivalence
$\|f\|_{1,s}\simeq\|f\|_{0,s}+\|\partial_{1}f\|_{0,s}$, we arrive at

$\begin{split}\frac{d}{dt}\|X(t)\|^{2}_{0,s}+2\mu\|\partial_{1}X(t)\|^{2}_{0,s}\leq
C\|\partial_{1}X(t)\|_{0,s}^{2}\|X(t)\|_{0,s}\\\
+C\|\partial_{1}X(t)\|_{0,s}\|X(t)\|_{0,s}^{2}.\end{split}$ (25)

Applying Young’s inequality

$\begin{split}C\|\partial_{1}X(t)\|_{0,s}\|X(t)\|_{0,s}^{2}\leq\mu\|\partial_{1}X(t)\|_{0,s}^{2}+\frac{C^{2}\|X(t)\|_{0,s}^{4}}{2\mu}\end{split}$
(26)

and redefining the constant $C$, we get

$\begin{split}\frac{d}{dt}\|X(t)\|^{2}_{0,s}+\mu\|\partial_{1}X(t)\|^{2}_{0,s}\leq
C\|\partial_{1}X(t)\|_{0,s}^{2}\|X(t)\|_{0,s}+C\|X(t)\|_{0,s}^{4}.\end{split}$
(27)

Let us define the energy functional

$E(t)=\sup_{0\leq\tau\leq
t}\|X(\tau)\|^{2}_{0,s}+\mu\int_{0}^{t}\|\partial_{1}X(\tau)\|^{2}_{0,s}d\tau.$

Integrating (27) on $[0,t]$ gives an equality

$E(t)=E(0)+CE(t)^{\frac{3}{2}}+CE(t)^{2},~{}\text{for all}~{}t>0.$ (28)

Now, we use the standard bootstrap argument and prove the global bound for the
quantity $E(t)$. Set

$\begin{split}\delta=\min\\{\frac{1}{4C},\frac{1}{16C^{2}}\\},~{}0<\epsilon\leq\frac{\delta}{4}.\end{split}$

Let

$\begin{split}\Gamma=\sup\\{t\in[0,T):\sup_{\tau\in[0,t]}E(\tau)\leq\delta\\}\end{split}$

for some $T>0$. Suppose $\Gamma<T$. Then, from (28), it yields

$\begin{split}E(t)\leq 2E(0)\leq
2\epsilon\leq\frac{\delta}{2},~{}\text{for}~{}t\leq\Gamma.\end{split}$

By the continuity of the solution,

$E(t)\leq\frac{\delta}{2},~{}\text{for}~{}t\leq\Gamma+\eta<T$ (29)

and for some small $\eta>0$. (29) contradicts the definition of $\Gamma$,
therefore, it must be $\Gamma=T$. Since $T$ is arbitray, we can take
$T\rightarrow\infty$ and

$E(t)<\delta$ (30)

for all $t\in[0;\infty)$, if the smallness condition (4) is satisfied. This
finish the proof.

##  4 Acknowledgements

This research was partially supported by the National Research Fund with
numbers 2022R1F1A10630071, 2022R1I1A1A01055459 and 2020R1I1A3071769.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

  * [1] D. Adhikari, C. Cao, H. Shang, J. Wu, X. Xu, and Z. Ye.  Global regularity results for the 2D Boussinesq equations with partial dissipation.  J. Differential Equations, 260(2):1893–1917, 2016.
  * [2] D. Adhikari, C. Cao, J. Wu, and X. Xu.  Small global solutions to the damped two-dimensional Boussinesq equations.  J. Differential Equations, 256(11):3594–3613, 2014.
  * [3] C. Cao and J. Wu.  Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation.  Arch. Ration. Mech. Anal., 208(3):985–1004, 2013.
  * [4] A. Castro, D. Córdoba, and D. Lear.  On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term.  Math. Models Methods Appl. Sci., 29(7):1227–1277, 2019.
  * [5] D. Chae and P. Constantin.  Remarks on type I blow-up for the 3D Euler equations and the 2D Boussinesq equations.  J. Nonlinear Sci., 31(5):Paper No. 77,16, 2021.
  * [6] D. Chae, P. Constantin, and J. Wu.  An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations.  J. Math. Fluid Mech., 16(3):473–480, 2014.
  * [7] D. Chae, S.-K. Kim, and H.-S. Nam.  Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations.  Nagoya Math. J., 155:55–80, 1999.
  * [8] D. Chae, Q. Miao, and L. Xue.  Global regularity of nondiffusive temperature fronts for the two-dimensional viscous Boussinesq system.  SIAM J. Math. Anal., 54(4):4043–4103, 2022.
  * [9] D. Chae and H.-S. Nam.  Local existence and blow-up criterion for the Boussinesq equations.  Proc. Roy. Soc. Edinburgh Sect. A, 127(5):935–946, 1997.
  * [10] D. Chae and J. Wolf.  On the regularity of solutions to the 2D Boussinesq equations satisfying type I conditions.  J. Nonlinear Sci., 29(2):643–654, 2019.
  * [11] D. Chae and J. Wu.  The 2D Boussinesq equations with logarithmically supercritical velocities.  Adv. Math., 230(4-6):1618–1645, 2012.
  * [12] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier.  Fluids with anisotropic viscosity.  volume 34, pages 315–335. 2000.  Special issue for R. Temam’s 60th birthday.
  * [13] K. Choi, A. Kiselev, and Y. Yao.  Finite time blow up for a 1D model of 2D Boussinesq system.  Comm. Math. Phys., 334(3):1667–1679, 2015.
  * [14] P. Constantin and C. R. Doering.  Heat transfer in convective turbulence.  Nonlinearity, 9(4):1049–1060, 1996.
  * [15] B. Cushman-Roisin and J.-M. Beckers.  Introduction to geophysical fluid dynamics, volume 101 of International Geophysics Series.  Elsevier/Academic Press, Amsterdam, second edition, 2011.  Physical and numerical aspects, With a foreword by John Marshall.
  * [16] R. Danchin and M. Paicu.  Global well-posedness issues for the inviscid Boussinesq system with Yudovich’s type data.  Comm. Math. Phys., 290(1):1–14, 2009.
  * [17] R. Danchin and M. Paicu.  Global existence results for the anisotropic Boussinesq system in dimension two.  Math. Models Methods Appl. Sci., 21(3):421–457, 2011.
  * [18] C. R. Doering, J. Wu, K. Zhao, and X. Zheng.  Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion.  Phys. D, 376/377:144–159, 2018.
  * [19] B. Dong, J. Wu, X. Xu, and N. Zhu.  Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation.  Calc. Var. Partial Differential Equations, 60(3):Paper No. 116, 21, 2021.
  * [20] P. Dreyfuss and H. Houamed.  Uniqueness result for the 3-D Navier-Stokes-Boussinesq equations with horizontal dissipation.  J. Math. Fluid Mech., 23(1):Paper No. 19, 24, 2021.
  * [21] T. M. Elgindi and K. Widmayer.  Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems.  SIAM J. Math. Anal., 47(6):4672–4684, 2015.
  * [22] T. Hmidi, S. Keraani, and F. Rousset.  Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation.  J. Differential Equations, 249(9):2147–2174, 2010.
  * [23] T. Hmidi, S. Keraani, and F. Rousset.  Global well-posedness for Euler-Boussinesq system with critical dissipation.  Comm. Partial Differential Equations, 36(3):420–445, 2011.
  * [24] T. Y. Hou and C. Li.  Global well-posedness of the viscous Boussinesq equations.  Discrete Contin. Dyn. Syst., 12(1):1–12, 2005.
  * [25] W. Hu, I. Kukavica, and M. Ziane.  Persistence of regularity for the viscous Boussinesq equations with zero diffusivity.  Asymptot. Anal., 91(2):111–124, 2015.
  * [26] W. Hu, Y. Wang, J. Wu, B. Xiao, and J. Yuan.  Partially dissipative 2D Boussinesq equations with Navier type boundary conditions.  Phys. D, 376/377:39–48, 2018.
  * [27] W. Huang.  Stability and exponential decay of the magnetic Bénard system with horizontal dissipation.  J. Math. Anal. Appl., 519(1):Paper No. 126767, 25, 2023.
  * [28] D. Iftimie.  A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity.  SIAM J. Math. Anal., 33(6):1483–1493, 2002.
  * [29] R. Ji, L. Yan, and J. Wu.  Optimal decay for the 3D anisotropic Boussinesq equations near the hydrostatic balance.  Calc. Var. Partial Differential Equations, 61(4):Paper No. 136, 34, 2022.
  * [30] X. Ji and D. Bian.  On the stability of hydrostatic equilibrium in a uniform magnetic field with partial dissipation.  Math. Methods Appl. Sci., 45(8):4312–4342, 2022.
  * [31] H. S. Jin, M. Kwak, and B. Lkhagvasuren.  Stability result for the surface quasi-geostropic equations with horizontal dissipation in anisotropic Sobolev space.  J. Math. Phys., 63(9):Paper No. 091507, 7, 2022.
  * [32] Q. Jiu, J. Wu, and W. Yang.  Eventual regularity of the two-dimensional Boussinesq equations with supercritical dissipation.  J. Nonlinear Sci., 25(1):37–58, 2015.
  * [33] A. Kiselev and C. Tan.  Finite time blow up in the hyperbolic Boussinesq system.  Adv. Math., 325:34–55, 2018.
  * [34] A. Larios, E. Lunasin, and E. S. Titi.  Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion.  J. Differential Equations, 255(9):2636–2654, 2013.
  * [35] L. Ma, L. Li, and D. Liu.  Stability of the 2D anisotropic Boussinesq equations with mixed partial dissipation.  J. Math. Phys., 63(10):Paper No. 101505, 17, 2022.
  * [36] A. Majda.  Introduction to PDEs and waves for the atmosphere and ocean, volume 9 of Courant Lecture Notes in Mathematics.  New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
  * [37] A. J. Majda and A. L. Bertozzi.  Vorticity and incompressible flow, volume 27 of Cambridge Texts in Applied Mathematics.  Cambridge University Press, Cambridge, 2002.
  * [38] C. Miao and L. Xue.  On the global well-posedness of a class of Boussinesq-Navier-Stokes systems.  NoDEA Nonlinear Differential Equations Appl., 18(6):707–735, 2011\.
  * [39] M. Paicu and N. Zhu.  On the striated regularity for the 2D anisotropic Boussinesq system.  J. Nonlinear Sci., 30(3):1115–1164, 2020.
  * [40] J. Pedlosky.  Geophysical Fluid Dynamics.  Springer New York, 2013.
  * [41] J. Rauch and M. Reed.  Nonlinear microlocal analysis of semilinear hyperbolic systems in one space dimension.  Duke Math. J., 49(2):397–475, 1982.
  * [42] A. Stefanov and J. Wu.  A global regularity result for the 2D Boussinesq equations with critical dissipation.  J. Anal. Math., 137(1):269–290, 2019.
  * [43] L. Tao, J. Wu, K. Zhao, and X. Zheng.  Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion.  Arch. Ration. Mech. Anal., 237(2):585–630, 2020.
  * [44] P. Wang.  Stability for the 2D anisotropic surface quasi-geostrophic equation with horizontal dissipation.  Appl. Math. Lett., 123:Paper No. 107577, 9, 2022.
  * [45] J. Wu, X. Xu, and Z. Ye.  Global regularity for several incompressible fluid models with partial dissipation.  J. Math. Fluid Mech., 19(3):423–444, 2017.
  * [46] X. Xu.  Global regularity of solutions of 2D Boussinesq equations with fractional diffusion.  Nonlinear Anal., 72(2):677–681, 2010.
  * [47] Z. Ye and X. Xu.  Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation.  J. Differential Equations, 260(8):6716–6744, 2016.
  * [48] C. Zillinger.  On enhanced dissipation for the Boussinesq equations.  J. Differential Equations, 282:407–445, 2021.