ON THE CONSTRUCTION AND INTEGRATION OF A HIERARCHY FOR THE PERIODIC TODA LATTICE WITH A SELF-CONSISTENT SOURCE Текст научной статьи по специальности «Математика»

ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫЕ УРАВНЕНИЯ И ФУНКЦИОНАЛЬНЫЙ АНАЛИЗ

INTEGRO-DIFFERENTIAL EQUATIONS AND FUNCTIONAL ANALYSIS

Серия «Математика»

2021. Т. 38. С. 3-18

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

ИЗВЕСТИЯ

Иркутского

государственного

университета

УДК 517.95 MSC 35Q51

DOI https://doi.Org/10.26516/1997-7670.2021.38.3

On the Construction and Integration of a Hierarchy for the Periodic Toda Lattice with a Self-Consistent Source

B.A. Babajanov1, M. M. Ruzmetov1

1 Urgench State University, Urgench, Republic of Uzbekistan

Abstract. In this paper, it is derived a rich hierarchy for the Toda lattice with a selfconsistent source in the class of periodic functions. We discuss the complete integrability of the constructed systems that is based on the transformation to the spectral data of an associated discrete Hill‘s equation with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy. At the end of the paper, we illustrate our theory on concrete example with analytical and numerical results.

Keywords: periodic Toda lattice hierarchy, Hill’s equation, self-consistent source, inverse spectral problem, trace formulas.

1. Introduction

Our goal is to construct a hierarchy for the periodic Toda lattice with a self-consistent source in the class of periodic functions that can be integrated via the inverse spectral method. Toda lattice [29] is the model of a

4

B. A. BABAJANOV, M. M. RUZMETOV

nonlinear one-dimensional crystal. It describes how the chain of particles with exponential interactions of the nearest neighbors move. It is widely known that, by means of the Flaschka variables [12], the Toda lattice has the form

/ — 4nipn+1 bn),

\bn — 2(a2n - a2n-1), n € Z.

The Toda lattice has several applications. For instance, the Toda lattice model of Deoxyribonucleic acid (DNA) in biology [26]. In addition, another significant property of the Toda lattice type equations is the existence of so-called soliton solutions. The presence of soliton solutions and the integrability of equations have tight connections. The research results present that all the integrable systems have soliton solutions [22]. For further development of the periodic Toda lattice we refer to [9; 11; 17; 18].

Over the last few years, the interest has been growing in the soliton equations with self-consistent source [1-4;8;20;23;27;28;32]. These equations have essential applications in plasma physics, hydrodynamics, solid state physics, etc. [7; 13; 14; 19;24;25].

The fisrt investigation of the discrete soliton equations with a self-consistent sources has been considered in [21]. In this work the Toda lattice with a self-consistent sources are formulated and calculated by using the Darboux transformation. In [6; 30; 31], a scattering method was developed for integrating the Toda lattice with a self-consistent source.

In [5], it is obtained algebro-geometric quasi-periodic finite-gap solutions of the sourceless Toda lattice hierarchy. In [1], the authors demonstrated integrability of the periodic Toda lattice hierarchy with an integral-type source which comes from the eigenfunctions of the continuous spectrum of the discrete Hill‘s equation.

In this work, we consider the N- periodic Toda lattice hierarchy with a self-consistent source where source is developed by the eigenfunctions of the discrete spectrum of the discrete Hill‘s equation.

The goal of this work is to obtain formulations for the solutions of the constructed new system in the framework of the inverse spectral problem for discrete Hill‘s equation. In the one-gap case, we write the explicit formulas for the solutions of under consideration problem, expressed in terms of Jacobi elliptic functions.

The new system, similarly to [13], [10] can be applied to some models of special types of electric transmission line.

2. Formulation of the problem

In the present paper, we consider the N-periodic Toda lattice hierarchy with self-consistent source

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Серия «Математика». 2021. Т. 38. С. 3—18

ON THE CONSTRUCTION AND INTEGRATION

5

' 2 N

an = Pm(an, b„) + an ^ 9N+i(\i, t) [(/Д+1)2 - (/Д)2] ,

i= 1 2 N

( bn = Qm(an,bn)+ 2 ^ 0N+i(\i,t)f^(anfln+i — On-i/A-i), (2Л)

i=1

an-ifn-i + bn& + On/A+i = A*/*, г = 1, 2, ..., 2W, v^n+w — &n, bn+N — bn, an > 0, (/n+_^) — (fn) , ^ G Z,t G R,

and the initial conditions

ara(0) = a°, bn(0) = 6°, n G Z, (2.2)

with the given N-periodic sequences a°n and b°n, n G Z. Where

Pm(^n, bn) = Ora[ Pn,m i^ra+1 ,m + bn+i®n+i,m]i

Qmbn) — Rra®ra+i,m ®ra—i^ra-1,m 2bnPn,m + bn&n,m, Ш G f G ^,

and |a„,s(t)}o<s<m, {^(f) }o<s<m satisfy the recursion relations

ora,o = 0, pn,o = co, an,i = 2co, co = const,

Pn,s—i fin-i,s-i — bn(Pn,s—2 Pn-i,s-2) ^n^ra+i,s-2 + ^n—1^'n—1,s—2,

&n,s = bnQ.n,s— i Pn-i,s-i Pn,s—1, 2 ^ S ft m ,

n2 n2 1,2

Pn,m — 2 ®-ra-i,m-i &ra+i,m-i + ^ ^ra,m-1 bn Pn,— i,m—i-

Varying m G N yields Toda lattice hierarchy with self-consistent source

(2.1). The function sequences

{o»(t)i“oo. [Ьп (О)?». {/ra(t)}-«. uvea. ..., uf (0)?

I ?

-?

- are unknown vector-functions, besides [f!n(f)}?? are the Floquet-Bloch solutions for the discrete Hill’s equation and

(Р(\Ь)У) ra — Чп—1Уп—1 + ЪпУп + Ч'пУ'п+1 — Pyra (2.3)

normalized by conditions

№) = 1, i = 1, 2, ..., 2N. (2.4)

The eigenvalues \i of (2.3) are solutions of equation Д2(А) — 4 = 0, where Д(А) = 6n(A,t) + +1(A,f), and 9ra(A,t), n G Z and <pra(\,t), n G Z are

solutions of equation (2.3) under the initial conditions

9o(A,t) = 1, 9i(\,t) = 0, <po(Kt)=0, <pi(\,t) = 1.

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B. A. BABAJANOV, M. M. RUZMETOV

The factor 9n+i(A,t) in system (2.1) is defined from the equality

N -1

eN+i(A ,t) = П(л - Hd

3 = 1

where g1(t), g2(t), ..., дм-1(t) are roots of the equation 9м+1 (A,t) — 0.

The main aim of this work is to construct the N - periodic Toda lattice hierarchy with integral-type self-consistent sources and derive representations for the solutions {an(t)}'°0°, {bn(t)}-°°, {/1(*)}_<», {/n(0}°°, ..., { f2(f)}°° of the N-periodic Toda lattice hierarchy with an integral source under the initial conditions (2.2) by means of the inverse spectral problem for discrete Hill equation.

3. The basic information about the theory of Direct and Inverse Spectral Problem for the discrete Hill’s equation

In this section we give basic information about the theory of direct and inverse spectral problem for the discrete Hill’s equation [2;29].

We start with the following discrete Hill’s equation

(Ly)n — ®п—1Уп— 1 + Ьпуп + чпуп+1 — А уп (3.1)

O-n+N — ^n, Ьп+м — bn, П G Z ,

with spectral parameter A, and with period N > 0. Let’s 9n(A), n G Z and ^ra(A), n G Z be the solutions of equation (3.1) under the initial conditions 0o(A) — 1, 01(A) —0, <^o(A) — 0, ^ч(А) — 1.

Let A1, A2, ..., A 2n be the roots of equation A2(A)-4 — 0. We define the auxiliary spectrum ^1, ц,2, ..., дм-1 as the roots of equation 0v+1(A) — 0. As it is known [29], all Aj, i — 1, 2, ..., 2N and gj, j — 1, 2, ..., N — 1 are real, the roots gj are simple, but among the roots Aj may occur the roots of multiplicity two. It is easy to show, that

A2(A) — 4

N

Wa3

-2

2N

П(А — )

3 = 1

@N+1(A)

We shall introduce

ao

N

П

з

N -1

П c

»3 )

sign

9n (gj)

1

9n (gj)

3 — 1, 2,

, N — 1.

(3.2)

(3.3)

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ON THE CONSTRUCTION AND INTEGRATION

7

Definition 1. The set of the numbers p.j, j = 1, 2, ..., N — 1 and sequences of signs Oj, j = 1, 2, ..., N — 1 is called spectral parameters of the discrete Hill’s equation (3.1).

Definition 2. System of spectral parameters {p,j, <jj }^=—1 and numbers \i, i = 1, 2, ..., 2N is called spectral data of the discrete Hill’s equation (3.1).

To find the coefficients an, bn, n € Z by the spectral data, we shift all the suffixes n by a constant к in (3.1) to get

Ora+fc-1 Уп—1 + bn+kУп + ®п+кУп+1 — Zyn,n € Z (3.4)

then we get the following trace formulas [2]

bk+1 = 1 2 2N + 2 ^ (^2j + ^2j'+1 — 2^>,k) , (34)

2 3=1

flfc

^1 + ^2n

1

+ 8

w- 1

^2 (л2j + A2j+1— 2t

■2,fc)

3=1

8

1

4

Ai + A

2 N

2

+ -

1 ^

2 u

(A2 j + A2J+1 — 2yj,k)

2

1 N-1 1e

J = 1

a3,k\j Y\i=ii^j,k ^i)

ГТ>=1 (^j,k f^i,k)

(3.6)

where p.j,fc, j = 1, 2, ..., N — 1 are the roots of equation вn+1,fc(A) = 0. Here 6n,fc(A), n € Z is the solution of (3.4), under the initial conditions 0o,k(A) = 1, 01,fc(A) = 0.

4. Constructing a hierarchy for the periodic Toda lattice with a

self-consistent source

In this section, we present our method for constructing a hierarchy of the periodic Toda lattice with a self-consistent source by using spectral theory of the discrete Hill‘s equation. We consider the system

' 2 N

an = Pm(an, bn) + an ^ dN+1(A*, t) [(/Д+1)2 — (/Д)2] ,

i= 1

2N

< bn = Qrn(an, bn) + 2 ^ 0JV+1(Ai, t) fa (anfn+1 — an— 1 /ra — 1), (41)

i=1

ara— 1/ra— 1 + brafra + arafn+1 = ^ifra, ^ = 1, 2, ..., 2^,

K&ra+N — &ra, bra+N — bra, &ra > 0, (fra+N) — (/ra) , ^ € ^, ^ € ^,

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B. A. BABAJANOV, M. M. RUZMETOV

and initial conditions

an(0) = a°n, bn(0) = b°n, n € Z, (4.2)

with the given N - periodic sequences a°n and b°n, n € Z. Where Pm and Qm are unknown functions of an and bn. The aim is to find all functions Pm and Qm, m € N so that the Cauchy problem (4.1)-(4.2) should be completely integrable in the framework of the inverse spectral problem of discrete Hill‘s equation

(L(t)y)n = 0,п—1Уп—1 + bnyn + &пУп+1 = Ayn, W € Zi. (4.3)

Let yj(t) = (y30(t), y{(t), ..., y3N(t))T, j = 1, 2, ..., N — 1 be the normalized eigenvectors for the corresponding eigenvalues A = yj (t), j = 1, 2, ..., N — 1, associated with the following boundary problem

(L(t)y)n = &п—1Уп—1 + Ьпyn + &пУп+1 = ^Уn, 1 ^ ^ « N У\ = 0, Ум+1 = °.

Using (4.1), we obtain the following equality

N 2N

i4 (t) = ^^Pm^ bn)yJn у3п+1 + Ъп)(у3п)2] + ^ F3 (t), (4.4)

n=1 i=1

where

N

F3(t) = dN+1(\i,t) ^ |2ara[(/^+1)2 — (fn)2}y3ny3n+1 +

n=1

+2an(fnfn+1 )(У]п)2 — 2а«-1(/^-1/^)(Уп)2} .

For convenience, let us put

Hn = 2Pm(an, bn)yityli+1 + Qm(an, bn)(y:l) .

We will find sequences un, that

Un+1 Ur>

H„

We seek for un as following

Un

= An(y3n)2 + 2an(t) ВпуЪу^+1 + a2n(t) Cn(y3n+l)‘2,

(4.5)

(4.6)

(4.7)

where An = An(yj(t)), Bn = Bn(yj(t)) and Cn = Cn(yj(t)) are unknown coefficients yet. Substituting (4.7) in (4.6) we get

An + йп Cn+1 = Qm,

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ON THE CONSTRUCTION AND INTEGRATION

9

An+i anCn + 2Bn+\(p,j bn+1) + Cn+\(^j bn+1) — 0,

лп

&nBn йпВп+1 йп Cn+i(pj bn+1) — Pn

Consiquently,

Pm — 0,n Bn Bn+1 4nCn+1(^j bn+1),

Qm — &nCn+1 An— 1&n-1 + 2Bn(^j bn) + Cn(^j Ьп)

(4.8)

(4.9)

The left side of (4.8) and (4.9) independent of , according to this we seek for Bn and Cn in the form

Bn — £ Pn,kV™-k, C* — E an,k

m—k

(4.10)

fc=0 fc=0

Putting (4.10) into (4.8)-(4.9) and comparing left and right sides of the last equality we find

&n,0 — ° Pn,0 — Co — const, OA,1 — 2Co,

Pn,k-1 Pn-1,k-1 —bn(Pn,k-2 Pn-1,k-2) ^n^n+1,k-2 +

+ a^-1an-1)fc-2, 2 < к < m,

&n,k — bnQ-n,k— 1 Pn— 1,k— 1 Pn,k—1, 2 < k < ^,

(4.11)

(4.12)

(4.13)

72

Mn— 1

b2

'-/n

Pn,m — 2 &n-1,m-1 2 ®-n+1,m-1 + 2 ®-n,m-1 ЪпPn—1,m—1, (4.14)

and so

I Рт(& n, bn) —a П ( - Pn ,m Pn+1 ,m + bn+1&n+1 ,m),

Qm(&n, bn) —&n&n+1,m &n— 1®n-1,m 2Ьп$п,ш + bn&n,m, Ш G N.

(4.15)

Then, varying m G N yields the Toda lattice hierarchy with a self-consistent source

2 N

'ln =Q'n( $n,m $n+1,m + ^n+l^n+l,m)+ ^ @N +1 [(/n+i) (/n) ]

i=1

Ьп an an+1,m an-1an—1,m 2Ьп$п,т + Ьп&п,т +

2 N

+ 2^Vw +i(Aj,t)f)(anf)+1 — On-if^-1), m G N.

i= 1

(4.16)

Explicitly, one obtains from (4.11)-(4.16) few equations of the periodic Toda lattice hierarchy with a self-consistent source, m — 1, c0 — —1,

2N

dn —dn(bn+1 — bn) + &n 'y \ @N+1(V t) [(fn+1) — (fn) ] ,

i=1 2N

Ьп 2(^n ®n— 1) + 2 ^ +1 (^i, f) fnfanfn+1 ^n—1fn-1),

i=1

2

10

B. A. BABAJANOV, M. M. RUZMETOV

m = 2,

o>n —c\dn(bn bn+1) + flra—1) + ^n(bn+1 bn)+

2 N

+ an ^ eN+i(Xi,t) [(/^+1)2 - (fn)2] ,

• 2=1 2 2 2 (4.17)

bn —2ci(Q>n— i &n) 2^n-1 (bn + bn—1) + 2,о,п(Ьп + bn+1) +

2 N

+ 2 ^ ' &N+i(Xi, t)fn (Q-nfn+i a,n-ifn-i),

i= 1

m — 3,

hn —C2 &п{Ьп bn+1) Cl &n(&n+1 ®‘n_1 + bn+1 bn) + &п(Ьп+1 bn) +

+ flra[flra+1(^ra+2 + 2Ьп+1) + an(bn+1 - Ъп) — a^_1(bn_1 + ‘2bn) +

2N

+ an^ On+1(Ai,t) [(/^+i)2 - (ti)2] ,

i=1

bn —2c2(al-i - al) + 2ci[a2n_i(bn + bn-1) - a2n(bn + 6„+i)j + 2a2n(a2n+

2

+ ara+1) + 2an(bn + bn bn+1 + &ra+1) — 2ara_1(^ra + ЬпЪп_1 + Ь^_1)-

^nv^n

b2

°n-

2N

2a2n_1a2n_2 - 2afl_1 + 2J2 ^>N+1(Xi,t)f^ (anfn+1 - an_1^n_1),

i=1

etc.

5. Evolution of spectral parameters

In this section, we prove the basic result of this paper.

Theorem 1. If the functions an(t), bn(t), f%(t), n € Z, к — 1, 2, ..., 2N are solutions of the problem (2.1)-(2.4), then the spectrum of discrete Hill operator (2.3) is independent of t, and the spectral parametrs p.j(t), j — 1, 2, ..., N — 1, satisfy the system of equations

, . °j (0 VП*=1 Chj(t) - xk)

H j (t) —2-rp-p-------------

m=Wj (Hj (o - h-k (t))

where

m

H(t)) - i ^n}

(5.1)

ci (hj о —E ^ i v?_k (c

k=0

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11

Proof. Putting (4.7) into (4.6) and summing over n, we get

N

У ' Bn — un+1 Ui — ^w+1 (V'n+i) + 2aN+i(t) Bn+iVn +iVn+2+

J „,3 +

n=i +2 (5.2)

+ a%(t) CN(y3N+2)2 - Ai(y{)2 - 2ai(t) Siy[у2 - a^t) Ci(yi,)2.

Due to

a-N+i(t)yJN+2 — (Pj(t) - bN+i(t))yJN+i - aN(t)y3N, and y{ — 0, y3N+1 — 0, from (5.2) we find

N

^ Hn — a2(t)Ci(У0)2 - а%(У)СМ+i(y3N)2 — a2(t)Ci[(y2)2 - (y^)2] (5.3)

n= i

Substituting (5.3) in (4.4) we obtain

2 N

fij(t) — al(t)Ci(y.j(t))[(y0)2 - (y3N)2] + ^ F3(t). (5.4)

i=i

Now, we will calculate second term of right-hand side of (5.4).

N

*i(t) — +i(^i,t)J2[2anfn+iy3n+i(y3nfn+i - vL+ifk)+

n= i

N

+ 2anfnУ3п(У3пPn+i - fhyi+i)} — 2®n+1(К t) ^ Fr+iV'n+iTn+

n=i

N N

2&N+i(^i, t) fn+iyi+iTn+i — 2@N+i(Aj, t) 1п+1Уп+1(Тп + Tn+i)

n=i

n=1

N

E29n+1(\i,t) frjl ^ л/гг 1ГгЛ_2вм+1(\i,t) /rr2 rji2\

\ ,, j+\ (Jra+1 — In)(J-n+1 + +n) — x (IN+1 — 11 ),

=1 Pj (t)

n=1

Xi j (t)

where Tn — an(y3nfln+1 - y3n+1f:f). It is easy to see that

F3 — 2®n+i(\t,t)a20(fi)2 < )2 (i)2]

^ — A, - H(t) [(Vn) (Уо)]

Therefore, according to (5.5), we have

(5.5)

(5.6)

2N

2N

(t) £ 2)n +* Ы)2 - «)2] (5.7)

i=1 i=1 C'jV /

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B. A. BABAJANOV, M. M. RUZMETOV

Substituting (5.7) in (5.4) we obtain

fij(t) = —2a(j [(y3N)2 - (y^2

<*0 )21 {« <*» — gА+щ) ■

(5.8)

where C1(pj(t)) = g)fc=o ai,fchg-(C The factors a1,k, к = 0, 1, ...,m are defined from recursion relations (4.11)-(4.14).

By virtue of the equalities

\03 II2 =

^ (%)

'N\ d\

we can write (5.8) in the form

A=^j , (y0) =

(^0)2 r,.j \2 (@N)2

Г II

(V*N )2 =

2 ’ V УМ

Г'I I 2

(5.9)

kj (0 =

2«q (03n(pj(t),t)

i^N+i)

{

2 N

OO,(0) — g %+?£$}■

^—№j (0

Using the equality

On (Л, t)^w+i(A, f) — 6V+i(A, t)^w (A, t) = 1,

(5.10)

i

we obtain

Chi(C0 — w, Vv .v = CTi(t^JA2(yj(C — 4 (5.11)

^ Chi (С0 V

where

^i(i) = (pj(t),t) - 1 ) ,j = 1, 2, ..., Ж — 1.

\ (hi W, f) /

It follows from expansion (3.3) that

N \ 1N-1

0N+i(A,i) = -ao aj I (A - pk(t)).

(5.12)

к 3=1

k=1

Differentiating expansion (5.12) with respect to A and assuming that A = pj (t), we find

°N +1(A)|A=Uj (t)

—ao

(n*)

N-1

П Chi (0 — hfc (t))

к = 1

к = 4

(5.13)

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13

Substituting (5.11), (3.2) and (5.13) in (5.10) we obtain equality (5.1).

We now show that Xfc(t) independent of t. Let {g*(f)} be the normalized eigenfunction of the operator L(t) corresponding to the eigenvalue- Xfc(t), к = 1, 2, ..., 2N, i.e.

an—i9n—i + bngn + an9n+1 — Xk 9n,.

By differentiating the last identity with respect to t, multiplying by gjfc and summing over n we get

^ (2an(t)9kngkn+1 + t)n(t) (gty ) . (5.14)

4= 1 ' '

Usining (2.1) we can write equality (5.14) in the following form

dXk

dt

where

N

n= 1

2 N

^[2Рт(ап, bn)gfcgfc+1 + Qm(an, bn)(g^)2] + ^Ffc(t), (5.15)

i= 1

N

Fi(t) = 9N+1 (Xi,t) ^ |2Ora[(/ra+1)2 - Ш)2]5n5n+1 + n=1

+2an(fn fn+1)(5n)2

k\2

— 2an-1(fn- Jn)(gn)

As (5.8), taking into account the periodicity of g/f, we can easily see that X к (t) — 0. □

Remark 1. The factors a1,k, к — 0, 1, ...,m are defined from recursion relations (4.11)-(4.14) and they depend on an(t), bn(t). It is easy to see that an(t), bn(t) are formulated by Xfc and pq(t) via trace formulas (3.5) and (3.6).

Remark 2. Theorem 1 provides the method for solving problem (2.1)-(2.4).

1. Solving the direct spectral problem for the discrete Hill’s equation with |a°} and {b°n} the spectral data Xi, i — 1, 2, ..., 2N and pj(0),aj(0), j — 1,2, ..., N — 1 are obtained.

2. Using the result of Theorem 1, we find the pj(t), Oj(t) ,j — 1,..., N-1.

3. Using the trace formulas (3.5) and (3.6), we calculate an(t), bn(t) and hence ffc(t), n € Z, к — 1, 2, ..., 2N.

Corollary 1. If N — 2p and the number p is the period of the initial sequences {a°n} and {b°n}, then all roots of the equation Д(А) + 2 — 0 are

14

B. A. BABAJANOV, M. M. RUZMETOV

double roots. Because the Lyapunov function corresponding to the coefficients an(t) and bn(t) coincides with A(A), according to the analogue of the Borg inverse theorem for the discrete Hill equation (see [15]), the number p is also the period of the solution an(t), bn(t) with respect to the variable n.

We now illustrate the use of Theorem 1 to solve the problem (2.1), (2.2) when m = 2 and m = 3.

Let us consider the following periodic initial value conditions (ara)2 = 2 - (-1)™3, 6° = 0, П € Z,

for the periodic Toda lattice hierarchy with a self-consistent source (4.17) (m = 2). In this case,

N = 2, Ai = -3, A2 = -1, As = 1, A4 =3, ^i(0) = 0, ^(0) = 1.

Using remark 2, we obtain

an(t) = ^2 _ 1 sn2 ^3°i) 0 _ (-1)™2cn 3^ dn, ^3°f, 3^

bn(t) = (_1)nsn (30t, ^ ,

fok (t)

A| — sn2 (30t, 3) — 3cn (30t, 1) dn (30f, |) 2ao(t) [Ak _ sn (30f,1)] fi (t) = 1, к = 1,2, 3,4,

where sn, cn and dn are the Jacobi elliptic functions. The graphs of these functions are listed below.

The graph of ao(t) and ai(t) on [0, 2].

The graph of bo(t) and bi(t) on [0, 2].

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We now illustrate the use of Theorem 1 to solve the problem when m = 3. In this case the solution of the problem (2.1), (2.2) has the form

an{t) = ^ - 1 sn2 (-30f, 3) - (-1)n3cn (-Ж, 0 dn (-Ж, 3), bn(t) = (-1)nsn ^-30^ 0 ,

fo (t) =

X2k - sn2 (-30t, 3) - 3cn (-30t, I) dn (-30t, |) 2ao (t) [Afe - sn (-30f,1)]

/i (*) =1Д = 1, 2, 3,4.

The graph of a0(t) and al(t) on [0, 2]. The graph of b0(t) and bl(t) on [0, 2].

6. Conclusion

In this paper, we have investigated the hierarchy of the periodic Toda equations with self-consistent sources, where the source is formed by eigenfunctions belonging to the discrete spectrum of the discrete Hill equation. We also presented a new method for constructing a hierarchy for the periodic Toda equations with a self-consistent source. Then the integrable a rich hierarchy for the periodic Toda lattice with a self-consistent source are obtained. We also presented an efficient method for solving the inverse spectral problem for the discrete Hill's equation which is very comfortable for numerical calculation. At the end of the paper, we have illustrated our theory on concrete examples with analytical and numerical results which can be used in some models of special types of electric transmission line.

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Bazar Babajanov, Doctor of Science (Physics and Mathematics), Associate Professor, Urgench State University, 14, Kh. Alimdjan st., Urgench, 220100, Republic of Uzbekistan, tel.:+9(9862)2246700, email: a.murod@mail.ru

ORCID iD https://orcid.org/0000-0001-6878-791X

Murod Ruzmetov, Senior Lecturer, Urgench State University, 14, Kh. Alimdjan st., Urgench, 220100, Republic of Uzbekistan,

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B. A. BABAJANOV, M. M. RUZMETOV

tel.:+9(9862)2246700, email: rmurod2002@gmail.com ORCID iD https://orcid.org/0000-0002-3572-3482

Received 26.07.2021

Построение и интегрирование иерархической цепочки Тоды с самосогласованным источником

Б. А. Бабаджанов1, М. М. Рузметов1

1 Ургенчский государственный университет, Ургенч, Республика Узбекистан

Аннотация. В статье выведена иерархия для цепочки Тоды с самосогласованным источником в классе периодических функций. Обсуждается полная интегрируемость построенных систем, основанная на преобразовании в спектральные данные связанного дискретного уравнения Хилла с периодическими коэффициентами. В частности, уравнения типа Дубровина выводятся для временной эволюции спектральных данных, соответствующих решениям любой системы в иерархии. Данная теория проиллюстрирована на конкретном примере с аналитическими и численными результатами.

Ключевые слова: периодическая цепочка Тоды, уравнение Хилла, самосогласованный источник, обратная спектральная задача, формулы следов.

Базар Атаджанович Бабаджанов, доктор физико-математических наук, Ургенчский государственный университет, Республика Узбекистан, 220100, г. Ургенч, ул. Х. Алимджана, 14, тел.: +9(9862)2246700, email: a.murod@mail.ru

ORCID iD https://orcid.org/0000-0001-6878-791X

Мурод Марксович Рузметов, старший преподаватель, Ургенчский государственный университет, Республика Узбекистан, 220100, г. Ургенч, ул. Х. Алимджана, 14, тел.: +9(9862)2246700, email: rmurod2002@gmail.com ORCID iD https://orcid.org/0000-0002-3572-3482

Поступила в редакцию 26.07.2021

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