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Научни трудове на Съюза на учените в България - Пловдив Серия В. Техника и технологии, том XIII., Съюз на учените, сесия 5 - 6 ноември 2015 Scientific Works of the Union of Scientists in Bulgaria-Plovdiv, series C. Technics and Technologies, Vol. XIII., Union of Scientists, ISSN 1311-9419, Session 5 - 6 November 2015.
SOLITON SOLUTION OF THE NONLINEAR SCHRODINGER EQUATION INCLUDING THIRD ORDER OF LINEAR DISPERSION AND DISPERSION OF NONLINEARITY
PetkS. Staykov,Ivan Bojikoliev, Diana Dakova, Anelia Dakova Faculty of PhysiRS^niversity of Plovdiv "PSisii HilendarSki", 24 Tsar Asen Str., 4000 Plovdiv, BULGARIA
Abstract
In recent years actively are studied the effects resulting from the propagation of optical pulses in nonlinear dispersive media. One of the most commonly used equation, for describing the evolution of laser pulses, is the nonlinear Schrodinger equation (NSE). It works very well for nano and picosecond laser pulses, but in the femto and attosecond region it is necessary to be included terms responsible for higher order of dispersion and dispersion of nonlinearity which are significant in the frames of ultra-short optics. In present paper it is presented a theoretical model of the propagation of light pulses. We found an exact analytical soliton solution of modified NSE, including third order of linear dispersion and dispersion of nonlinearity. The soliton is possible to be observed as a result of the dynamic balance between the effects of higher order of dispersion and nonlinearity. In our work losses and Raman scattering of the medium are neglected.
1. INTRODUCTION
In last two decades the evolution of femto and attosecond optical pulses with broad-band spectrum in nonlinear dispersive medium is of a considerable interest for the scientific community. Its study is a result of the growing needs of ultrafast high intensity optics. One of the most commonly used equations to describe the propagation of laser pulses in one-dimensional structures is the nonlinear Schrodinger equation [1-3]. In the frame of ultrashort optics (T0<1 ps) NSE is usually modified my adding a terms that govern the third order of linear dispersion (TOD) and the dispersion of nonlinearity [1,4]. It is well known that for such pulses it is necessary to be included (TOD) p3 even when (GVD) As a result of that the shape of the pulse becomes asymmetric with oscillatory structure on one of its edges, depending on the sign of p3 (fig. 1). Self-steepening (s=1/a0T0) is a higher-order nonlinear effect that results from the intensity dependence of group velocity and leads to an asymmetry in the shape and spectrum of ultrashort pulses. Their peaks shift toward the trailing edges, moving at lower speed than wings (fig. 2) [1]. It is interesting to know, is it possible to be formed a stable soliton pulse under influence of these effects?
Figure 1. Gaussian pulse in the presence Figure 2. Dispersionless case of self-of TOD, L'D=T03/\p3\. [1] steepening of Gaussian pulse, z=10LNL
and 20Lnl, s=1/a0T0h [1]
In [4] equation (1) is studied numerically. It is shown that under certain conditions it is possible to be observed a soliton as a result of the balance between the higher-order nonlinear and dispersive effects (fig. 3).
Figure 3. Intensity profile of sech input pulse (green line) z=0 and z=10LD, s=1/4n, T0=10fs, S3=0.02. Predictions of equation (1) are shown by the solid yellow curve. [4]
In present paper we propose a theoretical model based on the evolution of light pulses with a broad-band spectrum in optical fibers under the effects of third order of linear dispersion and dispersion of nonlinearity, using modified NSE. In our work losses and Raman scattering of the medium are neglected. It is found a new exact analytical soliton solution of NSE by using mathematical method described in [5].
2. BASIC EQUATION
We use the modified NSE written in local time coordinate system of the kind [1,4]:
dA 1 d2 A V3 A n T
l — + - ls3^T + &n2ld
dg 2 or or
where:
g _ A 3 '
T _ -
T_
T
|A|2 A + . A2 A)
T _ t - z / v ,
_ 0
z0 _ vT0
(1)
(2)
00
In the e quation above A is the amplitude function of the envelope of the pulse; LD is the di sp ersion length; t is time; a0, p0, v, n2 are respectively the carrier frequency, wave number, group velocity, nonlinear refractive index of the medium.
z
3. SOLITON SOLUTION OF THE NONLINEAR SCHRODINGER EQUATION
We make the following substitution:
A(g,r) _O0O(y)exp(iar + ibg), y _g-r, a, b - const (3)
0O is the initial amplitude function. We substitute the expression (3) in equation (1) and we obtain one complex nonlinear ordinary differential equation of the third order and third degree of the unknown function Q(y):
1 a 2
iO' - bO + - O'' - iaO'--O - ig,O" - 3agO" + i3ga 2O' + (4)
2 2 3 3 3 (4)
S3a3O + ]O>3 - 3isyO2O' - syaO3 _ 0 Since Q(y) is a real function and we need to define the constants a and b, we equalize the real and imaginary parts on both sides of our ordinary differential equation. Thus, we obtain the following two differential equations:
1 a 2
Re: (- - 3a£3)®''-O[b + — -S3a3] + Y3[1 - sa] _ 0 (5)
Im: -g3O'''+O'[1 - a + 3ga2] - 3ys®2O'_ 0 (6)
The coefficients in front of the various derivatives and degrees of the unknown function 0 are dimensionless in the both equations above. To lower the order of the equation (6) we integrate it with respect to the variable y. So, it takes the form of:
- g3O' '+O[1 - a + 3g3a2 ] - YsO3 _ 0 (7)
The equations (5) and (7), obtained by equalizing the real and imaginary parts on both sides of the output amplitude equation, are of the same type and they are referred to the same unknown function. For this reason, they should match. This means that the coefficients in front of the corresponding derivatives and degrees of 0 must be the same [6]. Since we need to define the two constants a and b, let first divide the two equations respectively by the coefficients in front of 0' and then to equalize the coefficients of 0 and 03. Thus, we obtain the following expressions for the two constants:
s - 2g , 6s2g3 - s3 - 2sg3 + 4g2
a _-3, b _-3-3-3-— (8)
4s g 4s g3
Once we have determined the two constants a and b in the way that the two equations (5) and (7) match, we obtain for the unknown real function 0=0(y) the following nonlinear ordinary differential equation of second order:
, „ , ys , 3 ^ 1 - a + 3—3a ,Q,
O "-n®+ — O3 = 0, n=-3—' (9)
—3 —3
where n has a meaning of an amplitude. The equation (9) has well known soliton solution [1-3]:
O = n sec h( y^/n) (10)
We substitute the constants a and b in the expression for n
1 1 1,1 2.2 (11) o3 s 16 o3
The solution (10) has a physical meaning when n>0 and it is real. This condition can be satisfied by the appropriate selection of the parameters of optical pulses and medium. The obtained result is quite different from the classical solution of NSE and can be used for more accurate description of the propagation of narrow-band as well as broad-band optical pulses.
4. CONCLUSION
In present paper is reviewed the evolution of optical pulses with broad-band spectrum in nonlinear dispersive medium. It is found an exact analytical soliton solution of modified NSE (1) in which the effects up to third order of the linear dispersion and dispersion of nonlinearity are included. The expression (10) differs from the classical soliton solution of NSE - the constant n depends on the coefficients, characterizing the second and third order of the linear dispersion and the nonlinearity of the medium. The numerical simulations of equation (1) show that the soliton solution is stable and the pulse keeps it shape as a result of the dynamic balance between the higher-order nonlinear and dispersive effects.
Acknowledgments: The present work is supported by projects kI15FFIT005 and SP15FFIT004, Faculty of Physics, Plovdiv University "Paisii Hilendarski".
REFERENCES
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[4] Y. Xiao, D. N. Maywar, G. P. Agrawal, New approach to pulse propagation in nonlinear dispersive optical media, JOSA B, Vol. 29, Issue 10, pp. 2958-2963, (2012).
[5] A. M. Dakova, D. Y. Dakova, Evolution of optical pulses with broadband spectrum in a nonlinear dispersive media, Bulg. J. Phys. 40, 182-185, PACS codes: 42.81.-I, (2013).
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