CC BY
25
% 1И..1..й11?
Серия «Математика»
2021. Т. 36. С. 84—94
Онлайн-доступ к журналу: http://mathizv.isu.ru
УДК 517.95 MSC 35Q51
DOI https://doi.org/10.26516/1997-7670.2021.36.84
The Soliton Solutions for the Nonlinear Schrodinger Equation with Self-consistent Source
A. A. Reyimberganov1, I. D. Rakhimov1
1 Urgench State University, Urgench, Republic of Uzbekistan
Abstract. In this paper by using Hirota's method, the one and two soliton solutions of nonlinear Schrodinger equation with self-consistent source are studied. We have shown the evolution of the one and two soliton solutions in detail by using graphics.
Keywords: soliton solution, Schrodinger equation, nonlinear equations, Hirota's method.
Integrable nonlinear evolution equations have various applications in many fields. It is known that the existence of multi-soliton solutions is an important feature of integrable nonlinear evolution equations, which play a main role in science. They describe nonlinear waves and have important applications in solid state physics, plasma physics and etc.
The nonlinear Schrodinger equation with self-consistent source (nlSES CS) describes the soliton propagation in a medium with both resonant and nonresonant nonlinearities [10]. It is also indicated as the nonlinear interaction of high-frequency electrostatic waves with ion acoustic waves in plasma [1]. Soliton equations with self-consistent source have important physical applications. Therefore, it is always interesting to find its soliton solutions. In 1971, Hirota [3] proposed the Hirota direct method for the Korteweg-de Vries (KdV) equation. Soliton equations with self-consistent source (SESCS) were discussed in [5;7;8;9]. In recent years, extensive research has been conducted on SESCS using the Hirota method [2;14;15;16].
1. Introduction
Apart from that, other methods exist to find solutions of SESCS, such as inverse scattering method and the special treatments of the singularity in the evolution of eigenfunctions [6;11;12], the binary Darboux transformations for the KdV hierarchies with self-consistent sources have been proposed [13].
Usually one-soliton and two-soliton solutions are found using the Hirota method and the next step the Wronskian technique is used for N-soliton solutions. Soliton solutions can be represented by Wronski determinant. In this work, we study one-soliton and two-soliton solutions of the nlSESCS through Hirota's method.
We consider the integration of the following system of equations
N
iut + 2 |u|2 u + uxx = 2i (<Pij - tâj),
3 = 1
(1.1)
(1.2)
..., N are the
¥lj,x = — +U(pij,
^2j,x = i ijf2j - U<fiij, j = 1, 2, ...N,
where the bar means complex conjugation and (j, j = 1,2, eigenvalues.
We assume that the solution u(x, t) of the system (1.1)-(1.2) exists possessing the required smoothness and tends to its limits sufficiently rapidly as |x| ^ to, i.e., for all t > 0 satisfies the condition
(1 + |œ|) |u(cc, t)ldx +
£
k=\
dku(x, t)
dxk
dx < to.
(1.3)
As shown in [4], under the condition shown below the system of equations (1.2) has a finite number of eigenvalues. In general, these eigenvalues can be multiples. Here, we assume that all the eigenvalues are simple and their numbers are equal to N. We also assume that the eigenfunctions = (tfij ,(p2j)T corresponding to this eigenvalues satisfy the following normalizing conditions
/œ
(fiij(fi2jdx = $(t), j = 1, 2,..., N. (1.4)
-<x
Here fij(t), j = 1,2, ...,N are given and the continuous functions of t.
0
oo
2. Bilinear form for the nlSESCS
We will find the solition solution of the nlSESCS by using of Hirota's method. With the help of the dependent variable transformations
« = 7> W = P~7, V23 = , J = l, 2,...,N (2.1)
(2.4)
the system (1.1)-(1.2) can be transformed into the bilinear forms
N
(iDt + D2x)g -f = 2i - h?), (2.2)
3 = 1
D2J -f = 2g ■ g, (2.3)
f DxPj ■ f = — CjPjf + ghj, \Dxhj ■ f = i {j hjf - gpj,
where g and h are the complex conjugation of the functions g and h, respectively and Hirota's bilinear operators Dt and Dx are defined by
id d \m id d \n D^D-g (x, t)-f ^ t) = - - —J - -J g^ t)f (x',f)lx=x',t=t'.
(2.5)
Here, the subscripts of the functions and define the order of the partial derivatives with respect to x and .
Equations (2.2)-(2.4) can be solved by introducing the following power series expansions for f, g, pj and hj:
f = 1+^/(1) + x4f(X + (2.6)
9 = X9(1) +X39(X) + ..., (2.7)
p3 =XP(1 +X3pf) + ..., (2.8)
h = hf) , (2.9)
where x is a formal expansion parameter. Substituting Eqs. (2.6)-(2.9) into Eqs.(2.2)-(2.4) and equating coefficients of the same powers of x to zero can yield the recursion relation for f(k), g(k), p^ and h^1), k = 1, 2,...
3. One-soliton solution
We will give the analytical expression of one-soliton solution (i.e. in the case N = 1) of the system (1.1)-(1.2). According to In the known Hirota's method, we consider for the one-soliton solution of nlSESCS in the form below
9 = x9(1), / = 1 + x2/(1).
Using the definition (2.5) the above (2.3) equation can be expressed in details. Substituting these expressions into (2.3) and equating the coefficients of the same powers of x, we have
№ = 9(1)9(1), (3.1)
№fw - = 0. (3.2)
If we take
g(1) = e^1, (3.3)
then
fW = eVl+Vl+aii (3.4)
satisfies equations (3.1) and (3.2). Here, r]1 = k\x + ry1(t) and an = ln (ki +-1)2, where k1 is constant and 71(i) is an arbitrary function of t.
The next step is to find functions p1 and h1 in case when one-soliton solution are
(1) h u(1) P1 = XPj , h1 = h) .
Based on the above, we collect coefficients of the same power in x according to the expression (2.4) and we get
P{S = —C1P? + g(1)h{1), (3.5)
p[^f(1) - p{1) fi1) = 1P{11)f(1). (3.6)
Using expressions (3.4), (3.3) and by solving (3.5) and (3.6), we have
fJ(1) = p(ki+h-ii;i)x+ni(t)
^ - , (3.7)
h11) = (h + jfe1) ^i^M^jI (t),
where Q1 is an arbitrary function of t.
Using expressions (3.4), (3.3) and (3.7), we can rewrite the functions f, g, p1 and h1 in the following form:
3 V1+V2 +«11
= i=
h1 =(k1 + jfe1) e(l1-iÎ1 ^^W^W.
/=1 + m
(k1+h-iiÏ1)x+n1(t) (3.8)
Substituting these expressions into (2.2), we can assure that the functions Q1(t), 71(i) and the constant k1 satisfy the following conditions
(7l(f ))t = -2e2n 1(i)-271(i)-71(i)-«11 + i k2,
(3.9)
k1 = -2 i £1.
Also, using transformations (2.1) and conditions (1.4), we obtain
Q1(t ) = lnft(i) + 2(71(i) + an ) + 71(1). (3.10)
Using expression (3.10) and solving the differential equation (3.9), we get the following:
71 (t) = -4iiXt - 2 j ft(r)dr + 7i(0). (3.11)
J 0
Thus, taking into account (2.1), (3.8), (3.10) and (3.11) we can write the one-soliton solution of nlSESCS in the following form
e-Xi£ix+~(i(t)
U = 1 + e(-2ifi+2i£i)x+7i(t)+7i(t)+aii , (3.12)
e( - Xiii+i£i)x+ji (t)+7i (t)+aii
011 = ^1 (t)
021 = ßl (t)
1 + e(-2i£i+2i£i)x+ji(t)+ji(t)+aii
(-2i & + 2 г 6) eß1x+^(t)+a11 1 + e(-2iÇi+2iÇi)x+7i(i)+7i(i)+an
The following figure shows one-soliton solution of the nlSESCS.
Figure 1. a) real part b) intensity profiles of the one-soliton solution (3.12) for
Ci = i, 71(0) = 0, ßi(t)
4. Two-soliton solution
In this section, we find two-soliton solution of nlSESCS (i.e. in the case N = 2). We take the functions f and g in the following form
9 = X9(1) +X39(X), / = 1 + XX/(1) +X4/(X).
By applying the same previous procedure, we obtain the set of equations from Eq. (2.3) corresponding to the different power of x
№ = 9(1)9(1), (4.1)
№ + (1) - (/P)2 = g(1)g(2) + g(2)s(1), (4.2)
№ f(2) - 2№ f(2) + f(1) № = g(2)g(2), (4.3)
(2) - (f(2))2 = 0. (4.4)
In order to find two-soliton solution, we utilize the superposition principle. We may use this principle since we are dealing with a bilinear equation and not a nonlinear one. As discussed in the one-soliton solution case, we can solve the equations (4.1)-(4.4) for getting the expression of f and g. In order to construct the two-soliton solution of the system (1.1)-(1.2) we assume 1 has the form
g(1) = eri + e'2, (4.5)
where r]j = kjx + 7j(t), (j = 1, 2). Therefore, the solution of the Eq. (4.1) is following
j(1) = eV1+'-1+a11 + eV1+'-2+ai2 + g T/2+fjl+a21 + er/2+fl2+a22 (4 6)
where
amn = ln (, + - , m,n = 1,2. With the help of Eqs. (4.2)-(4.4), we can obtain the functions f(2 and g(2
j(2) = ¿ri+m+m +m+r, (4.7)
g(2) = ¿ri+ri+m+di + eri+m+m+52, (4.8)
where the constants r, 5j, j = 1,2, are given by
* = ln (ff) • ^ 2
Q1 = ( h + h)(k2 + £1), q_2 = (k1 + foX k2 + h),
(h + h + k2 + k2) (k1 + h + k2 + h) = 71-, 7 xo/,-, r xo , B2 =
(*1 +^1)2(^2 +*1)2 2 (k1 -k2 )2(k2 + k2)2' ( k2 -h)2(h - h)2
=
(k1 + fc1)2(fc1 + ^)2( k2 + k1)2 The next step to find two-soliton solution is to determine the functions pj and hj (j = 1,2). The functions pj and hj (j = 1,2) for two-soliton solution are as follows
Pj = XP^ + X3pf\ hj = h^.
Based on the above, we collect coefficients of the same power in % according to the the (2.4), we have
= + ^^ (4.9)
?$/(1) - p?f21] + pQ = (1) + P?) + g{2Hl), (4.10)
(2) -+(1) -^=(.;1)/(2)+P?f(1)), (4.11)
(2) - = -^2)/(2). (4.12)
We solve differential equations (4.9)-(4.12) by using (4.5)-(4.8) and we get the following expressions
p(1) =e(fci+fci)2-72(i)-72(i)+7i(i)+7i (i)+Hj (t)-r+an + e(fcl+fc2)2-7l(i)-7lM+72(i)+72 (i)+^j (i)-r+ai2 + e(fc2+fcl-i$j )2-7l(i)-72(i)+7l(i)+72 (i)+^j (i)-r+«2l + e(fc2+fc2)2-7l(i)-7l(i)+72(i)+72 (i)+^j (i)-r+«22
(4.13)
p(2) _ e(fci+fci+fc2+fc2-i?j(t) (4 14)
h(1) = -_=__h
3 (kl + fci)(e^1 + e^2 )
e(fcl+fc2-iÇj )a;-72(i)-72(i)+7l (i)+7l(i)+^j (i)-r
(fci + foXe^ + e^2 )
e(fc2+fcl-iÇj )a;-72(i)-72(i)+7l (i)+7l(i)+^j (i)-r e(fc2+72 )ж-72 (i) -72 (i)+7l (i)+7l (t)+iïj (t)-r
+ +
(4.15)
(fo + + e^2 ) '
i = 1,2.
The time-dependent evolution of the functions 7j (t) , Qj (t) (j = 1, 2) can be found similarly as in one-soliton solutions. By substituting the defined functions
/ = 1 + /(1) + /(2), 5 = + 5(2), Pj = P? + pf, hj = hj^ (4.16) into Eq.(2.2), we get the following
(7i(t))t = -2e2^(i)-2(7l(i)+72(i))-7i (t)-r+Si + ik2,
ki = -2г&, j = 1, 2.
(4.17)
We know that, the condition (1.4) is assumed for the functions (p1j and (p2j (j = 1, 2), so, the function Qj (t) (j = 1,2) is defined as
Qj(t) = ln (t) + 2(7,(t) - r + Sj) + 7i(i) + I2(t), j = 1, 2, (4.18) therefore,
Ъ(t) = -4Й* - 2 ß2(r)dr + Ъ(0), j = 1, 2.
(4.19)
Thus, taking into account (4.5)-(4.8) and (4.13)-(4.15) we can write the solution in the following form
и
g(1) + g(2) 1 + /(1) + f(2) -
(4.20)
Vu =
(1) + (2)
1 + f(1) + f(2y V23 1 + f(1) + f(2) These functions are two-soliton solutions of the nlSESCS.
The following figure shows two-soliton solution of the nlSESCS.
h(1)
3 = 1, 2.
a) 0.5 Re(ü) 0
b)
тш\\\\
Figure 2. a) real part b) intensity profiles of the two-soliton solution (4.20) for
£1 = 1 6 = 1 г + 1, 7j (0) = 0, ßj (i) = 1, (j = 1, 2).
t
0
10
4
-10 0
|U|
10
2
x
5. Conclusion
In this paper, we have obtained the one-soliton and two-soliton solutions for the nlSESCS, by directly applying Hirota's bilinear method. Besides other soliton solutions can also be got by Hirota's bilinear method.
Acknowledgements
The authors would like to express their sincere thanks to the Doctor of Physical and Mathematical sciences G. U. Urazboev for valuable advices and helpful recommendations in completing this work.
References
1. Claude C., Latifi A., Leon J. Nonlinear resonant scattering and plasma instability: an integrable model. J. Math. Phys., 1991, vol. 32, pp. 3321-3330.
2. Deng S.F., Chen D.Y., Zhang D.J. The Multisoliton solutions of the KP equation with self-consistent sources. J. Phys. Soc., 2003, vol. 72, pp. 2184-2192.
3. Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett., 1971, vol. 27, pp. 1192-1194.
4. Khasanov A.B., Reyimberganov A.A. About the finite density solution of the higher nonlinear Schrodinger equation with self-consistent source. Ufimsk. Mat. Zh., 2009, vol. 1, pp. 133-143.
5. Leon J., Latifi A. Solution of an initial-boundary value problem for coupled nonlinear waves. J. Phys. A: Math. Gen., 1990, vol. 23, pp. 1385-1403.
6. Lin R.L., Zeng Y.B., Ma W.X. Solving the KdV hierarchy with self-consistent sources by inverse scattering method. Physica A, 2001, vol. 291, pp. 287-298.
7. Mel'nikov V.K. Capture and confinement of solitons in nonlinear integrable systems. Commun. Math. Phys., 1989, vol. 120, pp. 451-468.
8. Mel'nikov V.K. Integration method of the Korteweg-de Vries equation with a self-consistent source. Phys. Lett. A, 1988, vol. 133, pp. 493-496.
9. Mel'nikov V.K. Integration of the nonlinear Schroedinger equation with a self-consistent source.Commun. Math. Phys., 1991, vol. 137, pp. 359-381.
10. Mel'nikov V. K. Integration of the nonlinear Schrodinger equation with a source. Inverse Problems, 1992, vol. 8, pp. 133-147.
11. Urazboev G.U., Khasanov A.B. Integrating the Korteweg-de Vriese equation with a self-consistent source and "steplike" initial data. Theor. Math. Phys., 2001, vol. 129, pp. 1341-1356.
12. Zeng Y.B, Ma W.X., Lin R. Integration of the soliton hierarchy with self-consistent sources. J. Math. Phys., 2000, vol. 41, pp. 5453-5489.
13. Zeng Y.B., Ma W.X., Shao Y.J. Two binary Darboux transformations for the KdV hierarchy with self-consistent sources. J. Math. Phys., 2001, vol. 42, 2113-2128.
14. Zhang D. J. The N-soliton solutions for the modified KdV equation with self-consistent sources. J. Phys. Soc. Japan, 2002, vol. 71, pp. 2649-2656.
15. Zhang D. J. The N-soliton solutions of some soliton equations with self-consistent sources. Chaos Solitons Fractals, 2003, vol. 18, pp. 31-43.
16. Zhang D. J., Chen D. Y. The N-soliton solutions of the sine-Gordon equation with self-consistent sources. Physica A, 2003, vol. 321, pp. 467-481.
Anvar Reyimberganov, Candidate of Science (Physics and Mathematics), Associate Professor, Urgench State University, 14, Kh. Alimdjan st., Urgench, 220100, Republic of Uzbekistan, tel.:+9(9862)2246700, email: anvar@urdu.uz, ORCID iD https://orcid.org/0000-0001-7686-1032
Ilkham Rakhimov, Postgraduate, Department of Applied Mathematics and Mathematic Physics, Urgench State University, 14, Kh. Alimdjan st., Urgench, 220100, Republic of Uzbekistan, tel.:+9(9862)2246700, email: ilham.rahimov.87@mail.ru, ORCID iD https://orcid.org/0000-0002-1039-3616
Received 30.01.2021
Солитонные решения нелинейного уравнения Шредин-гера с самосогласованным источником
А. А. Рейимберганов1, И.Д.Рахимов1
1 Ургенчский государственный университет, Ургенч, Республика Узбекистан
Аннотация. Нелинейное уравнение Шредингера с самосогласованным источником преобразовано в билинейные формы и найдены односолитонные и двух-солитонные решения прямым билинейным методом Хироты. Подробно обсуждена эволюция солитона с помощью графики.
Ключевые слова: солитонные решения, уравнение Шредингера, нелинейные уравнения, метод Хироты.
Список литературы
1. Claude C., Latifi A., Leon J. Nonlinear resonant scattering and plasma instability: an integrable model //J. Math. Phys. 1991. Vol. 32. P. 3321-3330.
2. Deng S. F., Chen D. Y., Zhang D. J. The Multisoliton solutions of the KP equation with self-consistent sources // J. Phys. Soc. 2003. Vol. 72. P. 2184-2192.
3. Hirota R. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons // Phys. Rev. Lett. 1971. Vol. 27. P. 1192-1194.
4. Khasanov A. B., Reyimberganov A. A. About the finite density solution of the higher nonlinear Schrodinger equation with self-consistent source // Ufimsk. Mat. Zh. 2009. Vol. 1. P. 133-143.
5. Leon J., Latifi A. Solution of an initial-boundary value problem for coupled nonlinear waves //J. Phys. A: Math. Gen. 1990. Vol. 23. P. 1385-1403.
6. Lin R. L., Zeng Y. B., Ma W. X. Solving the KdV hierarchy with self-consistent sources by inverse scattering method // Physica A. 2001. Vol. 291. P. 287-298.
7. Mel'nikov V. K. Capture and confinement of solitons in nonlinear integrable systems // Commun. Math. Phys. 1989. Vol. 120. P. 451-468.
8. Mel'nikov V. K. Integration method of the Korteweg-de Vries equation with a self-consistent source // Phys. Lett. A. 1988. Vol. 133. P. 493-496.
9. Mel'nikov V. K. Integration of the nonlinear Schroedinger equation with a self-consistent source // Commun. Math. Phys. 1991. Vol. 137. P. 359-381.
10. Mel'nikov V. K. Integration of the nonlinear Schrodinger equation with a source // Inverse Problems. 1992. Vol. 8. P. 133-147.
11. Urazboev G. U., Khasanov A. B. Integrating the Korteweg-de Vriese equation with a self-consistent source and "steplike"initial data // Theor. Math. Phys. 2001. Vol. 129. P. 1341-1356.
12. Zeng Y. B., Ma W. X., Lin R. Integration of the soliton hierarchy with self-consistent sources //J. Math. Phys. 2000. Vol. 41. P. 5453-5489.
13. Zeng Y. B., Ma W. X., Shao Y. J. Two binary Darboux transformations for the KdV hierarchy with self-consistent sources // J. Math. Phys. 2001. Vol. 42. P. 2113-2128.
14. Zhang D. J. The N-soliton solutions for the modified KdV equation with self-consistent sources // J. Phys. Soc. Japan. 2002. Vol. 71. P. 2649-2656.
15. Zhang D. J. The N-soliton solutions of some soliton equations with self-consistent sources // Chaos Solitons Fractals. 2003. Vol. 18. P. 31-43.
16. Zhang D. J., Chen D. Y. The N-soliton solutions of the sine-Gordon equation with self-consistent sources // Physica A. 2003. Vol. 321. P. 467-481.
Анвар Акназарович Рейимберганов, кандидат физико-математических наук, Ургенчский государственный университет, Республика Узбекистан, 220100, г. Ургенч, ул. Х. Алимджана, 14, тел.: +9(9862)2246700, email: anvar@urdu.uz, ORCID iD https://orcid.org/0000-0001-7686-1032
Илхом Давронбекович Рахимов, аспирант, кафедра прикладной математики и математической физики, Ургенчский государственный университет, Республика Узбекистан, 220100, г. Ургенч, ул. Х. Алимджана, 14, тел.: +9(9862)2246700, email: ilham.rahimov.87@mail.ru, ORCID iD https://orcid.org/0000-0002-1039-3616
Поступила в 'редакцию 30.01.2021
