ON INTEGRATION OF THE LOADED MKDV EQUATION IN THE CLASS OF RAPIDLY DECREASING FUNCTIONS Текст научной статьи по специальности «Математика»

Серия «Математика»

2021. Т. 38. С. 19-35

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

ИЗВЕСТИЯ

Иркутского

государственного

университета

УДК 517.957 MSC 37K15

DOI https://doi.org/10.26516/1997-7670.2021.38.19

On Integration of the Loaded mKdV Equation in the Class of Rapidly Decreasing Functions

A. B. Khasanov1, U. A. Hoitmetov2

1 Samarkand State University, Samarkand, Republic of Uzbekistan

2 Khorezm Branch of the V. I. Romanovskiy Institute of Mathematics, Urgench State University, Urgench, Republic of Uzbekistan

Abstract. The paper is devoted to the integration of the loaded modified Korteweg-de Vries equation in the class of rapidly decreasing functions. It is well known that loaded differential equations in the literature are usually called equations containing in the coefficients or in the right-hand side any functionals of the solution, in particular, the values of the solution or its derivatives on manifolds of lower dimension. In this paper, we consider the Cauchy problem for the loaded modified Korteweg-de Vries equation. The problem is solved using the inverse scattering method, i.e. the evolution of the scattering data of a non-self-adjoint Dirac operator is derived, the potential of which is a solution to the loaded modified Korteweg-de Vries equation in the class of rapidly decreasing functions. A specific example is given to illustrate the application of the results obtained.

Keywords: loaded modified Korteweg-de Vries equation, Jost solutions, inverse scattering problem, Gelfand-Levitan-Marchenko integral equation, evolution of scattering data.

1. Introduction

The inverse scattering method (ISM) traces its origins to the work of Gardner, Greene, Kruskal and Miura [19]. They managed to find a global solution to the Cauchy problem for the Korteweg-de Vries (KdV) equation by reducing it to the inverse scattering problem for the Sturm-Liouville operator. In this direction, the following important result was obtained

20

A. B. KHASANOV, U. A. HOITMETOV

by V.E. Zakharov and A.B. Shabat [6]. They succeded to integrate the nonlinear Schrodinger equation (NLS). Soon M. Wadati [30], relying on the ideas of [6], proposed a method for solving the Cauchy problem for the modified Korteweg-de Vries equation (mKdV):

2

Uf + 6U Ux + UXxx = °

where subscripts denote the corresponding partial derivatives, и is a real scalar function.

The mKdV equation can be applied in many areas, including Alfven waves in collisionless plasma [21], hyperbolic surfaces [27], thin elastic rods [25], etc. Due to the simple expression and rich physical application of the mKdV equation, there are many results devoted to the integration of this equation [3;8;11;15; 24; 31; 34]. In addition, some authors have studied in detail the more extensive form of the mKdV equation, for example, the paired mKdV [28], the multicomponent form [26; 32], and the matrix form [12; 18; 29]. In the paper [22], a deformed mKdV equation with a nonholonomic constraint in the form

i

Ut — 6u2ux — Uxxx = ш(ж, t), ux — 2u (c2(t) — w2)2 = 0

is considered. V. E. Zakharov, L. A. Takhtadzhyan, L.D.Faddeev [5],

M. Ablowitz, D. Kaup, A. Newell and H. Sigur [17] showed that the ISM can also be applied to the solution of the sine-Gordon equation. The application of the ISM to the NLS equation, mKdV and sine-Gordon equations is based on the scattering problem for the Dirac operator on the entire axis. The inverse scattering problem for the Dirac operator on the whole axis was studied in the papers [13;14].

In the works of A.M. Nakhushev [10] the most general definition of a loaded equation is given and various loaded equations are classified in detail, for example, loaded differential, loaded integral, loaded integro-differential, loaded functional equations, etc., and numerous applications are described. Among the works devoted to loaded equations, one should especially note the works of A.M. Nakhushev [9; 10], A.I. Kozhanov [7] and others.

Note that in the papers [16;20] the loaded KdV equation was studied in the class of rapidly decreasing complex-valued and real-valued functions, respectively.

In this paper, we study the loaded mKdV equation, namely, consider the following equation

ut + 6u2ux + uXXx + 7(t)u(0, t)ux(x, t) = 0, (1.1)

where y(t) is a given continuously differentiable function. The equation

(1.1) is considered under the initial condition

u(x, 0) = uo(x) (1.2)

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Серия «Математика». 2021. Т. 38. С. 19—35

ON INTEGRATION OF THE LOADED MKDV EQUATION

21

where the initial function u0(x) (—то < x < to) has the following properties:

1) for some e > 0

|«о(ж)| e2e'lx'ldx < to; (1.3)

2) non-self-adjoint operator L(0) = i( dx, , Ио^ ] has exactly 2N

\ —u°(x) — dx J

eigen-values ^(0), £2(0),..., (0) with multiplicities mi(0), m2(0),...,

m2N (0).

Suppose that the function u(x, t) has the required smoothness and tends to its limits rather quickly as x ^ ±to i.e.

f ^ dju(x, t)

dx1

e2£'ix'ldx < to, j = 0,1,2,3.

(1.4)

The main goal of this work is to obtain representations for the solution u(x,t) of the problem (1.1) - (1.4) within the framework of the inverse scattering method for the non-self-adjoint operator

si — u(x,t)

“>=•( NAT

2. Preliminaries

System of equations

L(0)F = (Y, —to < x < to possesses Jost solutions with the following asymptotics

Ф& ~ ( 0)

1 j р-г£х

p{%,0 ~

ф(х,с) ф(х,£)

0 | pit*

1

0 I e^x

1

—x

Im£ = 0, x ^ —to,

Im£ = 0, x ^ to.

(2.1)

(2.2)

0

(Note that (p is not complex conjugation to <p). For real £, pairs of vector functions {<£, p} and {ф, ф} are pairs of linearly independent solutions for the system of equations (2.1). Therefore, the following relations take place:

p = + КОФ,

p = —а(ОФ TKOV’

ф = — a(£,)p + b(£,)p, and p

Ф = a(OP + KOp

(2.3)

22

A. B. KHASANOV, U. A. HOITMETOV

where a(£) = W {^,ф} , &(£) = W {^р,ф}. It is easy to see that

H£)|2 + |ЬШ|2 = 1, a(£) = a(-a &(£) = b(—f).

The coefficients a(£) and b(£) are continuous functions for Im £ = 0 and satisfy the asymptotic equalities:

a(0 = 1 + o{| e | -1), b(i) = o( | £ | -1), | £ | ^rc.

The function ^(ж,£) satisfies (see [1], p. 33) the following integral representation

i>(x, e) = (J)~ K (*,s)Sds, (2.4)

where K (x,s) =

Кi (x,s) K2 (x,s)

. In the representation (2.4), the kernel

K (x, s) does not depend on £ and the equality

и (ж) = — 2К1 (x, x)

(2.5)

holds.

Theorem 1. ( [4] p. 311) If the function u(x) satisfies condition (1.3), then the Jost solutions ip(x, £), ф(х, £) are analytic functions of the variable £ for Im £ > —e.

Corollary 1. ( [4] p. 314) If the function u(x) satisfies condition (1.3), then for each e > 0 there exists a constant C£ such that |К(ж,у)| < Сее-£(х+У).

Moreover, the function a(£) admits an analytic continuation to Im f > —e and has the asymptotics below

a(0 = 1 + o( | С |-1), | C | ^ ^

and has a finite number of (in the general case, multiple) zeros there. Further, let us denote by m,k the multiplicity of the root £& of the equation a(£) = 0. Nonreal zeros {£&}ь=1 of the function a(£) are eigenvalues of the operator L(0) in Im f > 0. The eigenvalues of the operator L(0) in Im f < 0 coincide with the zeros of the function a(£). The zeros of the function a(£) (a(£)) that lie in — e < Im£ < 0 (0 < Im£ < e) are not eigenvalues of the operator L(0). So, the set {£&, — £&}ь=1 is the eigenvalues of the operator L(0), and this operator has no other eigenvalues. The requirement that there are no spectral singularities for the non-self-adjoint operator L(0) means that the function a(£) does not have real zeros, i.e. a(£) = 0, £ € R.

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Серия «Математика». 2021. Т. 38. С. 19—35

ON INTEGRATION OF THE LOADED MKDV EQUATION

23

Definition 1. Functions

(s)

F (x,fk)

ds

5=5 к

s = 1 ,тк — 1,

are called associated functions to the eigenfunction p(x,fk)•

The associated functions to the eigenfunction ф(х, fk) are defined similarly.

The eigenfunctions and associated functions satisfy the equations

(s) (s) (s-1)

LF (x,fk)= fk <p (x,fk)+ s <P (x,fk), (0)

F (x,fk) = <p(x,£k), к = 1,N, s = 0,mk — 1

There is such a chain of numbers , x!,..., Xmk—1} that the relations

( [2], [14])

(0, . . ^ k v (x,Ck) = 22 xi-v

v=0

U M

Ф (x,(k),

к

1, N, l = 0, mk — 1

(2.6)

hold.

Definition 2. Sequence of numbers {%q, x\,..., Xmk—1} are called the normalizing chain of the non-self-adjoint operator L(0).

The components of the kernel K.(x,y) in the representation (2.4) for у > x are solutions of the Gelfand - Levitan - Marchenko system of integral equations

where

K2(x,y)+ Ki(x,s)F (s + y)ds = 0,

J X

-K1 (x,y) + F (x + y)+ K2(x,s)F (s + y)ds = 0,

F (1) = 2^/Г

r+(Q =

= a(()'

a(f) to Imz > (2.5).

N mk — 1

—i Y, E —„—1

k=1 v=0

1

v! dzv

(z

- &)mk a(z)

^izx

^=5fc

(|Im £| < e), a(z) is an analytic continuation of the function —e. Now the potential u(x) is determined from the equality

24

A. B. KHASANOV, U. A. HOITMETOV

Definition 3. A set of quantities

jr+ (£), £ € R; fk, Im fk > 0; \kj, к = 1, N; j = 0, mk — 1}

is called the scattering data for the (2.1) system. .

3. Evolution of scattering data

Let the potential u(x,t) in the system of equations (2.1) be a solution to the equation

ut + 6u2ux + uxxx = G(x,t), (3.1)

where G(x,t) = —^(t)u(0,t)ux(x,t).

Operator

A

(

4if3 + 2iu2f 4uf2 + 2iuxf — 2u3 — uxx

4uf2 + 2iuxf + 2u3 + uxx 4if3 — 2iu2f

(3.2)

satisfies the Lax relation

[L,A]

LA-AL =

■(

0

—6u2ux — uxxx

—6u2ux — uxxx 0

(3.3)

Therefore, the equation (3.1) can be rewritten as

Lt + [L, A] = iR,

(3.4)

where R

0 —G

-G 0

Differentiating the equality Lip = with respect to t, we obtain

Lt^ + L^t = CVt,

which according to (3.4) can be rewritten as

(L — £) (<pt — A<p) = —iRtp. (3.5)

Using the method of variation of constants, we can write

<ft — Aip = а(х)ф + P(x)<p, (3.6)

where the functions a(x) and (x) are to be defined. Then to determine a(x) and fi(x) we get

Махф + Mfixp = —R<p, (3.7)

where M =

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0 С-

ON INTEGRATION OF THE LOADED MKDV EQUATION

25

To solve the equation (3.7), it is convenient to introduce the following

notation ф =

P2

Pi

0 = ( ^2 ). According to (3.3) and the definition

of Wronskian, the following equalities are true

фт Мф = — фт Мф = а, фт Мф = фт Мр = 0. Multiplying (3.7) by фт and фт, we get

фт Rip фт Rip

&Х = , Рх = .

а а

According to (3.2) for х ^ —то we have

(3.8)

pt — Ар ^ — ^

—Ait,3 0^1

0 Агф

е—г^х = ( А^3 ) е

0 ^ ( 1 ^ = I A*U \ р-грх

W

therefore, based on (3.6) for x ^ —то, we have ф(х) ^ Ait,3, a(x) ^ 0. Hence, from (3.8) we can determine

1 fx - 1 fx

P (x) = — фтRpdx + 4г{3, a (x) = - pTRpdx.

a

a

Thus, the equality (3.6) has the form

1 rx

pt — Ap = -a

j ipTRpdxф + ^—1J фтRpdx + 4г£3^ p. (3.

.9)

According to (2.3), the equality (3.9) takes the form

1 rx

a .

1 rx

1 fx

гмф + Ьф — А (аф + Ьф) = - фт Rpdxф

a J—ж

фт Rp dx + Ai£{аир + Ьф) .

+ 1 — а

+ Ait, 3J {аф + I

Using (3.2) and passing in the last equality to the limit x ^ +то, we obtain

at = — I tp1 Rpdx, (3.10)

bt = —

фт

f —Ж

(Ж X b (Ж ^ 3 ,

pT Rpdx----- фт Rpdx + 8i£3b. (3.11)

J — c© & J — Ж

^ dr+ bta — atb . . ,,

Therefore, from the equality —r— =----------it follows that

dr+

dt

dt a2

1 Г Ж

1 fЖ

= 8it,3r+----2 G (p"2 + p2) dx.

& J — Ж

(3.12)

26

Lemma 1.

A. B. KHASANOV, U. A. HOITMETOV

If vector functions £ = ^ ^ 0) ) and ^ = ^ ^ (;c ' 0) ^

are

solutions of the equation (2.1), then their components satisfy the equalities

Г O

/ G (£101 + £202) dx = 0 , (3.13)

G (^1 + ^2) dx = 2*07(t)u(0, t)a(f)b(f).

(3.14)

Proof. Indeed, using the formulas (1.4), (2.1), (2.2) and (2.3), we have

/OO PO

G (£101 + £202) dx = -j(t)u(0, tW Ux (£101 + £202) dx

-oo J — oo

-Д

= - lim 7(t)u(0,t) [u(x,t)(^101 + £202)]

/га

U (^101 + £10' + ^202 + £202) dx

O

R

= - lim 7(0-u(0,0 [«(x,9(£101 + £2020

R^<x -R

+7(t>U(0, 0 / (V1 (-02 + *002) + 01 (-^2 + ^£2) + ^2 (01 + *£01

-O

+02 C1 + *£<£1) ) dx = - lim 7(0«(0, t) \u(x, t) (£101 + £202) ]

/ R^o

O

+ifj(t)u(0,t) (<£102 + £101)' dx

R

-R

= - lim 7(t)u(0,t) [u(x,t)(£101 + £202) + *£ (£102 + £101)]

R^o

The following integral is calculated in the same way:

R

-R

0.

/га /*га

G (£1 + £2) dx = -j(t)u(0, t) ux (£1 + £2) dx

o -o

/га

(■U£1 £1 + W£2£2) dx

-CO

= 27 (t)u(0,t)

-o

/*00

(-£2 + fe) £1 + (£1 + *£<£1) ^2

= 2ifj(t)u(0,t) (£1£2)' dx = 2ifj(t)u(0,t) lim (£1£2)

7-o

= 2*07(t)u(0,t)a (0) b(0).

dx =

R

-R

□

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ON INTEGRATION OF THE LOADED MKDV EQUATION

27

Corollary 2. According to Lemma 1 and the equality (3.10), we have at = 0, therefore

mk(t) = mk(0), fk(t) = fk(0), к = 1 ,N. (3.15)

By (3.12) and (3.14) we have

nV'

— = (8*03 - 2ifj(t)u(0,t)) r+, (|Im0| < e). (3.16)

Let us proceed to finding the evolution of the normalizing chain {%q , %™, ..., xfnk-1} corresponding to the eigenvalue fn, n = 1, N. To do this, we rewrite the equality (3.9) as

£* - ^£

1

a

G (£101 + £202) dx £

G (^1 + ^2) dx 0 j + 4*03£. > /

X

(3.17)

First, using the formulas (1.4) - (2.3), we calculate the following integral:

ГХ

G (£101 + £202) dx

/X

ux (£101 + £202) dx

= -j(t)u(0, t) lim и ■ (^101 + £202)

R^-те

-R

+7(t)u(0, t) U (<£101 + £101 + £202 + £202) dx

= -7(t)u(0,t)u(x,t)(<f1^1 + £202) + *07(t)u(0,t) (£102 + £201) dx

J —^

= -7(t)u(0, t)u(x, t) (£101 + £202) + 2ifx(t)u(0, t)£201.

The following equality is shown in the same way:

/X

G (£? + £2) dx = -7(t)u(0, t)u(x, t) (£? + £2) + 2*07(i)u(0, i>£1£2-

-те

Based on the above, the equality (3.17) can be rewritten as

£t - ^£ = ^ 7(i)«(0, 1>u(£20 + £20 - £202£ - £101£) +2*07(t)u(0, t) (£201£ - £1£20) + 4*03£

7 (t)u(0,t)^ -^2)- 2*07 (*mm ( ° J + 4*e3 (^) .

(3.18)

Ж

Ж

28

A. B. KHASANOV, U. A. HOITMETOV

Differentiating the equality (3.18) (mn — 1) times by £ and setting £ = £n, we get

(r^^n_1)

д pn (mn_l) (mn_2) (<Шп 1)(^n 2) A (^n_3)

——--------Ло pn —(mn — l)Ai pn--------------------2---------a2 Pn —

((________1)

(шГ_1)

^1n

—2i£7 (t)u(0,t)l (mn_i)) — 2i(mn — 1)q (t)u(0,t) I (m„_2)

у ^2n J \ Ф2n

)

(m„_ 2)

( ___ 1)

+4i£)) фп +12i£);(mn — 1)v' Фп' +12f£„(mn — 1)(mn — 2) v+

(m„_ 3)

+4i(mn — 1)(mn — 2)(mn — 3)( тфА,

(3.19)

dl |

-r-riLt , l = 0,1,2,3. Using (1.4), (2.6), taking into account a£‘ '«-«и

where Ai = —-y A dU

Corollary 1 of Theorem 1, passing in the equality (3.19) to a redistribution

at x —— то and equating the coefficients at ^ ° ^ (ix)1 ■ ег^пХ, l =

mn -

1, mn — 2, ..., 0, we get

dxо dt

= (8*Cn — 2*Cn7(t)u(°,t)) хф

= (8*Cn — 2*Cn7(t)u(0,t)) Xi + (24*£); — 2i7(t)u(0,t^ Xm

= (8*Cn — 2*Cn7(t)u(0,t)) x% + (24*£n — 2*7(t)u(0,t)) Xi +24i£„Xo,

= (8*Cn — 2*C 7 (*М0,9)Хз + (24^n — 2*7 (iM0,0)X2

dx|

dt

(3.20)

+24*£raXl + 8*Xo ,

= (8*Cn — 2*C7(t)u(0,t)) x? + (24*£;; — 2i(nX(t)u(0, t)) Xi-1

n

dx[ dt

+24г£„хГ- 2 + 8*хГ- з, = 1, 2,...,N, l = 4, 5,..., mn — 1.

Thus, we have proved the following theorem.

Theorem 2. If the function u(x,t) is a solution to the problem (1.1) -(1.4), then the scattering data of the non-self-adjoint operator L(t) with the potential u(x,t) satisfy the differential equations (3.15), (3.16) and (3.20).

The obtained equalities completely determine the evolution of the scattering data, which makes it possible to apply the inverse scattering method to solve the problem (1.1) - (1.4).

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ON INTEGRATION OF THE LOADED MKDV EQUATION

29

Example 1. Consider the following problem

ut + 6u2ux + uxxx + 7(t)u(0, t)ux = 0,

2

u(x, 0) = — t'2 + 1

ch 2ж’

(3.21)

(3.22)

where 7(t) = 2(t2 + 1) + 2^2+2'

It is not difficult to find the scattering data for the operator L(0): N = 1, r+ (0) = 0, 6 (0) = i, xo (0) = 2i.

By Theorem 1, we have

Ш = 6(0) = г; r+(t) = 0, xo(t) = 2ie8t+2?(t),

where

£(*)=/ 7(r)«(0,r)dr.

o

Consequently F(ж,£) = 2е(-ж+8*+2^(4)). Solving the integral equation Ki(x,y) — F (x + y)+ / Ki(x,z)F (z + s)F (s + y)dsdz = 0,

J X J X

we can get

Ki(x,y) =

2 exp {— x — у + 8t + 2fd (t)}

1 + exp {—4ж + 16t + 40(t)}'

Whence we find the solution to the Cauchy problem (3.21) - (3.22)

2

и (x, t) = —

ch2 (x + arcsh —2^

4. Conclusion

The article shows that the method of the inverse scattering problem can be applied to the integration of the loaded mKdV equation in the case of multiple eigenvalues of the corresponding spectral problem. Facts from the theory of inverse problems for the non-self-adjoint Dirac operator with multiple eigenvalues are presented. The evolution of the normalizing chains for the associated functions of the non-self-adjoint Dirac operator is determined.

References

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A. B. KHASANOV, U. A. HOITMETOV

2. Blashchak V.A. Analog obratnoy zadachi teorii rasseyaniya dlya nesamosoprya-jennogo operatora. I [An Analog of the Inverse Problem in the Theory of Scattering for a Non-Selfconjugate Operator. I] Differencialnye Uravneniya, 1968, vol. 4, no. 8, pp. 1519-1533. (in Russian)

3. Demontis F. Exact solutions of the modified Korteweg-de Vries equation. Teo-reticheskaya i Matematicheskaya Fizika, 2011, vol. 168, no. 1, pp. 886-897. https://doi.org/10.1007/s11232-011-0072-4

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9. Nakhushev A.M. Nagrujennye uravneniya i ix prilojeniya [Loaded equations and their applications]. Diff. Urav., 1983, vol. 19, no. 1, pp. 86-94. (in Russian)

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11. Urazboev G.U. O modifitsirovannom uravnenii KdF s samomoglasovannym is-tochnikom, sootvetstvuyushij kratnym sobstvennym znacheniyam [On modified KdV equation with a self-consistent source corresponding to multiple eigenvalues]. Doklady Akademii Nauk RUz, 2005, no. 5, pp. 11-14. (in Russian)

12. Urazboev G.U., Xoitmetov U.A., Babadjanova A.K. Integration of the matrix modified Korteweg-de Vries equation with an integral-type source. Teo-reticheskaya i Matematicheskaya Fizika, 2020, vol. 203, no. 3, pp. 734-746. https://doi.org/10.1134/S0040577920060033

13. Frolov I.S. Obratnaya zadacha rasseyaniya dlya sistemy Diraka na vsej osi [Inverse scattering problem for a Dirac system on the whole axis]. Doklady Akademii Nauk SSSR, 1972, vol. 13, pp. 1468-1472. (in Russian)

14. Khasanov A.B. Obratnaya zadacha teorii rasseyaniya dlya sistemy dvux nesamoso-pryajennyx differensial’nyx uravnenij pervogo poryadka [An inverse problem in scattering theory for a system of two first-order nonselfadjoint differential equations]. Doklady Akademii Nauk SSSR, 1984, vol. 277, no. 3, pp. 559-562. (in Russian)

15. Khasanov A.B., Urazboev G.U. Metod resheniya uravneniya mKdF s samosoglaso-vannym istochnikom [Method for solving the mKdV equation with a self-consistent source]. Uzbek Math. journal, 2003, no. 1, pp. 69-75. (in Russian)

16. Khasanov A.B., Hoitmetov U.A. Integrirovaniye uravneniya Kortevega-de Friza s nagrujennym chlenom v klasse bystroubyvayushix funksij [Integration of the Korteweg-de Vries equation with a loaded term in the class of rapidly decreasing functions]. Doklady Akademii Nauk RUz, 2021, 1, pp. 13-18. (in Russian)

17. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H. The Inverse Scattering Transform - Fourier Analysis for Nonlinear Problems. Studies in Applied Mathematics, 1974, vol. 53, no. 4, pp. 249-315. http://dx.doi.org/10.1002/sapm1974534249

Известия Иркутского государственного университета.

Серия «Математика». 2021. Т. 38. С. 19—35

ON INTEGRATION OF THE LOADED MKDV EQUATION

31

18. Chen X., Zhang Y., Liang J., Wang R. The N-soliton solutions for the matrix modified Korteweg-de Vries equation via the Riemann-Hilbert approach. Eur. Phys. J. Plus, 2020, 135, art. no. 574. https://doi.org/10.1140/epjp/s13360-020-00575-6

19. Gardner C.S., Greene I.M., Kruskal M.D., Miura R.M. Method for Solving the Korteweg-de Vries Equation. Phys. Rev. Lett., 1967, no. 19, pp. 1095-1097. https://doi.org/10.1103/Phys RevLett.19.1095

20. Hoitmetov U.A. Integration of the loaded Korteweg-de Vries equation in the class of rapidly decreasing complex-valued functions. Uzbek Math. Journal, 2020, no. 4, pp. 44-52. https://doi.org/10.29229/uzmj.2020-4-6

21. Khater A.H., El-Kalaawy O.H., Callebaut D.K. Backlund Transformations and Exact Solutions for Alfven Solitons in a Relativistic Electron-Positron Plasma. Physica Scripta, 1998, vol. 58, no. 6, pp. 545-548. https://doi.org/10.1088/0031-8949/58/6/001

22. Kundu A., Sahadevan R., Nalinidevi L. Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability. J. Phys. A: Math. Theor., 2009, vol. 42, no. 11, art. no. 115213. https://doi.org/10.1088/1751-8113/42/11/115213

23. Lax P.D. Integrals of Nonlinear Equations of Evolution and Solitary Waves. Comm. Pure and Appl. Math., 1968, vol. 21, no. 5, pp. 467-490. https://doi.org/10.1002/cpa.3160210503

24. Mamedov K.A. Integration of mKdV Equation with a Self-Consistent

Source in the Class of Finite Density Functions in the Case of

Moving Eigenvalues. Russian Mathematics, 2020, vol. 64, pp. 66-78.

https://doi.org/10.3103/S1066369X20100072

25. Matsutani S., Tsuru H. Reflectionless Quantum Wire. Journal of

the Physical Society of Japan, 1991, vol. 60, no. 11, pp. 3640-3644. https://doi.org/10.1143/JPSJ.60.3640

26. Sasa N., Satsuma J. New-type of soliton solutions for a higher-order nonlinear Schrodinger equation. Journal of the Physical Society of Japan, 1991, vol. 60, no. 2, pp. 409-417. https://doi.org/10.1143/JPSJ.60.409

27. Schief W. An infinite hierarchy of symmetries associated with hyperbolic surfaces. Nonlinearity, 1995, vol. 8, no. 1, pp. 1-9. https://doi.org/10.1088/0951-7715/8/1/001

28. Tian S.F. Initial-boundary value problems of the coupled modified Kortewegde Vries equation on the half-line via the Fokas method. J. Phys. A: Math. Theo., 2017, vol. 50, no. 39, pp. 395-204. https://doi.org/10.1088/1751-8121/aa825b

29. Urazboev G.U., Babadjanova A.K. On the Integration of the Matrix

Modified Korteweg-de Vries Equation with a Self-Consistent Source. Tamkang Journal of Mathematics, 2019, vol. 50, no. 3, pp. 281-291,

https://doi.org/10.5556/j.tkjm.50.2019.3355

30. Wadati M. The exact solution of the modified Korteweg-de Vries equation. Journal of the Physical Society of Japan, 1972, vol. 32, p. 1681.

https://doi.org/10.1143/JPSJ.32.1681

31. Wu J., Geng X. Inverse scattering transform and soliton classification of the coupled modified Korteweg-de Vries equation. Communications in Nonlinear Science and Numerical Simulation, 2017, vol. 53, pp. 83-93. https://doi.org/10.1016/j.cnsns.2017.03.022

32. Yajima N., Oikawa M. A class of exactly solvable nonlinear evolution equations. Pro. Theo. Phys., 1975, vol. 54, no. 5, pp. 1576—1577. https://doi.org/10.1143/PTP.54.1576

32

A. B. KHASANOV, U. A. HOITMETOV

33. Zhang D.-J., Wu H. Scattering of Solitons of Modified KdV Equation with Selfconsistent Sources. Commun. Theor. Phys., 2008, vol. 49, no. 4, pp. 809-814. https://doi.org/10.1088/0253-6102/49/4/02

34. Zhang G., Yan Z. Focusing and defocusing mKdV equations with nonzero

boundary conditions: Inverse scattering transforms and soliton interac-

tions. Physica D: Nonlinear Phenomena, 2020, vol. 410, art. no. 132521. https://doi.org/10.1016/j.physd.2020.132521

Aknazar Khasanov, Doctor of Sciences (Physics and Mathematics), Professor, Samarkand State University, 15, University Boulevard, Samarkand, 140104, Republic of Uzbekistan, tel.: (99866)2391436, email: ahasanov2002@mail.ru ORCID iD https://orcid.org/0000-0003-2571-5179

Umid Hoitmetov, Candidate of Sciences (Physics and Mathematics), Khorezm Branch of the V. I. Romanovskiy Institute of Mathematics, Ur-gench State University, 14, Kh. Alimjan st., Urgench, 220100, Republic of Uzbekistan, (99862) 2246700, email: x.umid@urdu.uz ORCID iD https://orcid.org/0000-0002-5974-6603

Received 14-06.2021

Об интегрировании нагруженного уравнения мКдВ в классе быстроубывающих функций

А. Б. Хасанов1, У. А. Хоитметов2

1 Самаркандский государственный университет, Самарканд, Республика Узбекистан

2 Хорезмское отделение Института математики им. В. И. Романовского, Ургенчский государственный университет, Ургенч, Республика Узбекистан

Аннотация. Работа посвящена интегрированию нагруженного модифицированного уравнения Кортевега - де Фриза в классе быстроубывающих функций. Хорошо известно, что нагруженными дифференциальными уравнениями в литературе принято называть уравнения, содержащие в коэффициентах или в правой части какие-либо функционалы от решения, в частности значения решения или его производных на многообразиях меньшей размерности. В настоящей работе рассматривается задача Коши для нагруженного модифицированного уравнения Кортевега - де Фриза. Поставленная задача решается с помощью метода обратной задачи рассеяния, т. е. выводится эволюция данных рассеяния несамосопряженного оператора Дирака, потенциал которого является решением нагруженного модифицированного уравнения Кортевега - де Фриза в классе быстроубывающих функций. Приведен конкретный пример, иллюстрирующий применение полученных результатов.

Ключевые слова: нагруженное модифицированное уравнение Кортевега -де Фриза, решения Йоста, обратная задача теории рассеяния, интегральное уравнение Гельфанда - Левитана - Марченко, эволюция данных рассеяния.

Известия Иркутского государственного университета.

Серия «Математика». 2021. Т. 38. С. 19—35

ON INTEGRATION OF THE LOADED MKDV EQUATION

33

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34

A. B. KHASANOV, U. A. HOITMETOV

Phys. J. Plus. 2020. Vol. 135. Art. N 574, https://doi.org/10.1140/epjp/s13360-020-00575-6

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21. Khater A. H., El-Kalaawy O. H., Callebaut D. K. Backlund Transformations and Exact Solutions for Alfven Solitons in a Relativistic Electron-Positron Plasma // Physica Scripta. 1998. Vol. 58. N 6. P. 545—548. https://doi.org/10.1088/0031-8949/58/6/001

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33. Zhang D.-J., Wu H. Scattering of Solitons of Modified KdV Equation with Selfconsistent Sources // Commun. Theor. Phys. 2008. Vol. 49, N 4. P. 809-814. https://doi.org/10.1088/0253-6102/49/4/02

Известия Иркутского государственного университета.

Серия «Математика». 2021. Т. 38. С. 19—35

ON INTEGRATION OF THE LOADED MKDV EQUATION

35

34. Zhang G., Yan Z. Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions // Physica D: Nonlinear Phenomena. 2020. Vol. 410. Art. N 132521. https://doi.org/10.1016/j.physd.2020.132521

Акназар Бекдурдиевич Хасанов, доктор физико-математических наук, профессор, Самаркандский государственный университет, Республика Узбекистан, 140104, г. Самарканд, Университетский бульвар, 15, тел.: (99866) 2391436, email: ahasanov2002@mail.ru ORCID iD https://orcid.org/0000-0003-2571-5179

Умид Азадович Хоитметов, кандидат физико-математических наук, доцент, Хорезмское отделение Института математики им. В. И. Романовского, Ургенчский государственный университет, Республика Узбекистан, г. Ургенч, 220100, ул. Х. Алимджана, 14, тел.: (99862)2246700, email: x.umid@urdu.uz ORCID iD https://orcid.org/0000-0002-5974-6603

Поступила в редакцию 14-06.2021