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Optical solitons and Peregrine solitons for nonlinear Schrödinger equation by variational iteration method
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Short note
Optical solitons and Peregrine solitons for nonlinear Schrödinger equation by variational iteration method
T
⁎
Abdul-Majid Wazwaza, , Lakhveer Kaurb a b
Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA Department of Mathematics, Jaypee Institute of Information Technology, Noida, UP, India
A R T IC LE I N F O
ABS TRA CT
Keywords: Nonlinear Schrödinger equation Akhmediev breathers Peregrine soliton G′ -Expansion method G2 Solitons
The current study is dedicated for operating the variational iteration method (VIM) to the selffocusing nonlinear Schrödinger (NLS) equation. Consequently, optical solitons and singular soliton solutions are formally derived. The VIM, representing the resultant solution in a series structure, does not require linearization of the equation and accelerates the convergence of resultant solution. The presented analysis shows the pertinent features of this method.
( )
Furthermore, the powerful
( )-expansion method has been adopted for the aforementioned G′
G2
equation for culminating hyperbolic, trigonometric and rational function solutions with rich mathematical structures, leading to diverse types of solitons.
1. Introduction The self-focusing nonlinear Schrödinger equation reads
iut +
1 u xx + u |u|2 = 0, 2
i=
−1 ,
(1)
such that development of slowly shifting waves for a physical system with tiny dissipation in stable dispersive are ruled by complex function u(x, t), with x as longitudinal variable parameter and t acting as co-moving instant of time. Various areas of physics for instance: transmission of light waves in nonlinear medium of optical fibers, plasma physics, super conductivity and quantum mechanics [1–6] extensively deals with Eq. (1). Eq. (1) has variety of implications in advancement of electro magnetic wave propagation, Bose–Einstein condensates, hydrodynamics, water waves, the harmonic oscillator, the hydrogen atom, and many others [7–13]. The intrinsic scientific implications boost up theoretical as well as experimental studies for characteristics of soliton solutions to NLS equation (1). These solitons enlighten new aspects of optical propagation in nonlinear waveguides in assorted models. More specifically, NLS equation (1) embraces only group momentum dispersion and self-phase inflection for picosecond light pulses. Peregrine [2] demonstrate that self-focusing nonlinear Schrödinger (NLS) equation own up an exact solution, depicting a rogue wave model with a significant amplitude that appears from nowhere and disappears with no trace. In the process of gradient catastrophe with semi-classical form of Eq. (1), Peregrine solitons (PSs) are created as a comprehensive by-product. Most importantly, Akhmediev breather have localization property in transverse dimension, in addition to, longitudinal dimension. Moreover, the Kuznetsov–Ma breather, monitored in fiber optics, is localized in x-direction and having periodic nature in the t-direction. The Akhmediev and Kuznetsov–Ma breathers [1] could be merged into a common term, so that we may recover one or the other breather
⁎
Corresponding author. E-mail addresses: [email protected] (A.-M. Wazwaz), [email protected] (L. Kaur).
https://doi.org/10.1016/j.ijleo.2018.11.004 Received 3 November 2018; Accepted 3 November 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
if the modulus of a spectral constraint is above or below a certain specific value [3]. Akhmediev breather is presented as
u (x , t ) =
(1 − 4A1 )cosh(A2 t ) +
2A1 cos(ωx ) + iA2 sinh(A2 t )
2A1 cos(ωx ) − cosh(A2 t )
ei t,
i=
−1
(2)
where
A2 =
8A1 (1 − 2A1 ) ,
ω = 2 1 − 2A1 ,
(3)
and 0 < A1 < 0.5. However, When A1 > 0.5, the Akhmediev breather describes the Kuznetsov–Ma soliton. The so-called Peregrine solitons (PSs) are given as
4 + 8i t ⎤ i t u (x , t ) = ⎡1 − e . 1 + 4x 2 + 4t 2 ⎦ ⎣
(4)
It was reported [3] that hierarchy of solutions with rational structure having increasing order for Eq. (1) exist, originating with the simplest outline, which all may be described by
G + i H⎤ it u (x , t ) = ⎡1 − e , D ⎦ ⎣
D ≠ 0.
(5)
A higher order rational solution (5) was derived by selecting the following [3]
(
G = t2 + x2 + H=
t (t 2
D=
1 2 (t 3
−
5x 2)
3 4
)(x
+
+ x 2 )3 +
2
+ 5t 2 +
3(t 2
+
1 (x 2 4
−
3 4
),
15 , 8 3 3t 2)2 + 64 (12x 2
x 2)2
−
+ 44t 2 + 1).
(6)
Nonlinear transmission of solitons attracted more scrutiny in an array of physical surroundings. Solitons [14] are classified as concentrated waves, propagating with identical shape and velocity assets, and stable under interacting collisions and result as a balance between dispersion and nonlinear effects. The interactions of any number of solitons do not transform the soliton forms, and phase shift is the only outcome of this interaction. The exclusive analysis of optical soliton solutions [15–18] to nonlinear partial differential equations happens to be foremost resource for exploring widely different physical phenomena. Recently, many influential and dominating techniques have been effectively implemented to decipher nonlinear partial differential equations having constant and variable coefficients as well. Diverse efficient modes are earlier used to deal with NLS equation (1), like: Symbolic methods, similarity and symmetry analysis, Hirota's bilinear terms, conservation laws [19–25]. We plan to discover exact optical soliton, Akhmediev breather, and Peregrine soliton solutions for (1) by means of variational iteration method (VIM). Secondarily, we attempt to showcase potential of the VIM method to hand out with nonlinear partial differential equations of technical and scientific situations in a consolidated fashion without requiring any extra limitation. The VIM method has immense capacity of minimizing size of computations along with retaining high level of accuracy for numerical solution. Moreover, hyperbolic, trigonometric and rational function solutions for G′ considered equation with enhanced mathematical structure are found via 2 -expansion method.
( ) G
2. Summary of variational iteration method (VIM) The variational iteration method (VIM) is well known in the literature, earnest reader may refer to conventional research for its exclusive study, hence we give brief summary of this method. The differential equation is considered as
L1 u + L2 u = p (x ),
(7)
with L2 and L1 defined as nonlinear and linear operators respectively and source inhomogeneous term is exhibited by p(x). The considered method (VIM) enroll corrective action functional for Eq. (7) as given:
un + 1 (x ) = un (x ) +
(∫ 0
x
)
λ (t )(L1 un (t ) + L2 u˜n (t ) − p (t )) dt ,
(8)
here, Lagrange multiplier λ could be fixed parameter or variable parameter of t and λ is depicted optimally via variational theory and by using restricted variation u˜n , which means δu˜n = 0 . Primarily, λ is determined, which is further carried out with iteration formula not having controlled variation. Then evaluation of succeeding approximations un+1(x), n ≥ 0 for solution u(x) is uniformly achieved. Any selective function can be objected as zeroth approximation u0 for VIM and based on this mode, the solution is structured as
u (x ) = lim un (x ).
(9)
n →∞
The VIM was proved by many to be very effective, can be used in a direct manner without any need to linearization of the nonlinear 805
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
terms. As stated earlier, we will use the VIM to determine optical solitons, Akhmediev breathers and Peregrine solitons. 3. Optical solitons Nonlinear Schrödinger equation, under review, as
iut +
1 u xx + u|u|2 = 0, with u (x , 0) = ( 8 sech( 8 x )). 2
(10)
We first set the correction functional for (10) as
un + 1 (x , t ) = un (x , t ) +
∫0
t
λ (θ) ⎛i ⎝ ⎜
here, u¯ as complex conjugate of u with The stationary constrain leads to
∂un (x , θ) 1 ∂2u˜n)(x , θ) + + u2u¯ ⎞ dθ ∂θ 2 ∂x 2 ⎠ ⎟
|u|2
(11)
= uu ¯ .
1 + iλ = 0, λ′ = 0,
(12)
result into (13)
λ = i. Simulating Lagrange multiplier λ = i for the functional (11), ensures the resulting iteration phrase (formula):
un + 1 (x , t ) = un (x , t ) + i
∫0
t
2 ⎛i ∂un (x , θ) + 1 ∂ (u˜n (x , θ)) + u2u¯ ⎞ dθ , ∂ θ 2 ∂x 2 ⎠ ⎝
⎜
⎟
n ≥ 0.
(14)
As revealed earlier, u 0 (x , 0) = 8 sech( 8 x ) could be adopted from the given initial condition (10). Applying this u0(x, 0) into (14), subsequent approximations are provoked:
u 0 (x , t ) =
8 sech( 8 x ),
u1 (x , t ) =
8 sech( 8 x )(1 + 4i t),
u2 (x , t ) =
8 sech( 8 x ) 1 + 4i t +
(4i t)2 2!
8 sech(
(4i t)2 2!
+
(4i t)3 3!
(4i t)2 2!
+
(4i t)3 3!
u3 (x , t ) =
( 8 x ) (1 + 4i t +
), ),
⋮ un (x , t ) =
(
8 sech( 8 x ) 1 + 4i t +
+
(4i t) 4 4!
)
+⋯ .
(15)
The exact soliton solution are accordingly furnished as (Fig. (1))
u (x , t ) =
8 sech( 8 x ) e 4i t .
(16)
Following the similar fashion, nonlinear Schrödinger equation
iut +
1 u xx + u|u|2 = 0, with u (x , 0) = ( 8 cosech( 8 x )), 2
(17)
is accompanied with singular soliton structure
Fig. 1. Soliton profile for (a) real part and (b) Imaginary part of solution (16). 806
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
Fig. 2. Three dimensional graphics of soliton for (a) real part and (b) Imaginary part of solution (20) with a = 1, b = 4.
u (x , t ) = ± i 8 cosech( 8 x ) e 4i t .
(18)
4. More optical soliton solutions Following the above mentioned analysis, we set a new u0(x, t) as
u (x , 0) = k e i ax sech(kx),
(19)
and we acquire the soliton solutions, utilizing the VIM method:
u (x , t ) = k sech(k (x − at)) ei (ax + bt),
(20)
valid upon using (Fig. (2))
k=
a2 + 2b .
(21)
5. Complex solutions The nonlinear cubic Schrödinger equation enjoy the following form
iut +
1 u xx + u|u|2 = 0, with initial constraint u (x , 0) = e i x . 2
(22)
We proceed as before for Eq. (22) and use the correction functional (11) with the Lagrange multiplier λ = i for attaining the subsequent iteration ’phrase (formula)
un + 1 (x , t ) = un (x , t ) + i
∫0
t
2 ⎛i ∂un (x , θ) + 1 ∂ (u˜n (x , θ)) + u2u¯ ⎞ dθ , ∂θ 2 ∂x 2 ⎝ ⎠
⎜
⎟
n ≥ 0.
(23)
ix
As revealed earlier, u0(x, 0) = e is selected for presenting the successive approximations:
u1 (x , t ) = e i x u2 (x , t ) = e i x u3 (x , t ) = e i x
( + 1), ( t − t + 1), ( t− t − t it 2 i 2
1 2 8
i 2
1 2 8
i 3 48
)
+1 ,
⋮
(
i
1
un (x , t ) = e i x 1 + 2 t − 8 t 2 −
i 3 t 48
+
1 4 t 384
) + ⋯.
(24)
The exact complex solution are accordingly furnished as 1
u (x , t ) = ei (x + 2 t ) .
(25)
It is remarkable that we may show that nonlinear Schrödinger equation gives the generalized complex solution
u (x , t ) = ei (Λ1x + Λ2t ),
(26)
where 807
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
Λ2 = 1 −
1 2 Λ1 . 2
(27)
6. Soliton solutions by
( )-expansion method G′ G2
This segment of article is devoted to construct some new optical solutions to Eq. (1) via
( )-expansion approach. The rapid G′ G2
( ) G′
review of 2 -expansion method purpose the solution composition for nonlinear Schrödinger equation (1) as follows: G First and foremost, subsequent wave transformation is executed for Eq. (1):
u (x , t ) = W (ζ )exp(iσ ),
(28)
along with σ = α1x + α2t, ζ = iα3(x + α4t). Simulating (28) for Eq. (1), ensures the resulting reduced ordinary differential equation as:
2(W (ζ ))3 + (−α12 − 2α2 ) W (ζ ) + (−2α1 α3 − 2α3 α4 ) W ′ (ζ ) − W ″ (ζ ) α32 = 0.
(29)
We purpose the following solution structure for Eq. (29): m
W (ζ ) =
∑ l =−m
G′ l kl ⎛ 2 ⎞ , ⎝G ⎠
(30)
in which kl, − m ≤ l ≤ m are evaluated and G = G(ζ) follows subsidiary ordinary differential equation: 2 ′ ⎛ G′ ⎞ − M − N ⎛ G′ ⎞ = 0. 2 2 ⎝G ⎠ ⎝G ⎠
(31)
The homogeneous balance among most nonlinear term and most highest order derivative term in Eq. (29) results into m = 1 in Eq. (30) and then formal solution is written as follows:
G′ G′ −1 W (ζ ) = k 0 + k1 ⎛ 2 ⎞ + k2 ⎛ 2 ⎞ , ⎝G ⎠ ⎝G ⎠
(32)
with parameters k1, k2, k3, which are yet to be identified. Eq. (32) is employed to Eq. (30) with help of Eq. (32) and after that followed
( ) G′
j
by assembling of different coefficients with like powers of G2 . Finally, a set of nonlinear equations are retrieved for engaged parameters by comparing each coefficient to zero, as given below:
− 2 MNα32k2 − α12k2 + 6 k 0 2k2 + 6 k1 k22 − 2 α2 k2 = 0, − 2 MNα32k1 − α12k1 + 6 k 0 2k1 + 6 k12k2 − 2 α2 k1 = 0, 2 Mα1 α3 k2 + 2 Mα3 α4 k2 + 6 k 0 k22 = 0, − 2 Nα1 α3 k1 − 2 Nα3 α4 k1 + 6 k 0 k12 = 0, − 2 M 2α32k2 + 2 k23 = 0, − 2 N 2α32k1 + 2 k13 = 0, − 2 k1 Mα1 (α3 + α4 ) + 2 k2 Nα1 α3 + 2 k2 Nα3 α4 − k 0 α12 + 2 k 03 + 12 k 0 k1 k2 − 2 k 0 α2 = 0.
(33)
The algebraic set of Eq. (33) are exercised to reveal following nontrivial cases for involved parameters, such that k0, k1, k2 are nonzero:
N=−
α12 + 2α1 α 4 + α 42 , 36α3 k2 1
k 0 = − 3 (α1 + α4 ),
α 2
α2 = − 181 + k1 =
8α1 α 4 9
α12 + 2α1 α 4 + α 42 , 36k2
+
4α 42 , 9
M=
k2 , α3
(34)
with α1, α3, α4, k2 as free constants. Considering the acquired values (34) and general solutions of (31) in (32), we have developed soliton solutions to Eq. (29): (35) Class 1: Solution with trigonometric function for MN > 0 i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ ⎞ (α1 + α 4 ) i ⎛K 0 cos ⎛ 1 ζ + K1 sin ⎛ 1 ζ ⎝ 6α3 ⎠ ⎝ 6α3 ⎠⎠ ⎝ ⎜
W (ζ ) =
1 (α 3 1
+ α4 ) +
⎟
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ ⎞ 6 ⎛K1 cos ⎛ 1 ζ − K 0 sin ⎛ 1 ζ ⎝ 6α3 ⎠ ⎝ 6α3 ⎠⎠ ⎝ ⎜
(α1 + α 4 ) i ⎛K1 cos ⎛ ⎝ ⎝ ⎜
−
⎟
i (α1 + α 4 ) ⎞ i (α + α 4 ) ⎞ ⎞ ζ − K 0 sin ⎛ 1 ζ ⎟ 6α3 ⎠ ⎝ 6α3 ⎠⎠
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ ⎞ 6 ⎛K 0 cos ⎛ 1 ζ + K1 sin ⎛ 1 ζ ⎝ 6α3 ⎠ ⎝ 6α3 ⎠⎠ ⎝ ⎜
,
⎟
(35)
(36) Class 2: Solution having hyperbolic function for MN < 0 808
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ (α1 + α 4 ) i ⎛K 0 cosh ⎛ 1 ζ + K 0 sinh ⎛ 1 ζ + K1 ⎞ ⎝ 3α3 ⎠ ⎝ 3α3 ⎠ ⎝ ⎠ ⎜
W (ζ ) =
1 (α 3 1
+ α4 ) −
⎟
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ 6 ⎛⎜K 0 cosh ⎛ 1 ζ + K 0 sinh ⎛ 1 ζ − K1 ⎟⎞ ⎝ 3α3 ⎠ ⎝ 3α3 ⎠ ⎝ ⎠
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ (α1 + α 4 ) i ⎛K 0 cosh ⎛ 1 ζ + K 0 sinh ⎛ 1 ζ − K1 ⎞ ⎝ 3α3 ⎠ ⎝ 3α3 ⎠ ⎝ ⎠ ⎜
+
⎟
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ 6 ⎛K 0 cosh ⎛ 1 ζ + K 0 sinh ⎛ 1 ζ + K1 ⎞ ⎝ 3α3 ⎠ ⎝ 3α3 ⎠ ⎝ ⎠ ⎜
,
⎟
(36)
(37) Class 3: Solution having rational function for M = 0, N ≠ 0
W (ζ ) =
(α 2 + 2 α1 α4 + α4 2)(K 0 ζ + K1 ) 1 K 0 α3 (α1 + α4 ) − − 1 , 36K 0 α3 3 K 0 ζ + K1
(37)
with K0 and K1 as free parameters. Slipping back to original variables x, t and by means of (35)–(37), we acquired solutions for Eq. (1) as
u (x , t ) = W (ζ )exp(iσ ), along with σ = α1x + α2t, ζ = iα3(x + α4t). 7. Conclusion The current work involves the systematic examination of self-focusing nonlinear Schrödinger equation, modeling the propagation of an ultrashort (femtosecond) pulses in nonlinear optical fibers. The most significant variational iteration method (VIM) has been utilized for fabricating optical soliton solution and singular soliton solutions. Additionally, the current study discloses the potential of VIM method, as an organized tool for unfolding the solutions of nonlinear equations by employing suitable Lagrange multipliers. G′ Furthermore, 2 -expansion method has been taken into account to produce hyperbolic, trigonometric and rational solutions of G aforementioned equation, which are accompanied by free involved parameters. Illustration of the behavior of solutions is portrayed by appointing several values to the arbitrary constants, which might be fruitful for clarification of numerous problems related to nonlinear optics.
( )
References [1] O.G. Gaxiola, A. Biswas, Akhmediev breathers, Peregrine solitons and Kuznetsov–Ma solitons in optical fibers and PCF by Laplace–Adomian decomposition method, Optik 172 (2018) 930–939. [2] D.H. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Aust. Math. Soc. Ser. B 25 (1983) 16–43. [3] N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere and disappear without a trace, Phys. Lett. A 373 (2009) 675–678. [4] H. Leblond, D. Mihalache, Models of few optical cycle solitons beyond the slowly varying envelope approximation, Phys. Rep. 523 (2013) 61–126. [5] G.A. El, E.G. Khamis, A. Tovbis, Dam break problem for the focusing nonlinear Schrödinger equation and the generation of rogue waves, Nonlinearity 29 (2016) 2798 (1-29). [6] A. Biswas, S. Arshed, Application of semi-inverse variational principle to cubic–quartic optical solitons with Kerr and power law nonlinearity, Optik 172 (2018) 847–850. [7] A. Hasegawa, F. Tappert, Transmission of stationary nonlinear optical physics in dispersive dielectric fibers. II. Normal dispersion, Appl. Phys. Lett. 23 (1973) 171–172. [8] M. Gedalin, T.C. Scott, Y.B. Band, Optical solitary waves in the higher order nonlinear Schrödinger equation, Phys. Rev. Lett. 78 (1997) 448–451. [9] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G.V. Shlyapnikov, M. Lewenstein, Dark solitons in Bose–Einstein condensates, Phys. Rev. Lett. 83 (1999) 5198–5201. [10] W.P. Hong, Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic–quintic non-Kerr terms, Opt. Commun. 194 (2001) 217–223. [11] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, C. Salomon, Formation of a matter-wave bright soliton, Science 296 (2002) 1290–1293. [12] K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Formation and propagation of matter-wave soliton trains, Nature 417 (2002) 150–153. [13] G. Assanto, T.R. Marchant, A.A. Minzoni, N.F. Smyth, Reorientational versus Kerr dark and gray solitary waves using modulation theory, Phys. Rev. E 84 (2011) 066602-1-12. [14] C.H. Gu, Soliton Theory and Its Application, Springer, Berlin, 1995. [15] H. Leblond, D. Mihalache, Few-optical-cycle solitons: modified Korteweg–de Vries sine-Gordon equation versus other non-slowly-varying-envelope-approximation models, Phys. Rev. A 79 (2009) 063835-1-7. [16] A.M. Wazwaz, Multiple soliton solutions for two integrable couplings of the modified Korteweg–de Vries equation, Proc. Rom. Acad. A 14 (2013) 219–225. [17] A. Biswas, Optical soliton perturbation with Radhakrishnan–Kundu–Lakshmanan equation by traveling wave hypothesis, Optik 171 (2018) 217–220. [18] A. Biswas, Chirp-free bright optical soliton perturbation with Fokas–Lenells equation by traveling wave hypothesis and semi-inverse variational principle, Optik 170 (2018) 431–435. [19] H. Hereman, A. Nuseir, Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. Comput. Simul. 43 (1997) 13–27. G′ expansion [20] L. Kaur, R.K. Gupta, Kawahara equation and modified Kawahara equation with time dependent coefficients: symmetry analysis and generalized G method, Math. Methods Appl. Sci. 36 (2013) 584–600. [21] C.M. Khalique, Solutions and conservation laws of Benjamin–Bona–Mahony–Peregrine equation with power-law and dual power-law nonlinearities, Pramana – J. Phys. 80 (2013) 413–427. [22] A. Biswas, Y. Yildirim, E. Yasar, M.M. Babatin, Conservation laws for Gerdjikov–Ivanov equation in fiber optics and PCF, Optik 148 (2017) 209–214. [23] L. Kaur, A.M. Wazwaz, Similarity solutions of field equations with an electromagnetic stress tensor as source, Rom. Rep. Phys. 70 (2018) 114-1-12. [24] L. Kaur, A.M. Wazwaz, Dynamical analysis of lump solutions for (3 + 1) dimensional generalized KP-Boussinesq equation and its dimensionally reduced equations, Phys. Scr. (2018), https://doi.org/10.1088/1402-4896/aac8b8. [25] L. Kaur, A.M. Wazwaz, Painlevé analysis and invariant solutions of generalized fifth-order nonlinear integrable equation, Nonlinear Dyn. (2018), https://doi. org/10.1007/s11071-018-4503-8.
( )
809
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Short note
Optical solitons and Peregrine solitons for nonlinear Schrödinger equation by variational iteration method
T
⁎
Abdul-Majid Wazwaza, , Lakhveer Kaurb a b
Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA Department of Mathematics, Jaypee Institute of Information Technology, Noida, UP, India
A R T IC LE I N F O
ABS TRA CT
Keywords: Nonlinear Schrödinger equation Akhmediev breathers Peregrine soliton G′ -Expansion method G2 Solitons
The current study is dedicated for operating the variational iteration method (VIM) to the selffocusing nonlinear Schrödinger (NLS) equation. Consequently, optical solitons and singular soliton solutions are formally derived. The VIM, representing the resultant solution in a series structure, does not require linearization of the equation and accelerates the convergence of resultant solution. The presented analysis shows the pertinent features of this method.
( )
Furthermore, the powerful
( )-expansion method has been adopted for the aforementioned G′
G2
equation for culminating hyperbolic, trigonometric and rational function solutions with rich mathematical structures, leading to diverse types of solitons.
1. Introduction The self-focusing nonlinear Schrödinger equation reads
iut +
1 u xx + u |u|2 = 0, 2
i=
−1 ,
(1)
such that development of slowly shifting waves for a physical system with tiny dissipation in stable dispersive are ruled by complex function u(x, t), with x as longitudinal variable parameter and t acting as co-moving instant of time. Various areas of physics for instance: transmission of light waves in nonlinear medium of optical fibers, plasma physics, super conductivity and quantum mechanics [1–6] extensively deals with Eq. (1). Eq. (1) has variety of implications in advancement of electro magnetic wave propagation, Bose–Einstein condensates, hydrodynamics, water waves, the harmonic oscillator, the hydrogen atom, and many others [7–13]. The intrinsic scientific implications boost up theoretical as well as experimental studies for characteristics of soliton solutions to NLS equation (1). These solitons enlighten new aspects of optical propagation in nonlinear waveguides in assorted models. More specifically, NLS equation (1) embraces only group momentum dispersion and self-phase inflection for picosecond light pulses. Peregrine [2] demonstrate that self-focusing nonlinear Schrödinger (NLS) equation own up an exact solution, depicting a rogue wave model with a significant amplitude that appears from nowhere and disappears with no trace. In the process of gradient catastrophe with semi-classical form of Eq. (1), Peregrine solitons (PSs) are created as a comprehensive by-product. Most importantly, Akhmediev breather have localization property in transverse dimension, in addition to, longitudinal dimension. Moreover, the Kuznetsov–Ma breather, monitored in fiber optics, is localized in x-direction and having periodic nature in the t-direction. The Akhmediev and Kuznetsov–Ma breathers [1] could be merged into a common term, so that we may recover one or the other breather
⁎
Corresponding author. E-mail addresses: [email protected] (A.-M. Wazwaz), [email protected] (L. Kaur).
https://doi.org/10.1016/j.ijleo.2018.11.004 Received 3 November 2018; Accepted 3 November 2018 0030-4026/ © 2018 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
if the modulus of a spectral constraint is above or below a certain specific value [3]. Akhmediev breather is presented as
u (x , t ) =
(1 − 4A1 )cosh(A2 t ) +
2A1 cos(ωx ) + iA2 sinh(A2 t )
2A1 cos(ωx ) − cosh(A2 t )
ei t,
i=
−1
(2)
where
A2 =
8A1 (1 − 2A1 ) ,
ω = 2 1 − 2A1 ,
(3)
and 0 < A1 < 0.5. However, When A1 > 0.5, the Akhmediev breather describes the Kuznetsov–Ma soliton. The so-called Peregrine solitons (PSs) are given as
4 + 8i t ⎤ i t u (x , t ) = ⎡1 − e . 1 + 4x 2 + 4t 2 ⎦ ⎣
(4)
It was reported [3] that hierarchy of solutions with rational structure having increasing order for Eq. (1) exist, originating with the simplest outline, which all may be described by
G + i H⎤ it u (x , t ) = ⎡1 − e , D ⎦ ⎣
D ≠ 0.
(5)
A higher order rational solution (5) was derived by selecting the following [3]
(
G = t2 + x2 + H=
t (t 2
D=
1 2 (t 3
−
5x 2)
3 4
)(x
+
+ x 2 )3 +
2
+ 5t 2 +
3(t 2
+
1 (x 2 4
−
3 4
),
15 , 8 3 3t 2)2 + 64 (12x 2
x 2)2
−
+ 44t 2 + 1).
(6)
Nonlinear transmission of solitons attracted more scrutiny in an array of physical surroundings. Solitons [14] are classified as concentrated waves, propagating with identical shape and velocity assets, and stable under interacting collisions and result as a balance between dispersion and nonlinear effects. The interactions of any number of solitons do not transform the soliton forms, and phase shift is the only outcome of this interaction. The exclusive analysis of optical soliton solutions [15–18] to nonlinear partial differential equations happens to be foremost resource for exploring widely different physical phenomena. Recently, many influential and dominating techniques have been effectively implemented to decipher nonlinear partial differential equations having constant and variable coefficients as well. Diverse efficient modes are earlier used to deal with NLS equation (1), like: Symbolic methods, similarity and symmetry analysis, Hirota's bilinear terms, conservation laws [19–25]. We plan to discover exact optical soliton, Akhmediev breather, and Peregrine soliton solutions for (1) by means of variational iteration method (VIM). Secondarily, we attempt to showcase potential of the VIM method to hand out with nonlinear partial differential equations of technical and scientific situations in a consolidated fashion without requiring any extra limitation. The VIM method has immense capacity of minimizing size of computations along with retaining high level of accuracy for numerical solution. Moreover, hyperbolic, trigonometric and rational function solutions for G′ considered equation with enhanced mathematical structure are found via 2 -expansion method.
( ) G
2. Summary of variational iteration method (VIM) The variational iteration method (VIM) is well known in the literature, earnest reader may refer to conventional research for its exclusive study, hence we give brief summary of this method. The differential equation is considered as
L1 u + L2 u = p (x ),
(7)
with L2 and L1 defined as nonlinear and linear operators respectively and source inhomogeneous term is exhibited by p(x). The considered method (VIM) enroll corrective action functional for Eq. (7) as given:
un + 1 (x ) = un (x ) +
(∫ 0
x
)
λ (t )(L1 un (t ) + L2 u˜n (t ) − p (t )) dt ,
(8)
here, Lagrange multiplier λ could be fixed parameter or variable parameter of t and λ is depicted optimally via variational theory and by using restricted variation u˜n , which means δu˜n = 0 . Primarily, λ is determined, which is further carried out with iteration formula not having controlled variation. Then evaluation of succeeding approximations un+1(x), n ≥ 0 for solution u(x) is uniformly achieved. Any selective function can be objected as zeroth approximation u0 for VIM and based on this mode, the solution is structured as
u (x ) = lim un (x ).
(9)
n →∞
The VIM was proved by many to be very effective, can be used in a direct manner without any need to linearization of the nonlinear 805
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
terms. As stated earlier, we will use the VIM to determine optical solitons, Akhmediev breathers and Peregrine solitons. 3. Optical solitons Nonlinear Schrödinger equation, under review, as
iut +
1 u xx + u|u|2 = 0, with u (x , 0) = ( 8 sech( 8 x )). 2
(10)
We first set the correction functional for (10) as
un + 1 (x , t ) = un (x , t ) +
∫0
t
λ (θ) ⎛i ⎝ ⎜
here, u¯ as complex conjugate of u with The stationary constrain leads to
∂un (x , θ) 1 ∂2u˜n)(x , θ) + + u2u¯ ⎞ dθ ∂θ 2 ∂x 2 ⎠ ⎟
|u|2
(11)
= uu ¯ .
1 + iλ = 0, λ′ = 0,
(12)
result into (13)
λ = i. Simulating Lagrange multiplier λ = i for the functional (11), ensures the resulting iteration phrase (formula):
un + 1 (x , t ) = un (x , t ) + i
∫0
t
2 ⎛i ∂un (x , θ) + 1 ∂ (u˜n (x , θ)) + u2u¯ ⎞ dθ , ∂ θ 2 ∂x 2 ⎠ ⎝
⎜
⎟
n ≥ 0.
(14)
As revealed earlier, u 0 (x , 0) = 8 sech( 8 x ) could be adopted from the given initial condition (10). Applying this u0(x, 0) into (14), subsequent approximations are provoked:
u 0 (x , t ) =
8 sech( 8 x ),
u1 (x , t ) =
8 sech( 8 x )(1 + 4i t),
u2 (x , t ) =
8 sech( 8 x ) 1 + 4i t +
(4i t)2 2!
8 sech(
(4i t)2 2!
+
(4i t)3 3!
(4i t)2 2!
+
(4i t)3 3!
u3 (x , t ) =
( 8 x ) (1 + 4i t +
), ),
⋮ un (x , t ) =
(
8 sech( 8 x ) 1 + 4i t +
+
(4i t) 4 4!
)
+⋯ .
(15)
The exact soliton solution are accordingly furnished as (Fig. (1))
u (x , t ) =
8 sech( 8 x ) e 4i t .
(16)
Following the similar fashion, nonlinear Schrödinger equation
iut +
1 u xx + u|u|2 = 0, with u (x , 0) = ( 8 cosech( 8 x )), 2
(17)
is accompanied with singular soliton structure
Fig. 1. Soliton profile for (a) real part and (b) Imaginary part of solution (16). 806
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
Fig. 2. Three dimensional graphics of soliton for (a) real part and (b) Imaginary part of solution (20) with a = 1, b = 4.
u (x , t ) = ± i 8 cosech( 8 x ) e 4i t .
(18)
4. More optical soliton solutions Following the above mentioned analysis, we set a new u0(x, t) as
u (x , 0) = k e i ax sech(kx),
(19)
and we acquire the soliton solutions, utilizing the VIM method:
u (x , t ) = k sech(k (x − at)) ei (ax + bt),
(20)
valid upon using (Fig. (2))
k=
a2 + 2b .
(21)
5. Complex solutions The nonlinear cubic Schrödinger equation enjoy the following form
iut +
1 u xx + u|u|2 = 0, with initial constraint u (x , 0) = e i x . 2
(22)
We proceed as before for Eq. (22) and use the correction functional (11) with the Lagrange multiplier λ = i for attaining the subsequent iteration ’phrase (formula)
un + 1 (x , t ) = un (x , t ) + i
∫0
t
2 ⎛i ∂un (x , θ) + 1 ∂ (u˜n (x , θ)) + u2u¯ ⎞ dθ , ∂θ 2 ∂x 2 ⎝ ⎠
⎜
⎟
n ≥ 0.
(23)
ix
As revealed earlier, u0(x, 0) = e is selected for presenting the successive approximations:
u1 (x , t ) = e i x u2 (x , t ) = e i x u3 (x , t ) = e i x
( + 1), ( t − t + 1), ( t− t − t it 2 i 2
1 2 8
i 2
1 2 8
i 3 48
)
+1 ,
⋮
(
i
1
un (x , t ) = e i x 1 + 2 t − 8 t 2 −
i 3 t 48
+
1 4 t 384
) + ⋯.
(24)
The exact complex solution are accordingly furnished as 1
u (x , t ) = ei (x + 2 t ) .
(25)
It is remarkable that we may show that nonlinear Schrödinger equation gives the generalized complex solution
u (x , t ) = ei (Λ1x + Λ2t ),
(26)
where 807
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
Λ2 = 1 −
1 2 Λ1 . 2
(27)
6. Soliton solutions by
( )-expansion method G′ G2
This segment of article is devoted to construct some new optical solutions to Eq. (1) via
( )-expansion approach. The rapid G′ G2
( ) G′
review of 2 -expansion method purpose the solution composition for nonlinear Schrödinger equation (1) as follows: G First and foremost, subsequent wave transformation is executed for Eq. (1):
u (x , t ) = W (ζ )exp(iσ ),
(28)
along with σ = α1x + α2t, ζ = iα3(x + α4t). Simulating (28) for Eq. (1), ensures the resulting reduced ordinary differential equation as:
2(W (ζ ))3 + (−α12 − 2α2 ) W (ζ ) + (−2α1 α3 − 2α3 α4 ) W ′ (ζ ) − W ″ (ζ ) α32 = 0.
(29)
We purpose the following solution structure for Eq. (29): m
W (ζ ) =
∑ l =−m
G′ l kl ⎛ 2 ⎞ , ⎝G ⎠
(30)
in which kl, − m ≤ l ≤ m are evaluated and G = G(ζ) follows subsidiary ordinary differential equation: 2 ′ ⎛ G′ ⎞ − M − N ⎛ G′ ⎞ = 0. 2 2 ⎝G ⎠ ⎝G ⎠
(31)
The homogeneous balance among most nonlinear term and most highest order derivative term in Eq. (29) results into m = 1 in Eq. (30) and then formal solution is written as follows:
G′ G′ −1 W (ζ ) = k 0 + k1 ⎛ 2 ⎞ + k2 ⎛ 2 ⎞ , ⎝G ⎠ ⎝G ⎠
(32)
with parameters k1, k2, k3, which are yet to be identified. Eq. (32) is employed to Eq. (30) with help of Eq. (32) and after that followed
( ) G′
j
by assembling of different coefficients with like powers of G2 . Finally, a set of nonlinear equations are retrieved for engaged parameters by comparing each coefficient to zero, as given below:
− 2 MNα32k2 − α12k2 + 6 k 0 2k2 + 6 k1 k22 − 2 α2 k2 = 0, − 2 MNα32k1 − α12k1 + 6 k 0 2k1 + 6 k12k2 − 2 α2 k1 = 0, 2 Mα1 α3 k2 + 2 Mα3 α4 k2 + 6 k 0 k22 = 0, − 2 Nα1 α3 k1 − 2 Nα3 α4 k1 + 6 k 0 k12 = 0, − 2 M 2α32k2 + 2 k23 = 0, − 2 N 2α32k1 + 2 k13 = 0, − 2 k1 Mα1 (α3 + α4 ) + 2 k2 Nα1 α3 + 2 k2 Nα3 α4 − k 0 α12 + 2 k 03 + 12 k 0 k1 k2 − 2 k 0 α2 = 0.
(33)
The algebraic set of Eq. (33) are exercised to reveal following nontrivial cases for involved parameters, such that k0, k1, k2 are nonzero:
N=−
α12 + 2α1 α 4 + α 42 , 36α3 k2 1
k 0 = − 3 (α1 + α4 ),
α 2
α2 = − 181 + k1 =
8α1 α 4 9
α12 + 2α1 α 4 + α 42 , 36k2
+
4α 42 , 9
M=
k2 , α3
(34)
with α1, α3, α4, k2 as free constants. Considering the acquired values (34) and general solutions of (31) in (32), we have developed soliton solutions to Eq. (29): (35) Class 1: Solution with trigonometric function for MN > 0 i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ ⎞ (α1 + α 4 ) i ⎛K 0 cos ⎛ 1 ζ + K1 sin ⎛ 1 ζ ⎝ 6α3 ⎠ ⎝ 6α3 ⎠⎠ ⎝ ⎜
W (ζ ) =
1 (α 3 1
+ α4 ) +
⎟
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ ⎞ 6 ⎛K1 cos ⎛ 1 ζ − K 0 sin ⎛ 1 ζ ⎝ 6α3 ⎠ ⎝ 6α3 ⎠⎠ ⎝ ⎜
(α1 + α 4 ) i ⎛K1 cos ⎛ ⎝ ⎝ ⎜
−
⎟
i (α1 + α 4 ) ⎞ i (α + α 4 ) ⎞ ⎞ ζ − K 0 sin ⎛ 1 ζ ⎟ 6α3 ⎠ ⎝ 6α3 ⎠⎠
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ ⎞ 6 ⎛K 0 cos ⎛ 1 ζ + K1 sin ⎛ 1 ζ ⎝ 6α3 ⎠ ⎝ 6α3 ⎠⎠ ⎝ ⎜
,
⎟
(35)
(36) Class 2: Solution having hyperbolic function for MN < 0 808
Optik - International Journal for Light and Electron Optics 179 (2019) 804–809
A.-M. Wazwaz, L. Kaur
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ (α1 + α 4 ) i ⎛K 0 cosh ⎛ 1 ζ + K 0 sinh ⎛ 1 ζ + K1 ⎞ ⎝ 3α3 ⎠ ⎝ 3α3 ⎠ ⎝ ⎠ ⎜
W (ζ ) =
1 (α 3 1
+ α4 ) −
⎟
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ 6 ⎛⎜K 0 cosh ⎛ 1 ζ + K 0 sinh ⎛ 1 ζ − K1 ⎟⎞ ⎝ 3α3 ⎠ ⎝ 3α3 ⎠ ⎝ ⎠
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ (α1 + α 4 ) i ⎛K 0 cosh ⎛ 1 ζ + K 0 sinh ⎛ 1 ζ − K1 ⎞ ⎝ 3α3 ⎠ ⎝ 3α3 ⎠ ⎝ ⎠ ⎜
+
⎟
i (α + α 4 ) ⎞ i (α + α 4 ) ⎞ 6 ⎛K 0 cosh ⎛ 1 ζ + K 0 sinh ⎛ 1 ζ + K1 ⎞ ⎝ 3α3 ⎠ ⎝ 3α3 ⎠ ⎝ ⎠ ⎜
,
⎟
(36)
(37) Class 3: Solution having rational function for M = 0, N ≠ 0
W (ζ ) =
(α 2 + 2 α1 α4 + α4 2)(K 0 ζ + K1 ) 1 K 0 α3 (α1 + α4 ) − − 1 , 36K 0 α3 3 K 0 ζ + K1
(37)
with K0 and K1 as free parameters. Slipping back to original variables x, t and by means of (35)–(37), we acquired solutions for Eq. (1) as
u (x , t ) = W (ζ )exp(iσ ), along with σ = α1x + α2t, ζ = iα3(x + α4t). 7. Conclusion The current work involves the systematic examination of self-focusing nonlinear Schrödinger equation, modeling the propagation of an ultrashort (femtosecond) pulses in nonlinear optical fibers. The most significant variational iteration method (VIM) has been utilized for fabricating optical soliton solution and singular soliton solutions. Additionally, the current study discloses the potential of VIM method, as an organized tool for unfolding the solutions of nonlinear equations by employing suitable Lagrange multipliers. G′ Furthermore, 2 -expansion method has been taken into account to produce hyperbolic, trigonometric and rational solutions of G aforementioned equation, which are accompanied by free involved parameters. Illustration of the behavior of solutions is portrayed by appointing several values to the arbitrary constants, which might be fruitful for clarification of numerous problems related to nonlinear optics.
( )
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