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у. О
2024. Т. 50. С. 51-65
Онлайн-доступ к журналу: http: / / mathizv.isu.ru
Серия «Математика»
Research article УДК 517.957
MSC 35P25, 35P30, 35Q51, 35Q53, 37K15
DOI https://doi.org/10.26516/1997-7670.2024.50.51
Integration of the Loaded Negative
Order Nonlinear Schrodinger Equation in the Class
of Periodic Functions
Muzaffar M. Khasanov1^, Ilkham D. Rakhimov1, Donyor B. Azimov1
1 Urgench State University, Urgench, Uzbekistan И [email protected]
Abstract. In this paper, we consider the loaded negative order nonlinear Schrodinger equation (NSE) in the class of periodic functions. It is shown that the loaded negative order nonlinear Schrodinger equation can be integrated by the inverse spectral problem method. The evolution of the spectral data of the Dirac operator with a periodic potential associated with the solution of the loaded negative order nonlinear Schrodinger equation is determined. The results obtained make it possible to apply the inverse problem method to solve the loaded negative order nonlinear Schrodinger equation in the class of periodic ones. Important corollaries are obtained about the analyticity and period of the solution concerning the spatial variable.
Keywords: loaded negative order nonlinear Schrodinger equation, soliton, Dirac operator, inverse spectral problem, Dubrovin's system of equations, trace formulas
For citation: KhasanovM. M., Rakhimovl. D., AzimovD. B. Ilntegration of the Loaded Negative Order Nonlinear Schrodinger Equation in the Class of Periodic Functions. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 50, pp. 51-65. https://doi.org/10.26516/1997-7670.2024.50.51
Научная статья
Интегрирование нелинейного уравнения Шредингера отрицательного порядка с нагруженным членом в классе периодических функций
М. М. Хасанов1 И, И. Д. Рахимов1, Д. Б. Азимов1
1 Ургенчский государственный университет, Ургенч, Узбекистан И [email protected]
Аннотация. Рассматривается нелинейное уравнение Шредингера отрицательного порядка с нагруженным членом в классе периодических функций. Показано, что такое уравнение может быть проинтегрировано методом обратной спектральной задачи. Определена эволюция спектральных данных оператора Дирака с периодическим потенциалом, связанного с решением нелинейного уравнения Шредингера отрицательного порядка с нагруженным членом. Полученные результаты позволяют применить метод обратной задачи для решения нелинейного уравнения Шре-дингера отрицательного порядка с нагруженным членом в классе периодических функций. Получены важные следствия об аналитичности и о периоде решения по пространственной переменной.
Ключевые слова: нагруженное нелинейное уравнение Шредингера отрицательного порядка, солитон, оператор Дирака, обратная спектральная задача, система уравнений Дубровина, формулы следов
Ссылка для цитирования: KhasanovM. M., RakhimovI. D., AzimovD.B. Integration of the Loaded Negative Order Nonlinear Schrodinger Equation in the Class of Periodic Functions // Известия Иркутского государственного университета. Серия Математика. 2024. Т. 50. C. 51-65. https://doi.org/10.26516/1997-7670.2024.50.51
1. Introduction
One of the representatives of the class of completely integrable nonlinear partial differential equations, which has great applied significance, is the nonlinear Schrodinger equation (NSE). The complete integrability of this equation by the inverse problem method, in the classes of periodic and finite-zone functions, was first established in [1;8].
In [7; 26], other integrable nonlinear evolution equations with loaded terms in the class of periodic functions were studied using the inverse spectral problem method. The use of the (G'/G)-expansion method for integrating the loaded Korteweg-de Vries equation (KdV), the loaded modified Korteweg-de Vries equation (mKdV) and the loaded Burger's equation is discussed in [2; 20; 23].
In 1991, J.M. Verosky [25], when studying symmetries and negative powers of the recursive operator, derived the following KdV equation of negative order:
(^j +2ppx=0. (1.1)
S.Y. Lou [15] introduced additional symmetries based on the invert-ibility of the recursive operator for the KdV equation, and in particular showed that the negative order KdV equation is equivalent to the system
of equations
qt = 2ppx,
(1 2)
pxx+qp = 0.
In [19], using the inverse scattering problem method, the KdV equation of negative order was integrated in the class of rapidly decreasing functions.
The KdV equation of negative order with a self-consistent source in the class of periodic functions was studied in [12;21;22], and in [6] the negativeeven mKdV hierarchy and its soliton solutions were studied. Mixed positive and negative hierarchies were studied in [5; 13].
Hierarchies of the negative-order mKdV equation were studied using recurrent methods in [17].
In [9], breather solutions of the following mKdV equation of negative order were studied
+(2<?)t=0,
q x
or
2
Pxx = q , qxt + 2qpxt + aq = 0.
It is easy to see that by replacing q by iq and using transformation^ = px + §i the mKdV equation of negative order will take the following simpler form
' qxt = —2q* 't> 0, x € R.
Vx = —q2
The mKdV equation of negative order in the class of periodic functions was studied in [24], and in [18], the negative order mKdV equation was integrated in the class of rapidly decreasing functions using the inverse scattering problem method.
In this paper, the inverse spectral problem method is applied to the integration of the loaded negative order Schrodinger equation (NSE) in the class periodic functions.
We consider the following loaded negative order nonlinear Schrodinger equation
Pxt = 2vp — qx — Y(t)(p2 (0,t) + q2(0,t))qx,
qxt = 2^q + px + Y(t)(p2(0,t) + q2(0,t))px t> 0, x € R (1.3) Vx = 2qqt + 2ppt
with conditions
q(x,t)|t=0 = qo(x), p(x,t)|t=0 = Po(x^ V(x,t)|x=0 = VO^ (1.4)
where, V0(t) € C 1[0, to), p0(x),q0(x) and Y(t) € C[0, to) - given real functions with period n, and the function y(î) is bounded. It is required to find the real functions p(x, t), q(x, t) and v(x, t) that are periodic in variable x, where
p(x+n, t) = p(x, t), q(x+n, t) = q(x, t), v(x+n, t) = v(x, t), t > 0, x € R,
and satisfying smoothness conditions:
p(x, t) € Cl(t > 0) П C/(t > 0) П C(t > 0), q(x,t) € CX(t > о) П C/(t > о) П C(t > 0), ^(x, t) € Cl(t > 0) П C/(t > 0) П C (t > 0).
(1.5)
When studying problem (1.3)-(1.5), we use the spectral problem for the following Dirac operator
where
B _
10
L(t)y = B^I + tt(x,t)y = Xy, x e R
Л 0(x t) _( p(x,t) q(x,tM y _( yi(x,t)
V ' 0(X,t)_^ q(x,t) -p(x,t)J ' У y2(x,t)
(1.6)
The purpose of this work is to give a procedure for constructing a solution (q(x, t),p(x, t),^(x, t)) to problem (1.3)-(1.5), within the framework of the inverse spectral problem for the Dirac equation (1.6).
0
2. Direct and inverse spectral problems for the Dirac operator
with a periodic coefficient
In this section, we present some basic information concerning the inverse spectral problem for the Dirac operator with a periodic coefficient [3;4; 10; 11; 14; 16]. Let us consider the system of Dirac equations on the entire straight-line
Ly K--) (¡¡I)+( $ 4XX)) (y:)=*(£)•*e R ^
where p(x) and q(x) are real continuous functions from class C:(R), which has period n, and a complex parameter A. Let us denote by
c(x, A) = (c:(x, A), c2(x, A))T and s(x, A) = (s:(x, A), s2(x, A))T
solutions to equation (2.1) satisfying the initial conditions c(0, A) = (1, 0)T and s(0,A) = (0, 1)T. Function A(A) = c:(n, A) + s2(n, A) is called the Lyapunov function or the Hill discriminant for the Dirac operator (2.1). The spectrum of the operator (2.1) consists of the following set
E = {A € R : -2 < A(A) < 2 } = R\ j q (A2n-i,A2„)| .
n= —oo
The intervals (A2n-1, A2n), n € Z are called lacunes.
The roots of the equation s1(n,A) = 0 we denote by n € Z. The numbers n € Z coincide with the eigenvalues of the Dirichlet problem y1(0) = 0,y1(n) = 0 for system (2.1) and the relations € [A2n-1,A2n], n € Z are satisfied.
Numbers € [A2n-1, A2n], n € Z and signs
= sign |s2(n,{„) — C1(n,£n)} ,
n € Z are called spectral parameters of problem (2.1). Spectral parameters tfn, n € Z and spectrum boundaries An, n € Z are called spectral data of problem (2.1). Finding the spectral data of problem (2.1) is called a direct problem, and recovering the coefficients p(x) and q(x) from spectral data is called an inverse problem.
If in the problem (2.1) we consider p(x + t) and q(x + t) instead of p(x) and q(x), then the spectrum of the resulting problem does not depend on the parameter t: A„(t) = An, n € Z, and the spectral parameters depend on the parameter t:{n(T), ct„(t), n € Z. These spectral parameters satisfy an analog of the Dubrovin's system of equations:
oo
^ = (-1 )ra-Vra(T)M£(r)) 2e„(r) (A2fc-i + A2fc — 2{fc(r)) \,n&Z,
L fc=—œ J
where
g (A_2fc-1 - Cra(T))(A2fc - C„(T))
(Ck(T) - Cn(r))2
k=n
hn(0 = V(6*0") - A2ra_i)(A2ra - £„(T))
\
The sign (t) — changes to the opposite at each collision of ) with the boundaries of its lacuna [A2n-1, A2n].
Dubrovin's system of equations, as well as the following trace formulas
P(r) = Efcl-co " ' = )n-^n(r)hn(ar))
give the method for solving the inverse problem.
Lemma 1. If the vector function (y1, y2)T is a solution to the system (2.1), then the following identities hold:
22/22/1 = ^\vl ~vl]' + \q(yl + Vl) (2-2)
y\ - vl = \[vm\' + \p(yi + y22) (2.3)
\[yl+yl\' = Q{y\- vl) -Zpym- (2.4)
3. Evolution of spectral parameters
The main result of this work is the following theorem.
Theorem 1. Let (p(x,t),q(x,t),^(x,t)) be the .solution to problem (1.3)-(1.5). Then the spectrum of operator (1.6) does not depend on the parameter t, and the spectral parameters £n = £n(t), n € Z\{0} satisfy an analogue of the Dubrovin system of equations:
in = ^(-l)n<Tn(t)hn(Ox
x {qt(0, t) + M0,t) + (—p(0, t) - £n)(1 + Y(t)(p2(0,t) + q2(0,t)))} . (3.1)
The signs an(t) = ±1 change each time a point £n(t) collides with the boundaries of its lacuna [A2n-i, A2n]. Moreover, the following initial conditions are satisfied
£n(t)|i=0 = £n, *n(t)|t=0 = ^ , n € Z\{0} (3.2)
where £n, , n € Z\{0} are the spectral parameters of the Dirac operator with coefficients p0 (x), q0 (x).
Proof. Let us denote by yn(x,t) = (yn>i(x,t), yn,2(x,t))T, n € Z, the orthonormal eigenvector functions of the Dirichlet problem for equation (1.6), corresponding to the eigenvalues of £n(t), n € Z.
Differentiating the identity £n(t) = (L(t)yn,yn) with respect to t and using the symmetry of the operator L(t), we have
£n = (ii(x,t)yn,yn). (3.3)
Using an explicit dot product
(y, z) = (yi(x)^i(x) + y2(x)*2(x)] dx, y = ^ yi(X) ) , z = ( Zl(X) ) ,
we rewrite equality (3.3) in the form
£n = [(yn,i - yn,2)pt + 2yn,iyn,2 qt]dx. (3.4)
0
Using formula (2.2) and (2.3) we obtain the following equality
1 fn 1 fn in = — (yn,iVn,2)'ptdx + t (Vn, 2 + yl,i)Wtdx+
£n ,/0 £n ,/0
/ (Уп,2 - yl,i)'4tdx + — / (y2)2 + ylA)qqtdx, n € Z\{0}, (3.5) JO sra .70
The equality (3.5) can be rewritten as
1 1 fn Cn = T^rbl^^) - vl,2(9>t)]qt(0,t) - J- (yn,lVn,2)Pxtdx-2Sn Sn JO
1 fn 1 fn
T^r / (Vn,2 - yl,i)Qxtdx + — / (yn,2 + Vn,i)(PPt +qqt)dx. 2Sn JO JO
I /*n 1 /*n
II
?ra JO Cra ./0
From the system of equation (1.3) we have
Hence,
PPt + qqt = y, Pxt = 2pß -qx - j(t)(p2(0,t) + q2(0,t))qx,
qxt = + px + Y (t)(p2 (0, t) + q 2 (0,t))p*. (3.6)
Cn = -y2;2(o,i)]<?t(o,i)-
2 fn 1 /"n
— / Vn,iyn,2Pßdx - — / (Vn,2 - Vn,i)qßdx+ JO Sn J0
1 1
Cra JO 2Sn JO
11
1 1
+7- / (yn,iyn,2)qxdx - — / (y2)2 - Un,i)Pxdx— JO 2Sn JO
+Y(i)(p2(0,i) +g2(0,i)) I <j -¿-(yn,iyn,2)qx - - yl,i)Px \dx+
JO I2Sn J
1 /■n
/ {yl,2 + yl,i)ßxdx (3.7)
O
We integrate the last integral by parts
Il = ^T (^n.,2 + yl,l)ßxdx = 2Sn JO
1 1 /"n = i) - i/n,2(0. i)M0, ^-^r + yl,n)'ßdx. (3.8)
2Sn 2Sn JO
Based on (2.4), equality (3.8) will take the form
h = -Uy2,2(M) -y2,2(0,i)M0,i)+
2Sn
1 2 7- / (i/2,n - Vl,n)q^dx + -Sn J0 Sn J0
Now consider the third and fourth integrals in equality (3.7):
1 rn 1 en
Cn ./0 J0
1 2
/ {yl,n ~ yl,n)qßdx + — / yi,ny2,nPßdx. (3.9)
JO Sn JO
1 1
h = — (yn,iyn,2)qxdx - — / (y2;2 - yltl)pxdx
JO 2Sn JO
2Sn
1 fn
+T {(vn,2p-vn,iq)yn,2-(vn,ip + vn,2q)vn,i)dx- (3.io)
J0
From equation (1.6) the following equalities follow:
yn,1 + = qyn,i - pyn,2 - yn,2 = pyn,i + qyn,2 •
Using these identities, we get
1 fn 1 fn h = — / (yn,iyn,2)qxdx - — / {ylt2 - ylA)pxdx =
Sra JO JO
= ^[y2,2(M) -y2;2(0,i)](-p(0,i) -Cn) (3.11)
Now let's calculate the fifth integral in equality (3.7): rn i 1 1
— {yn,iyn,2)qx-^r<v2 "'2
/0 I sra 2Sn
1
h = I \ j-(Уп,1Уп,2) qx ~ 2 - vl,i)Px\ dx =
J0 lSn 2Цп J
= 2(M) -Уп,2(0,i)](-p(0,i) -Cn) (3.12)
2Sn
2 jA „,2
n
2(r\ 1 „2 /
From (3.7), (3.9), (3.11) and (3.12) we deduce that
Cn(t) = ^[y2,2(M)-y2,2(o,i)]x
x {qt(0,i) + ^(0,t) + (—p(0, t) - £„)(1 + 7(i)(p2(0,t) + q2(0,t)))}
n e z\{0}. (3.13)
Let us denote by s(x, A, t) = (s1(x,A,i), s2(x,A,t))Tthe solution of equation (1.6) satisfying the initial conditions s(0, A, t) = (0, 1)T. From equality
r n
22
[s|(x, A, t) + s2(x, A, t)]dx
/0
( , .ds2(n,A,t) dsi(n, A, t) = Sl(7T, A, t)-—--s2(7T, A, i)-—-
we find a formula for the norm of the eigenvector function s(x,{n(t), t) of the Dirichlet problem (1.6), (2.4), corresponding to the eigenvalue £n(t):
cn(t)= / [s2(x,e„(t),t) + s2(x,e„(t),t)]dx = 0
dsi(n,{„(t),t)
dA
s2(n,Cn(i),i) (3.14)
Using equality yn(x,t) = ^-jpjs(x,(n(t),t) and (3.13) we get 2 / 2 /n si(vr,C„(i),t) - 1
c„(t)
(3.15)
ax
Substituting values x = n and A = {„(t) into identity
ci(x, A, t)s2(x, A, t) — c2(x, A, t)si(x, A, t) = 1,
we find
cifr,Ut),t) = —,r^TTwv (3-16)
S2(n,4„ (t),t)
Taking into account the equality (3.16) and the following identity
[ci(n, A, t) — s2(n, A, t)]2 = (A2(A) — 4) — 4c2(n, A, t)si(n, A, t), we write
S2(TT,Ut),t) ~ e , ) m .. = <Tn(tWA2(e„(i))-4, (3.17)
where
A(A) = ci(n, A, t) + S2(n, A, t), a„(t) = sign {s2(n, {„(t), t) — ci(n, {„(t), t)} • From (3.15) and (3.17) we deduce
dX
Using the following expansions
A2(A) — 4 = —4-/r2 f] (A"A2fc"fA"A2fc)^i(vr,A,i)=vr fl — >
where a0 = 1 and ak = k when k = 0, the equality (3.18) can be rewritten as follows:
y„,2(n, t) — y„,2(0, t) = 2(—1)n (t)h„ ({)• (3.19)
At the same time, we used the equality
n tt {fc— sign \ —— 11
k = —TO
k=n
= ( — 1)n-
Substituting expression (3.19) into identity (3.13) we derive (3.1).
If we replace Dirichlet boundary conditions with periodic y(n) = y(0) or antiperiodic y(n) = —y(0) boundary conditions, then instead of equation (3.13) we have An = 0. This means that the eigenvalues An, n € Z of the periodic and anti-periodic problems do not depend on the parameter t. □
Corollary 1. If we consider p(x,t) and q(x,t) instead of p(x + t, t) and q(x + t, t), then the eigenvalues of the periodic and antiperiodic problem do not depend on the parameters t, t, and the eigenvalues £n of the Dirichlet problem and the signs depend on t, t: £n = £n(t, t), = (t, t) = ±1, n € Z. In this case, the signs ct„(t, t) = ±1 change, when a point £n(t, t) collides with the boundaries of its lacuna [A2n-1, A, and system (3.1) will take the form
= yn^(r,t)hn(0x
x {qt(T, t) + Mt, t) + (—p(T, t) — C„(t, t))(1 + Y(t)(p2(0, t) + q2(0, t)))} ,
n € Z\{0}, (3.20)
Cn(T,t)|i=0 = en(t), ^n(T,t)|t=0 = ^(t) ,n € Z\{0} (3.21)
Using the trace formula
<x
q(T,t) = E (—1)n-1^n(T,t)hn(e) (3.22)
n=—oo
P(r,i)= E (^T^-Ur.i)) (3-23)
and the equality = 2ppt + 2qqt, we get
oo
ft(r,f)= E (3.24)
n=—oo
p.«)
n=
^(r, t) = ^o(i) + 2 f (p(s, t)pt(s, t) + q(s, i)qt(s, t))ds (3.26)
J0
Corollary 2. This theorem provides a method for solving problem (1.3)-(1.5). To do this, we first find the spectral data An, {П(т), (т), n € Z of the Dirac operator corresponding to the potential q0(x + т), p0(x + т). Next, by solving the Cauchy problem (3.20)-(3.21) when т = 0 we find £n(0, t) and an(0, t), n € Z. Using these data we will find q(0, t), p(0, t). After this, we
substitute the expression for q(0, t), p(0, t), into equation (3.20) and solving the Cauchy problem for an arbitrary t value, we find {„(t, t) and a„(r, t), n e Z. After this, using the trace formulas (3.22) and (3.23), we find the solutions p(x,t) and q(x,t) of problem (1.3)-(1.5), and then from formula (3.26) we determine ^(x,t).
Corollary 3. Using the results of work [10], we conclude that if the initial functions p0(x) and q0(x) are real analytical functions, then the components of the solution p(x, t) and q(x, t) are real analytical functions in x.
Corollary 4. If the number ^ is the period for the initial function po(x) and q0(x), then all the roots of the equation A(A) + 2 = 0 are twice multiple. Since the Lyapunov function corresponding to the coefficients p(x, t) and q(x,t) coincides with A(A), then according to Borg's converse theorem [11], the number ^ is also a period for both the solution p(x, t) and q(x, t) in the variable x.
Corollary 5. If the number | is an antiperiod for the initial function po(x) and q0(x), then all the roots of the equation A(A) — 2 = 0 twice multiple. Since the Lyapunov function corresponding to the coefficients p(x, t) and q(x,t) coincides with A(A), then [3] it follows that the number is also an antiperiod for solutions p(x,t) and q(x,t) in the variable x.
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Об авторах Хасанов Музаффар Машарипович, канд. физ.-мат. наук, доц., Ургенчский государственный университет, Ургенч, 220100, Узбекистан, [email protected], http://orcid.org/0000-0002-2347-1484
Рахимов Илхом Давронбекович,
канд. физ.-мат. наук, Ургенчский государственный университет, Ургенч, 220100, Узбекистан, [email protected],
https://orcid.org/0000-0002-1039-3616
Азимов Дониёр Бахром угли,
магистр, младший научный сотрудник, Ургенчский государственный университет, Ургенч, 220100, Узбекистан, [email protected]
About the authors Muzaffar M. Khasanov, Cand. Sci. (Phys.-Math.), Assoc. Prof., Urgench State University, Urgench, 220100, Uzbekistan, [email protected], http://orcid.org/0000-0002-2347-1484
Ilkham D. Rakhimov, Cand. Sci. (Phys.-Math.), Urgench State University, Urgench, 220100, Uzbekistan, [email protected], https://orcid.org/0000-0002-1039-3616
Donyor B. Azimov, Master (Phys.-Math.), Junior Researcher, Urgench State University, Urgench, 220100, Uzbekistan, [email protected]
Поступила в 'редакцию / Received 03.05.2024 Поступила после рецензирования / Revised 14.06.2024 Принята к публикации / Accepted 17.06.2024
