INTEGRATION OF EQUATIONS OF KAUP SYSTEM KIND WITH SELF-CONSISTENT SOURCE IN CLASS OF PERIODIC FUNCTIONS Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 12. № 1 (2020). С. 104-114.

INTEGRATION OF EQUATIONS OF KAUP SYSTEM KIND WITH SELF-CONSISTENT SOURCE IN CLASS OF PERIODIC FUNCTIONS

A.B. YAKHSHIMURATOV, B.A. BABAJANOV

Abstract. In this paper, we consider the equations of Kaup system kind with a self-consistent source in the class of periodic functions. We discuss the complete integrability of the considered nonlinear system of equations, which is based on the transformation to the spectral data of an associated quadratic pencil of Sturm-Liouville equations with periodic coefficients. In particular, Dubrovin-tvpe equations are derived for the time-evolution of the spectral data corresponding to the solutions of equations of Kaup system kind with self-consistent source in the class of periodic functions. Moreover, it is shown that spectrum of the quadratic pencil of Sturm-Liouville equations with periodic coefficients associated with considering nonlinear system does not depend on time. In a one-gap case, we write the explicit formulae for solutions of the problem under consideration expressed in terms of the Jacobi elliptic functions. We show that if p0(x) and q0(x) are real analytical functions, the lengths of the gaps corresponding to these coefficients decrease exponentially. The gaps corresponding to the coefficients p(x, t) and q(x, t) are same. This implies that the solutions of considered problem p(x, t) and q(x, t) are real analytical functions in x.

Keywords: equations of Kaup system kind, quadratic pencil of Sturm-Liouville equations, inverse spectral problem, trace formulas, periodical potential.

Mathematics Subject Classification: 39A23, 35Q51, 34K13, 34K29

is completely integrable. The system describes the waves propagation in a shallow water. In [2], complex finite-gap multiphase solutions expressed in terms of the Eiemann theta functions were considered. Multi-soliton solutions were found and the asymptotic behavior of these solutions was studied. In [3], [4] and [5], a real finite-gap regular solutions of Kaup system were studied. In [6], the 'Inverse Scattering Transform' was employed to solve a class of nonlinear equations associated with the inverse problem for the one-dimensional Schrodinger equation with the energy-dependent potential.

It is not difficult (see [2]) to confirm that after the transformations

A.B. Yakhshimuratov, B.A. Babajanov, Integration of equations of Kaup system kind with

self-consistent source in class of periodic functions.

© A.B.Yakhshimuratov, B.A.Babajanov 2020. Поступила 25 февраля 2019 г.

1. Introduction In [1], D.J. Kaup proved that the nonlinear system of equations

4ß\ o 1 — (q + 3p2) + -,

£ £

4ß

§x = —P,

%£

t = ißr,

the system of Kaup equations easts into a simpler form

{

Pt = -6ppx - Qx

Qt = Pxxx - 4qpx - 2pqx

We shall also call it the Kaup system.

The Kaup system can be considered as a compatibility condition (see [2])

Vxxt - Vtxx = [(qt - Pxxx + 4qpx + 2pqx) + 2\(pt + 6ppx + qx)]y = 0

for the system of the linear equations

-Vxx + qy + 2\py - X2y = 0 yt + 2Xyx + 2pyx - pxy = 0.

The first of these equations is called the quadratic pencil of Sturm-Liouville equations.

In [7], the Kaup system with self-consistent sources was studied by means of the inverse problem for the quadratic pencil of Sturm-Liouville equations with periodic potential. The inverse problem for the quadratic pencil of Sturm-Liouville equations with periodical potential on a half-line and the entire line was solved in the works [8]-[13].

Recently, nonlinear evolution equations with self-consistent sources have received much attention in the scientific literature. Physically, the sources appear in solitary waves with non-constant velocity and lead to a variety of dynamics in physical models. They have important applications in plasma physics, hydrodynamics, solid-state physics, etc, [14]-[19]. For example, the KdV equation, which is included an integral type self-consistent source, was considered in [17], This type equation can be employed to describe the interaction of long and short capillarv-gravitv waves [18], Other important soliton equations with self-consistent source are the nonlinear Schrodinger equation, which describes the nonlinear interaction of an ion acoustic wave in the two component homogeneous plasma with the electrostatic high frequency wave [19], Other aspects on integration of nonlinear systems were presented in [20]-[25],

In this paper, the method of the inverse spectral problem for the quadratic pencil of Sturm-Liouville equations with periodic coefficients is used to integrate the equation of Kaup system kind with a self-consistent source in the class of periodic functions. In the one-gap case, we write the explicit formulae for solutions of the problem under consideration, expressed in terms of the Jacobi elliptic functions.

The paper is organized as follows. In section 2, we present the formulation of the considered problem and we provide some basic information about the spectral theory of the quadratic pencil of Sturm-Liouville equations with periodic coefficients. Section 3 is devoted to describing the evolution of the spectral data corresponding to the problem in question. In section 4, we illustrate the application of the main result for the one-gap case.

2. Formulation of problem We consider the system of equations with a self-consistent source

Pt =Pxxx - 6pxq - 6pqx - 30p2px + ak(t)s(-,Xk,t)('tp2+ (x,Xk,t))x,

k=—<x

qt = qxxx + 6ppxxx + 18pxpxx - 6 qqx - 24pqpx - 6p2qx (2.1)

+ ak(t)s(7T,Xk, t) (—px (x,Xk, t) + (Xk — 2p)(ip\(x,X k,t))x)

k=—oo

in the class of real-valued --periodic in the spatial variable x functions p = p(x, t) and q = q(x, t) possessing the following regularity

p(x, t), q(x, t) eC3(t ^ 0) n Cl(t > 0) n C(t ^ 0)

and satisfying the initial conditions

P(x, t)\t=o = Po(x), q(x, t)\t=0 = qo(x). (2.2)

Here po(x) and go(x) are the given real-valued --periodic functions such that for each nontrivial function y(x) E W2[0,k] satisfying the identities

y'(0)y(0) - y'(n)y(n) = 0, (y, y) = \,

the following inequality holds:

(PoV, y)2 + (qoy, y) + (y', y') > ° where (■, ■) is a scalar product of the space L2(0, k). We observe that for p(x) = 0 the equation (2.1) reduces to the Korteweg-de Vriez equation with a self-consistent source. In system (2.1), ak(t), k E Z is a given sequences of real-valued continuous functions having a uniform asymptotic behavior

ak = O ( — ) , k ^

GO

and ^+ (x,Xk, t) is the Floquet solution normalized by the condition ^+(0,Xk, t) = 1 of the quadratic pencil of Sturm-Liouville equations

T(X, t)y = —y" + qy + 2Xpy — X2y = 0, x E R. (2.3)

Here Xfc are the zeroes of the function A2(X) — 4, where A(X) = c(n,X, t) + s'(n,X, t). We

( x, X, ) ( x, X, )

c(0,X, t) = 1, c'(0, X, t) = 0, s(0,X, t) = 0, s'(0, X, t) = 1,

respectively.

The spectrum of the quadratic pencil (2.3) is real and coincides with the set [8-9] a(T) = {X E R | — 2 ^ A(X) ^ 2} = R \ |J (X2n-i, X^).

n=—<x

The intervals (X2n-1, X2n), n E Z, are called the gaps or lacunas. The numbering is introduced in such a way that X-1 < 0 < X0.

We denote by £n(t), n E Z \ {0} the eigenvalues of the Diriehlet problem (y(0) = y(n) = 0) for equation (2.3). The inclusions Çn(t) E [X2n-1, X2n] and the identity

s(7,K t)=7 ft ^^ (2.4)

0=k=-<x

are satisfied.

The numbers = £n(i) with the signs

On = &n(t) = sign Cn) - c(n, Cn)} , n e Z \ {0},

are called the spectral parameters of quadratic pencil (2,3),

The boundaries \n of the spectrum and the spectral parameters an are called the spectral data of problem (2,3),

The aim of this work is to develop a procedure for constructing the solution (p(x,t), q(x,t), ^+(x,\k,t)) of problem (2,l)-(2,3) bv means of the inverse spectral problem for the quadratic pencil of Sturm-Liouville equations (2,3),

3. Main Result The main result of the paper is presented in the following theorem.

Theorem 3.1. Letp(x,t), q(x,t) and^+(x,Xk,t) be solution of problem (2.1)-(2.3). Then the spectrum of problem (2.3) is independent of t, and the spectral parameters £n (t), n e Z\{0} satisfy the following analogue of the system of Dubrovin equations

£n(t) =2(-\)nan(t)sign(n) •sj(in(t) - A2n-l)(A2n - in(t))

\

(C (+\ \ \(C (+\ \ \ TT (Cn{t) - X2k-l){Cn{t) - À2fc)

tut)- ^ut)- a0) ¡A (m-öM-2

4Cn(i) + 2(A-i + Ao) + 4 £ iA2fc-l9+ ^ - &(t))] Ut)

o=k=—r ^ 2 H

(A-l + An ^^ ( À2fc-l + A2, , \ \

+ ^ ^ 2 6 (t)

+ £ (M-i+A2, - 2m + jrak^^> )

0=k^=—oo k^=—oo /

2

+ 21 + £ (X2k—12+A2 k - & (t^ + A—1 + A2

0=k=—r

k-1 + A2k - 2^)+ ^ -

0=k=-<^ snw k

The sign an(t) = ±1 changes at each, collision of the point £n(t) with the boundaries of its gap [A2n-1, A2n]. Moreover, the following initial conditions are satisfied

in(t)\t=0 = en, ^n(t)|t=0 = °n, n e Z \{0}, (3.2)

where ^^ n e Z \ {0} are the spectral parameters of the quadratic pencil of Sturm-Liouville equations corresponding to the coefficients p0(x) and q0(x).

Proof. Let yn(x,t) be the normalized eigenfunction of the Dirichlet problem for equation (2.3) corresponding to the eigenvalue £n = £n(i). It is easy to see that

y,n(x,t) = -^T s(x,Cn (t),t), (3.3)

Cn(t)

where

k

C2n(t) = J S2(X, £n(t), t)dx. 0

Differentiating the identity

-(y'n, Vn) + fay^ Vn) + 2ín(PУn, Vn) - & = 0,

with respect to t, we get

- (У"n, yn) - , iln) + (qtyn + qyn, Vn) + (qy^ yn) + 2in(pyn,yn)

+ 2Cn(Ptyn + PУn, yn) + 2Cn(PУn, i)n) 2 £,n £,n — By the last identity we obtain

(-y"n + Q-Vn + 2£nPVn, yn) + (-y'n + QUn + 2CnPУn, yn) + (qtyn + 2CnPtУn, yn) + 2Cn(PУn, n) - 2 n ,n = 0, 2in[Cn - (pyn, yn)] = (Qtyn + 2 CnPty n , yn) ,

and hence,

2in I Cn - py2adx I = (It + 2CnPt)yn^x. (3-4)

According to (2.1) we have

(3.5)

qt + 2£nPt = Qxxx + 6ppxxx + 18pxpxx - 6qqx - 24pqpx - 6p qx

+ 2Cn(Pxxx - 6pxq - 6pqx - 30'P2PX) + G2 + 2£nGi,

where

<x

Gi(x, t)= ak(t)s(-,Xk, t)(ip+(x,\k, t))x

k=—<x

and

<x

G2(x, t) = 2 ^ ak(t)s(-,Xk, t) {-px l(x,Xk, t) + (Xk - 2p)(^2+(x,Xk, t))x} .

k=—<x

We introduce the polynomial

c(x,t, Cn) = Co(x, t)C + ci(x, t)Cn + c2(x, t) and we rewrite identity (3.5) as

qt + 2inPt = + 2 d ■ (gn - 2pin - q) - c ■ (2p'^ + q') + G + 2£nGu (3.6)

where

c0(x, t) = 4, c\(x, t) = 4p(x, t), c2(x, t) = 2[q(x, t) + 3p2(x, t)]. Using the identity

qyn = £yn + y'n - 2CnPyn

it is easy to show that

^c" + (^ - 2Pin - q^j y2n - dyny'n + c(y2ny^j

(1c"' + 2c' ■ (C - 2pCn - q) -€■ (2p'+ q')^ y'n.

(3.7)

w

w

Substituting expressions (3.6) into formula (3.4) and taking into consideration (3.7), we arrive at the identity

2in | Cn - I pyldx | = ^ Qc" + c • (& - 2'Pin - q^j yn - c1 yny'n + c(y2n

0

+ y (G2 + 2CnGi)y2ndx (3.8)

0

k

=c(0, t, in) {y'2n(tt, t) - y'2n(0, t)) + J(G2 + 2£nGi)y2ndx.

0

Now we calculate the last integral in (3.8)

k

( G2 + 2CnGi) y2ndx

0

<x

= ^ akS(n,\k, t) (3.9)

k=—^o

k

•J ( - 2Pxyn • ^+(x, xk, t) + 2( Cn + Ak - 2P) yn • (x, Xk, t))x) dx. 0

It is easy to confirm that

k

J = - 2 J Pxyn^ldx + 2 J (Cn + Ak - 2P)yn(^2k)xdx 00

k k

= - 2 JPxyn^ldx + J(in + Ak - 2P)yn(^2k)xdx + J (Cn + Ak - 2P)ynd(ip^) 0 0 0

k k

2 2 2

n

- 2 J pxy^^dx + J 2(£n + Ak - 2p) y^^k^kdx (3.10)

00 k

J ( - 2Pxy2n + 2(Cn + Ak - 2P)yny'n)^ldx 0

k

2(in + Ak - 2P) yn^k(yn^'k - y'n^k )dx, where = ^+(x, Xk, t). Using the identity

(6. + Ak - = <!^k - ''

n

0

in - Xk by (3.10) we obtain that

J = t\- yn(*, t) - y'2n(0, t)). (3.11)

<:,n - Ak

k

0

k

Substituting (3.11) into (3.9), we get

ak(t)s(n, Xk, t)

/= £ "k ('" •(V'l t)-v'l(0. t)). (3.12)

, — in - M

k=—^o

Hence, by means of expression (3.12), we conclude that

2Cn ( Cn -J py^dx) = (y'^TT, t) - y'2n (0, i))^ Ck (0, t) C-"

V 0 / k=0 (3.13)

, f '2( '2fn ^ ak (t)S(7V,Xk, t)

+ {y n (к, t) -y n(0, V) y, —t-\-.

, — Kn - a"

k=—^o

By virtue of (3.3), identity (3.13) can be written as

2in(t) ( Ut)c2n(t) - ps2(x, Ut), t)dx J = (s'2(tt, Ut), t) - 1)^ Ck(0, t)C—k

0 k=0

+ (s'2(n, Ut), t) - 1) ± ak(f-\Xk, t].

It follows from the identity

2Cn(t)c2n(t) - 2 ip(x, t)s2(x, Ut), t)dx = 8'(n, Ut), t)dS^ ^t]

, in - 'Xk

k=

dX

0

that

d S(7Г, in(t), t) f , 1 .Nt2-k

= 6. M, 0 - £ Ck tK

+ ('- ) t , t]

\ SI(K, in(t), t)J in

(3.14)

Now, substituting the values x = n and A = £n(i) into the identity

c(x, X, t)s'(x, X, t) - c'(x, X, t)s(x, X, t) = 1,

we find that

c(n, Ut), t)= 1 . (3.15)

S , in(t), t)

By (3.15) and the identity

[c(-k, X, t) - s'(-k, X, t)]2 = (A2(A) - 4) - 4c'(n, A, t)s(n, X, t), we arrive at the identity

s'(tt, Ut), t) - 1 , ) = On(t)^A2(Ut)) - 4. (3.16)

Using (2.4), (3.16) and the expansion

A2( A) - 4= -4tT2( A -A—i)(A - A0) ft (A - ^^ - ^),

0=k=—ro

we find:

^(TT, Ut), t) - ^\{t),) = an(t)^A2(&(*)) - 4

d s (ft, (t), t) a s(tt,h n(t), t)

ax ax

^n7n(t)Slgn (n)

-- 2(-l)nan(i)slgn(n) V(în(t) -A2n-l)(A2n - în(t)) \

(C \ \(C \ \ TT (Cn(t) - A2k-l)(Cn(t) - A2k)

( - ^ )(Î"(Î) - A0) U, (&«)-'

Here we have also employed the Identity

(3.17)

I t TT ik(t) - Cn(iM , nn • / X

sign I -n 11 -k-) = (-1) sign(n).

k=n,0 J

From (3.14), (3.17) and the trace formulae

f(i) = + £ (A2k-12+Aîk -&m)

0=k=-<x ^ '

m+2p2(t) = + £ (^^^ - ek(t))

0=k=-^ we get (3.1).

We note that if instead of the Dirichlet boundary conditions we consider periodic or anti-periodic boundary value conditions, then equation (3.13) becomes Xn(t) = 0, n E Z. Hence, the spectrum of problem (2.3) is independent of the parameter t, and this completes the proof. □

Remark 1. If instead of p(x, t) and q(x, t) we consider the functions p(x+r, t) and q(x+r, t), then, as we have seen in the previous section, the eigenvalues of the periodic and antiperiodic problems are independent of the parameters r and t. However, the eigenvalues £n of the Dirichlet problem and the signs an do depend on r and t: £n = £n(r, t), an = an(r, t) = ±1.

Remark 2. The theorem gives a method for solving problem (2.1)-(2.3). First we find the spectral data Xn, n E Z, ^(T), ^(T), n E Z \ {0} of the quadratic pencil of Sturm-Liouville equations corresponding to the coefficients p0(x + r) and q0( x + r). Then we solve the Cauchy problem for Dubrovin system (3.1) with the initial conditions

UT, t) 11=0 = en(r), *n(T, t) \t=0 = <J0n(r) , n E Z \ {0}. Finally, by using the trace formulae

( +\ X— + X0 i ^ f X2k-1 + X2k c , A

p(^ t) = —2— + I-2--^k(^ V),

0=k=-<x ^ '

g(r, t) + 2p2(r, t) = ^^ + £ (A2k-12+A2k - &-, t))

we get the expressions of p(r, t) and q(r, t). After that the Floquet solutions vp+(x,Xk, t) of equation (2.3) can be found easily.

Remark 3. If the number of zones is finite, that is, there are two nonnegative integer numbers N and M such that X2k-1 = X2k = for all k > N and — M < k, then system (3.1) reads as

=2(—1)nan(r, i)sign(n) W(Cn — X2n-i)(X2n — Cn)

\

(U - A-,)«,, - Ac) ^ - A2'"')(e" - A2')

k=-M (^n £k)2

k=n, 0

+ 2( A_i + Ao) + 4 £ (A2k-12+A2k - ôk(r, t))) &

0=k=-M ^ ' )

(N \ 2

A_^+A° + ^ (A2k-12+A2k - & ( T, ,)) + A-, + A2

0=k=-M ^

+ £ (A2k-1 + A2k - 2&T, i)) + £ Qk(t);(TJAAk^ T) ) ,

0=k=-M k=-M Ç,n k J

where n = -M,..., -1,1,..., N.

Remark 4. In [13], there was proved the theorem stating that the lengths of the gaps of the quadratic pencil of Sturm-Liouvelle equations with n-periodic real-valued coefficients decrease exponentially if and only if the coefficients are analytic. From this theorem we conclude that if p0(x) and q0(x) are real analytical functions, then the lengths of the gaps corresponding to these coefficients decrease exponentially. The gaps corresponding to the coefficients p(x, t) and q(x, t) are same. Hence, the solutions p(x, t) and q(x, t) of problem (2.1)-(2.3) are real analytical functions on x.

Remark 5. In [26], an analogue of Borg inverse theorem was proved: the number | is a period of the coefficients of the quadratic pencil of Sturm-Liouvelle equations with n-periodic real-valued coefficients if and only if all eigenvalues of antiperiodic problem are double. By this theorem we conclude that if the functions p0(x) and q0(x) have the period |, then all eigenvalues of antiperiodic problem corresponding to these coefficients are double. The gaps corresponding to the coefficients p(x, t) and q(x, t) are same. Hence, the solutions p(x, t) and q(x, t) of problem (2.1)-(2.3) are 2-periodic functions inx.

4. Example

We now illustrate the application of Theorem to solve the problem (2.1)-(2.2).

Let us consider the following initial value conditions

P(x, t)lt=0 = P0(x)

q(x, t)l t=0 = ®(x)

3 - 4 s n2 (3x, I)

1 + 2 cn2 (3x, | 81 - 156 s n2 (3x, | ) + 72 s n4 (3x, | )

(1 + 2cn2 (3x, 2)) 2

for equations of Kaup system kind (2.1) with (t) = . Let us find the spectral data of the problem (2.3) for p0(x) and q0(x). It has the form

A-i = -1, A0 = 1, Ai = 2, A2 = 4, £?(0) = 2, ct?(0) = 1.

In this case the system (3.1) reads as

^ = -226an(T, t) • V(4 -ei)(£i- 2)(£i- 1)(6 + 1). (4.1)

We consider this system subject to the initial condition

6(T,t) |i=0 = £(r), ^(r, t) U = a\(r) , (4.2)

where ^ (r) solves the differential equation

^ = 2a0(r)^(4 - e°(r))(e°(r) - 2)(£0(r) - 1)(£(r) + 1) and satisfies the initial condition

£(T) \ r=0 = 2, -0(r)\T=0 =1.

Solving the Cauchy problem (4.1)-(4.2) we find that (see [27])

6 - 2sn2 (-339i + 3t, §)

T, i) = 1 + 2cn2 (-339i + 3r, §).

Substituting this into the first and second trace formulae, we obtain the solution of the given problem:

P(r, t) =

3 - 4sn2 (-339Î + 3t, |)

q(r, t)

1 + 2cn2 (-339Î + 3r, I) ' 81 - 156sn2 (-339Î + 3t, I) + 72sn4 (-339i + 3t, §)

(1 + 2cn2 (-339Î + 3t, 2))2 where sn and cn are the Jacobi elliptic functions.

Acknowledgements

The authors express their gratitude to Prof. Aknazar Khasanov (Samarkand State University, Uzbekistan) for discussion and valuable advice, as well as to the International Erasmus+ Program KA106-2, Keele University, UK.

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Alisher Bekchanovich Yakhshimuratov, Urgench Branch of Tashkent University of Information Technologies named after Muhammad al-Khwarizmi, Al-Khwarizmi street, 110, 220100, Urgench city, Uzbekistan E-mail: [email protected]

Bazar Atajanovich Babajanov, Urgench State University, Hamid Alimjan street, 14, 220100, Urgench city, Uzbekistan E-mail: [email protected]