INTEGRATION OF THE MATRIX NONLINEAR SCHRO¨DINGER EQUATION WITH A SOURCE Текст научной статьи по специальности «Математика»

% 1И..1..й11?

Серия «Математика»

2021. Т. 37. С. 63—76

Онлайн-доступ к журналу: http://mathizv.isu.ru

УДК 517.957

MSC 34L25, 35Q41, 35R30, 34M46

DOI https://doi.org/10.26516/1997-7670.2021.37.63

Integration of the Matrix Nonlinear Schrodinger Equation with a Source

G. U. Urazboev1, A. A. Reyimberganov1, A. K. Babadjanova2

1 Urgench State University, Urgench, Uzbekistan,

2 V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Urgench, Uzbekistan

Abstract. This paper is concerned with studying the matrix nonlinear Schrodinger equation with a self-consistent source. The source consists of the combination of the eigenfunctions of the corresponding spectral problem for the matrix Zakharov-Shabat system which has not spectral singularities. The theorem about the evolution of the scattering data of a non-self-adjoint matrix Zakharov-Shabat system which potential is a solution of the matrix nonlinear Schroodinger equation with the self-consistent source is proved. The obtained results allow us to solve the Cauchy problem for the matrix nonlinear Schroodinger equation with a self-consistent source in the class of the rapidly decreasing functions via the inverse scattering method. A one-to-one correspondence between the potential of the matrix Zakharov-Shabat system and scattering data provide the uniqueness of the solution of the considering problem. A step-by-step algorithm for finding a solution to the problem under consideration is presented.

Keywords: matrix nonlinear Schrodinger equation, self-consistent source, inverse scattering method, scattering data.

The inverse scattering transform method was first proposed by Gardner, Greene, Kruskal and Miura (GGKM) [5] in 1967 for solving the Cauchy problem for the Korteweg-de Vries (KdV) equation. Their approach was based on the connection between the KdV equation and the spectral theory for the Sturm-Liouville operator on the line. Shortly thereafter, P. Lax [9]

1. Introduction

pointed out the general character of the inverse scattering method. A few years later, V.E. Zakharov and A.B. Shabat [14] managed to solve another important nonlinear evolution equation, the so-called nonlinear Schrodinger (NLS) equation, using a nontrivial extension of the methods used in [5; 9].

The inverse scattering problem for the Dirac operator on the entire line was studied by V.E. Zakharov, A.B. Shabat [14], L.A. Takhtadzhyan, L.D. Faddeev [6], A.B. Khasanov [7] and others. The work relevant to the necessity and sufficient conditions for the solvability of the inverse scattering problem for the matrix Sturm-Liouville operator on the axis was studied in [1]. In the matrix case, the inverse scattering theory for the matrix Zakharov-Shabat system [3] was investigated by F. Demontis and C. Van der Mee and applied for the integration of the matrix NLS equation [4].

The NLS equation with the self-consistent sources in various classes of functions were considered by V.K. Melnikov [10], A.B. Khasanov, A.A. Reyimberganov [8], I.D. Rakhimov [11], A.B. Yakhshimuratov [13]. In this work, we consider the matrix NLS equation with the self-consistent source in the class of rapidly decreasing matrix functions. Other matrix nonlinear evolution equations with the self-consistent sources were integrated via the inverse scattering method in the works [2; 12].

2. Scattering theory for the matrix Zakharov-Shabat system

In this section, we give well-known [3], necessary information concerning the theory of direct and inverse scattering problems for the operator

d

L = -iJ-f - - V(x) dx

on the line (—to < x < to) with the rapidly decreasing potential by x. We consider the following matrix Zakharov-Shabat system

-iJX' - FX = XX,

(2.1)

where X(A, x) is 2m x m matrix function,

J

Im, °m 0 —T

V =

0m iU (x) iU *(x) 0m

and U(x) is an m x m matrix valued function, U*(x) denotes the complex conjugate of U(x), Im and 0m are the identity and zero matrices of order m, respectively.

We assume that the function U(x) satisfies the following condition

Hi :

(ж)У dx < то,

оо

where \\U(z)|| = max£™=1 lUjk(x)|.

3

For A G R the Jost matrices F (X, x) and G(X, x) as the 2m x 2m matrix solutions of (2.1) satisfy the following asymptotic conditions:

F(X,x) = [ip(X, x) tp(X, x)] G(X,x) = [0(A, x) <p(X, x)\

„iXJx t e 12m,

„iXJx t e 12m,

X

x -

—oo

(2.2)

Here ^(X, x), ip(X, x), 0(A, x) and 0(A, x) are the submatrices with 2m rows and m columns, which are usually called Jost solutions. Here and below bar does not mean complex conjugation.

The Jost solutions ip(X, x) and ^(X, x) of the equation (2.1) at any A G R can be represented in the following form

ip(X, x) = e

iXx

Irr

—iXx

where

^(X, x) = e

[K(x, y) K(x, y)] =

+ fx°° eiXyK(x, y)dy,

+ f™e-iX*K (x,y)dy,

Ki(x,y) K2(x, y) K3(x,y) KA(x, y)

(2.3)

Ks(x, y), s = 1,4 are m xm matrices. Here the kernels have relations with the potential

U (x) = 2iK2(x,x) = 2iK3(x,x). We also consider the following auxiliary equation

i Y 'J — YV = ^Y, where Y is an m x 2m matrix function.

(2.4)

Lemma 1. Let X (X,x) andY (y,x) be solutions of the equations (2.1) and (2.4), respectively, then the following relation holds

i(X - ß)Y(ß, x)X(A, x) = (YJX)'.

(2.5)

For X £ R there exists 2m x 2m matrix A(X) such that

G(X, x) = F(X,x)A(X), F (X,x) = G(X,x)C (X). Here A(X) and C(X) consist of block matrices such as

A(x)=(Ai(X) A2(X)\ A(A) = ^ A3(A) AUW ) '

(2.6)

0

m

0

m

m,

As(A), s = 1,4 are m xrn matrices.

Assuming that the potential U(x) have entries in %1, we can say that for each fixed x £ R the matrix functions ip(X,x)e-lXx and <(X,x)elXx (^(X,x)elXx and <(X,x)e-lXx) can be continued to the half-plane ImX > 0 (ImX < 0) and for all ImX > 0 (ImX < 0) the matrix functions ip(X, x)e-lXx and <(X,x)elXx (^(X,x)elXx and <(X,x)e-lXx) are bounded. Invertible matrix function A1 (A) (A4(A)) can be analytically continued to the half-plane ImX > 0 (ImX < 0) and there the equation det A1(A) = 0 (detA4(A) = 0) has a finite number of zeros Xj, j = 1, N (Aj, j = 1, NV) which correspond to the eigenvalues of the operator L.

Definition 1. The real zeros of the equation det A1(A) = 0 (det A4(A) = 0) will be called the spectral singularities of the equation (2.1).

Remark 1. If \\U(x)\\dx < ^ then there do not exist neither spectral singularities of the operator L [23].

We assume that the operator L has no spectral singularities and all the eigenvalues of the operator L are simple.

The matrix functions (A1(A))-1 and (A4(A))-1 have simple poles on the points Xj, j = 1, N in ImX > 0 and Aj, j = 1, N in ImX < 0, respectively. Let Nj = Ees(A1(A))-1, j = 1,N and Nj = Res(A4(X))-1, j = 1,N, then

X=Xj X=Xj

there are matrices Rj and RVj such that

ф(Хз, x)Nj = tp(Xj, x)Rj, j = 1, N,

(2.7)

(2.8)

¿(Xj, x)N3 = ^(Xj, x)Rj, j = 1, X. Definition 2. The following matrices for X G R

R(X) = C2 (X)C-1(X) = —A-1(X)A2 (X),

R(X) = C3(X)C-1(X) = -A4"1(X)AS (X) are called reflection coefficients. As V*(x) = — V (x), we have

(X, x) = aii>(X*,x), (X, x) = gi ¿(X*, x), (2.9)

where g1 ^ j™ \ X* is the complex conjugation of X. Then, from

\ 1 m "m /

it yields that,

and

— T 0

1 m 0m

N = N, Xj = X*, Nj = N*, Rj = R* R(X) = -R*(X).

Definition 3. The set {R(X), X1, X2, XN, R\, R2, ■ ■■, RN } is called the scattering data associated with the equation (2.1).

The direct scattering problem is to find the scattering data via the given potential U(x) of the equation (2.1) and inverse scattering problem is to find the potential U(x) of the equation (2.1) via the given scattering data.

The kernels of the representation (2.3) satisfy the following Gelfand-Levitan-Marchenko integral equations for x > y

K(x, y) + K(x, y) +

0.

Ù(x + y) + fx°° K (x, z)Ù(z + y)dz = 0, fi(x + y) + /x°° K(x, z)Ü(z + y)dz = 0,

where

N

Q(x) = 2k

I R(X) eiXxdX + -™ 3=1

Q(x) = —Q*(x).

Here, p(x) = I R(X)eiXxdX and Ce-xAB = ^N=1 RjeiXjX.

In the work [3], it was proven the uniquely determining of the potential U(x) by the scattering data.

3. Integration of the matrix NLS equation with a self-consistent

source

We consider the integration of the following problem

N

iUt + 2UU*U + Uxx = 2 ^($i,n$2,n + $2,n$i,n), (3.1)

n=1

—i J$n — v$n = Xn$n, n = 1,2,..., N,

under the initial condition

U | t=o = Uo(x).

(3.2)

(3.3)

Here $3)n = $j,n(x, t), j = 1,2 are m x m matrix functions, columns

of $n =

1,n

2,n

matrices are linearly independent eigenfunctions corre-

sponding to the eigenvalue Xn, n = 1,N and normalized by the following conditions

/œ

$n (x, t)$n(x, t)dx = a2n(t) Im, n = 1,2,...,N. (3.4)

m

m

0

m

OO

where = 1,N satisfy the equation (2.4), Im is the m x m identity

matrix (note that t)&n(x, t) are normal matrices). Here U = U(x, i) is an m x m matrix function,

V(x, t) =

0 iU(x, t) iU*(x, t) 0

is 2m x 2m matrix, a2n(t), n = 1, N are nonzero continuous scalar functions.

The matrix function Uo(x) satisfies the following properties:

1)

/(

\\Uo(x)\\dx< to (3.5)

(

2) Operator L(0) = —iJ^ — V0(x) possesses exactly 2N eigenvalues Ai(0), A2(0), ..., X2n(0), every eigenvalue has m linearly independent corresponding eigenfunctions and linearly independent eigenfunctions corresponding to these eigenvalues don't have associated vector functions.

Our main purpose is to obtain the time evolution equations of the scattering data for finding the solution of the problem (3.1)-(3.5) which is a collection

{U(x, t), $i(Ai,x, t), $2(A2,x, t), ..., <&N(XN,x, t) } under assumption of existence in the following sense:

1) for all t > 0,

E

r=0 '

д r

— U(x, t) dxr v '

dx < to; (3.6)

2) the columns of the $ra(x, t), n = 1,2N matrices belong to the domain of L2(R,C2m), which is the space of complex-valued vector functions of size 2m with components belonging to L2(R).

In the current section we will derive the representations for the evolution equations of the scattering data with which it is available to find the collection of solution of the problem (3.1)-(3.5) in the class of the rapidly decreasing functions (3.6) via the inverse scattering method for the operator L(t) = —J± — V(x, t).

Under the assumption Xn+N = X^, n = 1,N and $1,ra+w = — $2,n+N = ^in the equation (3.1) can be represented as a Lax operator equality:

2N

Lt + [B,L]+ £ [J, $n$nl=0, (3.7)

n=1

oo

where [B, L] = BL — LB and

( iUU* + 2iImiUx + 2iU^ \ B = I I . (3.8)

V i U* + 2iU * £ —iUU * — 2i Im¿2 J

Here, both sides of the equality (3.7) turn out to be operators of multiplication by a matrix function.

Lemma 2. Let F0(X,x, t) be any 2m x 2m matrix .solution of the equation

LFo = XFo, X e R (3.9)

and let Fn, n = 1,2,..., 2N be any matrix functions m x 2m, which satisfy

dF,

n =i$*nFo, n = 1, 2,..., 2N. (3.10)

dx

Then, the matrix function

2N

Ho = Fo + BFo — ^ $nFn (3.11)

n=1

is also a matrix solution of equation (3.9).

Proof. We take the derivative from equation (3.9) with respect to t

LFo + LFo = XFo.

Here, we will find LFo

2N

LFo = XFo — LFo = XFo + BLFo — LBFo + ^ [J, Fo.

n=1

Using this equality now we calculate L Ho

2N 2N

LHo = XHo + ^ (X — Xn) §nFn — Y, *nKJFo

n=1 n=1

2N / 2

= XHo + Y (X — Xn) Fn — >

n=1 2N

2N / 2N \

Y M (X — Xn) Fn — Y KJFo

n=1 \ n=1 J

2N

XHo + Y ®nHn.

n=1

Here, we introduce Hn, n = 1,2,..., 2N

Hn = (X — Xn)Fn — $*nJFo, n = 1, 2,..., 2N. (3.12)

According to the lemma 1, we can show that

dHn

dx

0 (3.13)

and \\Hn\\ ^ 0 as x ^ Hence, it follows that Hn = 0 for all X e R. This means that the matrix function H0 for any X satisfies the equation (3.9). The proof of lemma is complete. □

Let us take F(X, x, t) and G(X, x, t) Jost matrices as the solution F0 of the equation (3.9) and let for n = 1, 2,2N hold the following expressions

Fn- = i $n(x, i)G(X, x, t)dx ,

(3.14)

Fn + = — XT $n(x, t)F(X, x, t)dx.

Here $n(x, t) belong to the L2(R, C2m), Jost matrices F and G are bounded for ImX = 0. Using lemma 1, we can show that these integrals are convergent:

$n(x, t)JG(X,x, t)

/X

ФП(х, t)G(X,x, t)dx = , . ,

-те (X — Xn)

■ f^x ^ л фП(x, t)JF(X, x, t) -г ФП(х, t)F(X,x, t)dx = n J J

X

(X - Xn)

Moreover, F± G L2(R,C2m).

Substituting (2.2), (3.14) into the expressions (3.11) and (3.12), we define H—, H+, H—, H+. According to (3.13), it is easy to show for n = 1,2,..., 2N that Hn = H£ = 0. Therefore, we can conclude that the matrix functions

H+ = F + BF - Zn=i $nFn+ ,

(3.15)

H- = G + BG - En=i ^n Fn are solutions of the equation (3.9).

Remark 2. Using the asymptotics (2.2) for the Jost solutions in (3.15) we obtain

Я+ 0\2 / & Im 0m It ™ ,

0+ ^ —2iX\ 0 iXxT )J, x

0m m

Ho- ^ -2iX2 ( °xmxT )j, x ^ -to.

0m m

By the uniqueness of the Jost solutions we get

H0+ = -2iX2FJ, H0- = -2iX2GJ. (3.16)

Lemma 3. For all X e R the following equality holds

R(X) = 4iX2R(X). (3.17)

Here the dot means the derivative respect to the parameter t. Proof. We introduce the following matrix function

H = Ho" — H+A(X). (3.18)

Substituting (3.16) into the expression (3.18), we receive

H = —2iX2FA(X)J + 2iX2FJA(X) = 2iX2F [J, A(X)]. (3.19)

Using the representations (3.15) and the expression (3.18), we have

2N

H = FA(X) — Y ^n(F~ — F+A(X)).

n=1

Here,

f+x

$*nGdx.

-OD

Using lemma 1 and since, $n(x,t) belong to the L2(R,C2m), Jost matrix G is bounded for ImX = 0, we find that

QlGdx =

n

-x

ex

= 0.

—x

i(X — Xn) So, we get

H = FÄ(X). (3.20)

Comparing (3.19) and (3.20) we find

2 iX2 [J,Ä(X)] = Ä(X). (3.21)

Particularly,

Äi(X) = 0, Ä2(X) = 4X2Ä2(X), (3.22)

Äs(X) = -4iX2Ä3(X), Ä4(X) = 0. (3.23)

According to R(X) = —Ä—1(X)Ä2(X), we can find

Äi(X)R(X) = —Ä2(X).

Taking the derivative by t from the last equality, we obtain

Äi(X)R(X) = —4iX2Ä2(X)

and we find that R(X) = -4iX2Ä—1(X)Ä2(X), which is (3.17). The proof of lemma is complete. □

Corollary 1. Since, Ai(A) does not depend on t, its determinant det Ai(A) and its zeros Xj, j = 1, 2, ..., N also do not depend on t.

Lemma 4. The matrix functions Rj(t), j = 1, 2, ..., N satisfy the following equations

d-R^ = (UX) -a2(t))R3(t). (3.24)

Proof. For ImXj > 0 , j = 1, 2, N we denote

h-(Xj ,x, t) = 0(Xj, x, t) + B4>(Xj ,x, t) - $n(x, t) f- (Xj ,x, t),

h+(Xj ,x, t) = i)(Xj, x, t) + B$(Xj,x, t) - $n(x, t) f+(Xj, x, t),

(3.25)

where the vector functions f-(Xj,x, t), f+(Xj,x, t) are defined as follows f~ (Xj,x, ^ =i $*(x, t)(j(Xj,x, t)dx,

f+(Xj ,x, t) = — /+M Ф*п(х, t)tp(Xj,x, t)dx .

(3.26)

Here, the functions under the integrals belong to the class L2 (R,C2m), which provide the convergence of the integrals. We now introduce the following matrix functions

hj = h- (Xj ,x, t)Nj - ih+(Xj, x, t)Rj (t), j = 1,2,...,N. (3.27)

Using the expressions (3.25) we can rewrite (3.27) as

hj = (j(Xj,x, t)Nj + B(j)(Xj, x, t)Nj - itj(Xj, x, t)Rj(t) +

2 N

-iBtj(Xj,x, t)Rj(t) - Y Mfn-Nj -if+Rj(t)), 3 = 1, 2,..., N. (3.28)

n=1

Differentiating (2.7) with respect to t and taking account of the independence of Nj from we obtain

(j(Xj, x, t)Nj = itj(Xj, x, t)Rj (t) + iij(Xj, x, t)R,j (t), j = 1,2,..,N.

(3.29)

Substituting (3.26) and (3.29) into (3.28) we get for j = 1, 2,..., N

2N

hj = itj(Xj ,x, t)Rj (t) + i'Y^n(x, t) $n(x, t)tj(Xj ,x, t)dxRj (t).

n=i J-™

(3.30)

If n = j, according to lemma 1 we get

fM

ФП(х, t)ip(Xj,x, t)dx = 0.

In the case of n = j, we receive for j = 1, 2,..., N

/ro

$**(x, t)tp(Xj,x, t)dxRj(t). (3.31)

-ro

We know that

(x, t)=ip(\j ,x, t) Cj (t), j = 1, N. (3.32)

Here Cj(t), j = 1,N are m x m matrices as supposing that aj(t) = 0, j = 1,N. Moreover, the columns of the matrix ip(Xj,x, t)Rj(t) are also eigenfunctions, therefore, Rj (t) matrix can be represented as linear combinations of columns of j( ), i.e., exist mx m matrices j( ) that the following equality holds

Rj (t) = Cj (t) ej (t),j = 1;N.

Using this representation and the relation (3.32) in the second term of the expression (3.31) we have

/ro

$*(x, t)ip(Xj,x, t)dxRj(t)

-ro

/ro

$*(x, i)$j (x, t)dxej (t) =

-ro

= iip(Xj,x, t)Cj(t)a2(t)Imej(t) = ia2(t)ip(Xj,x, t)Rj(t).

Hence, we get

hj = itp(Xj, x, t)R j (t) + itp(Xj, x, t)a2(t)Rj (t), j = 1,2,..,N. (3.33)

tj - Mj/y^/V^^^y-itj

According to (3.15) we get

hj = -4X2^(Xj ,x, t)Rj (t). (3.34)

By comparing (3.33) and (3.34) it yields that Rj(t), j = 1,N satisfy the equation (3.24) for ImXj > 0, j = 1,2,...,N. The proof of lemma is complete. □

Thus, we have proved the following theorem.

Theorem 1. If the collection {U (x, t), (Xj ,x, t),j = 1,N} is a solution of the problem (3.1)-(3.6), then the scattering data for the operator

d

L(t) = -iJ— - V(x, t), dx

satisfy the following relations

R(X) = 4fX2R(X),X e R,

^ = 0, ^ = (4X2 - a?(i))Rj(t), J = TTN.

Using the following algorithm we can find the solution. Let us given the functions Uo(x) and a2(i), n = 1, N.

— Solving the direct scattering problem for the initial matrix U0(x), obtain the scattering data { R(X), Xi, X2, ..., XN, Ri ,R2,...,Rn} of the

d

operator L(0) = — iJ---V(x).

dx

— Using the results of the Theorem 1, find the scattering data for t > 0 {R(X, t), Xi (t), X2(t), ..., Xn (t), Ri(t), R2(t),..., Rn (t)}.

— Using the method based on the Gelfand-Levitan-Marchenko integral equation, solve the inverse scattering problem, i.e. from the scattering data { R(X, t), Xi(t), X2 (t), ..., X2N (t), Ri(t), R2CO,..., Rn (t)} determine U(x, ).

— Find the Jost solutions of the operator L(t) with the potential U(x, t) and then using (2.3), construct the matrix $„(x, t).

4. Conclusion

In this work, we have deduced the evolution of the scattering data of a non-self-adjoint matrix Zakharov-Shabat system. The obtained results completely specify the time evolution of the scattering data for L( ) and satisfy the condition of the one-to-one correspondence between the potential of the matrix Zakharov-Shabat system and scattering data. This allows to find solution of the problem (3.1)-(3.6) in the class of the rapidly decreasing functions via the inverse scattering method.

Acknowledgements

The authors would like to gratitude to an anonymous referee for the valuable recommendations and corrections.

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Gayrat Urazboev, Doctor of Sciences(Physics and Mathematics), Associate Professor, Urgench State University, 14, Khamid Alimdjan st., Urgench, 220100, Uzbekistan, tel.: (99 891) 4281571, email: [email protected],

ORCID iD https://orcid.org/0000-0002-7420-2516

Anvar Reyimberganov, Candidate of Sciences (Physics and Mathematics), Associate Professor, Urgench State University, 14, Khamid Alimdjan st., Urgench, 220100, Uzbekistan, tel.: 9(9862) 2246700, email: [email protected], ORCID iD https://orcid.org/0000-0001-7686-1032

Aygul Babadjanova, Candidate of Sciences (Physics and Mathematics), Senior Researcher, Khorezm Branch of Uzbekistan Academy of Sciences V.I. Romanovskiy Institute of Mathematics, 14, Khamid Alimdjan st., Urgench, 220100, Uzbekistan, tel.:(99897) 5116440, email: [email protected], ORCID iD https://orcid.org/0000-0003-3458-3913

Received 08.05.2021

Интегрирование матричного нелинейного уравнения Шредингера с источником

Г. У. Уразбоев1, А. А. Рейимберганов1, А. К. Бабаджанова2

1 Ургенчский государственный университет, Ургенч, Узбекистан

2 Институт математики им. В. И. Романовского АН РУз, Ургенч, Узбекистан

Аннотация. Работа посвящена исследованию матричного нелинейного уравнения Шредингера с самосогласованным источником, состоящим из комбинаций собственных функций соответствующей спектральной задачи для матричной системы Захарова - Шабата и не имеющей спектральных особенностей. Доказана теорема об эволюции данных рассеяния несамосопряженной матричной системы Захарова -Шабата, потенциал которой является решением матричного нелинейного уравнения Шредингера с самосогласованным источником. Полученные результаты позволяют решить задачу Коши для матричного нелинейного уравнения Шредингера с самосогласованным источником в классе быстроубывающих функций методом обратной задачи 1.1 — соответствие между потенциалом матричной системы Захарова - Ша-бата и данными рассеяния обеспечивает однозначность решения рассматриваемой задачи. Приведен пошаговый алгоритм поиска решения рассматриваемой задачи.

Ключевые слова: матричное нелинейное уравнения Шредингера, метод обратной задачи рассеяния, самосогласованный источник, данных рассеяния.

Гайрат Уразалиевич Уразбоев, доктор физико-математических наук, доцент, Ургенчский государственный университет, Узбекистан, 220100, г. Ургенч, ул. Хамида Алимджана, 14, тел.: (99891)4281571, email: [email protected],

ORCID iD https://orcid.org/0000-0002-7420-2516.

Анвар Акназарович Рейимберганов, кандидат физико-математических наук, доцент, Ургенчский государственный университет, Узбекистан, 220100, г. Ургенч, ул. Хамида Алимджана, 14, тел.: 9(9862)2246700, email: [email protected], ORCID iD https://orcid.org/0000-0001-7686-1032.

Айгул Камилджановна Бабаджанова, кандидат физико-математических наук, старший научный сотрудник, Хорезмский отдел, Институт математики им. В. И. Романовского АН РУз, Узбекистан,220100, г. Ургенч, ул. Хамида Алимджана, 14, тел.: (99897)5116440, email: [email protected],

ORCID iD https://orcid.org/0000-0003-3458-3913.

Поступила в 'редакцию 08.05.2021