SOLITON SOLUTIONS OF THE CUBIC-QUINTIC NONLINEAR
SCHRÖDINGER EQUATION WITH VARIABLE COEFFICIENTS
HOURIA TRIKI1 , ABDUL-MAJID WAZWAZ2,∗
1
Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar
University, P. O. Box 12, 23000 Annaba, Algeria
E-mail: [email protected]
2
Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
∗
Corresponding author E-mail: [email protected]
Received January 4, 2016
In this work, a nonlinear Schrödinger equation with variable coefficients incorporating cubic-quintic nonlinearities, self-steepening, and self-frequency shift is investigated by application of the trial equation method. The model describes the propagation of femtosecond light pulses in optical fibers. Exact soliton-like solutions including
bright, dark, and singular solutions are derived. Parametric conditions for the existence
of envelope soliton solutions are given.
Key words: Trial equation method; cubic-quintic Schrödinger equation;
Travelling wave solutions.
1. INTRODUCTION
Nonlinear propagation of solitons has drawn considerable attention in a range
of physical settings, including fluid dynamics and plasma physics [1], atomic physics and Bose-Einstein condensates [2]-[7], nonlinear optics and photonics [8]-[16],
and so on. Solitons are defined as localized waves that propagate without change of
their shape and velocity properties and are stable against mutual collisions [17]. The
distinction between solitary wave and soliton solutions is that when any number of
solitons interact they do not change form, and the only outcome of the interaction is a
phase shift [18]. The existence of solitary wave solutions implies perfect balance between nonlinearity and dispersion, which usually requires rather specific conditions
and cannot be established in general [19].
As well known, the nonlinear Schrödinger (NLS) equation is a generic model
for describing the dynamics of light pulses in optical fibers [8]. For picosecond light
pulses, the NLS equation includes only the group velocity dispersion (GVD) and the
self-phase modulation, well known in fibers, and it admits bright and dark solitontype pulse propagation in anomalous and normal dispersion regimes, respectively
[20]. However, as one increases the intensity of the incident light power to produce
shorter (femtosecond) pulses, additional nonlinear effects become important and the
dynamics of pulses needs to be described in the frame of a generalized NLS equation
that includes higher-order nonlinear terms [21].
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Journ. Phys.,
3-4),
Vol.
360–366
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Soliton solutions of the cubic-quintic nonlinear Schrödinger equation
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The investigation of exact soliton-like solutions to nonlinear wave equations
is of great value in understanding widely different physical phenomena. Recently,
many powerful methods such as the sine-cosine methods [22]-[24], the subsidiary
ordinary differential equation method [25]-[27], the Hirota’s method [28], the PetrovGalerkin method [29], the collocation method [30], the solitary wave ansatz method
[31, 32], the Exp-function method [33], the trial equation method [34, 35], and many
others, have been successfully applied to exactly solve nonlinear wave equations with
constant and variable coefficients.
Recently, Liu [36] proposed a trial equation method which can be suitable to
both real equations and complex equations with variable coefficients. Liu [36] used
this method to obtain some exact envelope traveling wave solutions for the timevarying NLS equation with GVD and Kerr nonlinearity. Subsequently, the same
method has been applied to construct envelope traveling wave solutions of the generalized NLS equation with time-dependent coefficients involving an external potential term in addition to GVD and nonlinearity terms [37]. The trial equation method
is a useful systematic method capable of solving nonlinear equations by using linear
methods.
In this paper, the trial equation method will be extended to cubic-quintic NLS
equation with self-steepening and self-frequency shift [38]:
iqt + f (t)qxx + g(t) |q|2 + σ |q|4 q = ih(t) |q|2 q + ip(t) |q|2 q,
x
x
(1)
where f (t), g(t), h(t), and p(t) are time-dependent functions.
In Eq. (1), q(x, t) is the complex envelope of the electric field, f (t) is the timedependent dispersion coefficient, h(t) is the self-steepening coefficient, and p(t) is
the self-frequency shift coefficient. The third and fourth terms represent cubic and
quintic nonlinearities, respectively, σ is a constant, and the subscripts x and t denote
the spatial and temporal partial derivatives, respectively.
Recently, Green and Biswas [38] studied Eq. (1) by the ansatz method and
obtained the exact one-soliton solution under certain parametric conditions. To the
best of our knowledge, no attempt was made regarding soliton-like solutions of the
NLS equation with parabolic law nonlinearity and variable coefficients (1) by using
the trial equation method. Here, we investigate the applicability and effectiveness
of the trial equation method that was recently proposed by Liu [36] on the cubicquintic NLS equation with time-dependent coefficients. We will show that a subtle interplay between the group velocity dispersion, self-steepening, self-frequency
shift, and cubic-quintic nonlinearities, can result in a rich variety of shape-preserving
waves with interesting properties.
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2. EXACT SOLUTIONS TO EQUATION (1)
To seek the envelope traveling wave solutions of (1), we assume the solution in
the form [36]
q(x, t) = u (ξ) eiζ , ξ = k(t)x + ω (t) , ζ = s(t)x + r(t).
(2)
Here u is a function of ξ to be determined, and k(t), ω (t) , s(t), and r(t) are timedependent parameters that will be also determined.
Substituting Eq. (2) into Eq. (1) and separating the real and imaginary parts
leads to
0
k (t) x + ω 0 (t) + 2f (t)s(t)k(t) u0 − [3h(t) + 2p(t)] k (t) u2 u0 = 0,
(3)
f (t)k 2 (t)u00 − s0 (t) x + r0 (t) + f (t)s2 (t) u + [g(t) + h(t)s(t)] u3
+σg(t)u5 = 0.
(4)
Now we suppose that the exact solution of Eq. (1) satisfies the following trial
equation [36]
u0
2
= F (ξ) =
m
X
ai ui ,
(5)
i=0
where ai (i = 0, ..., m) are constants and m is an integer to be determined later.
Considering homogeneous balance between u5 and u00 in Eq. (4), we can determine the value of m in Eq. (5) as m = 6. Inserting the resulting trial Eq. (5) into
(4), and setting each coefficients of u0 , u2 u0 , and ui (ξ) (with i = 0, ..., 6) in Eqs. (3)
and (4) to zero yields
k 0 (t) x + ω 0 (t) + 2f (t)s(t)k(t) = 0,
(6)
[3h(t) + 2p(t)] k (t) = 0,
s0 (t) x + r0 (t) + f (t)s2 (t)
a2 −
= 0,
f (t)k 2 (t)
g(t) + h(t)s(t)
2a4 +
= 0,
f (t)k 2 (t)
3a6
g(t)
+
= 0,
σ
f (t)k 2 (t)
a1 = 0, a3 = 0, a5 = 0,
(7)
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(8)
(9)
(10)
(11)
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Solving Eqs. (6)-(11) yields
Z
Z
k(t) = k, s(t) = s, ω(t) = −2sk
f (t)dt, r(t) = c1
3
g(t) = c2 k 2 f (t), h(t) = c3 k 2 f (t), p(t) = − c3 k 2 f (t),
2
c2 + sc3
c1 + s2
σc2
a4 = −
, a2 =
, a6 = −
,
2
k2
3
a1 = a3 = a5 = 0, a0 = c4 ,
f (t)dt,
(12)
(13)
(14)
(15)
where k, s, c1 , c2 , c3 , and c4 are arbitrary constants.
Under the conditions (15), Eq. (5) with m = 6 is reduced to the following
expression
2
u0 = a0 + a2 u2 + a4 u4 + a6 u6 ,
(16)
The integral form of Eq. (16) is
Z
1
du.
± (ξ − ξ0 ) = √
2
a0 + a2 u + a4 u4 + a6 u6
(17)
In what follows, we discuss the case where a0 = 0. Then equation (17) is
reduced to the following form
Z
1
√
± (ξ − ξ0 ) =
du.
(18)
u a2 + a4 u2 + a6 u4
Denote ∆ = a24 − 4a2 a6 , according to Yomba [39], we obtain four families of soliton
solutions:
Family 1: ∆ = a24 − 4a2 a6 > 0 and a2 > 0. Then the exact solution of Eq. (1)
is a bright-soliton-type solution of the form
"
#1/2
2a2
u1 = √
exp (iζ) .
(19)
√ ∆ cosh 2 a2 ξ − a4
Family 2: ∆ = a24 − 4a2 a6 < 0 and a2 > 0. Then one obtains an exact singularsoliton-type solution for Eq. (1) as
"
#1/2
2a2
u2 = √
exp (iζ) .
(20)
√ −∆ sinh 2 a2 ξ − a4
Family 3: ∆ = a24 − 4a2 a6 = 0 and a2 > 0. Then Eq. (1) has a dark-solitontype solution given by
√
1/2
a2
a2
1 + tanh
ξ
exp (iζ) .
(21)
u3 = −
a4
2
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Houria Triki, Abdul-Majid Wazwaz
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Family 4: a2 > 0. We obtain another bright-type soliton-like solution for Eq.
(1) as
(
u4 =
)1/2
√ −a2 a4 sech2 a2 ξ
exp (iζ} ,
√ 2
a24 − a2 a6 1 + tanh a2 ξ
(22)
where = ±1.
Note that solutions (19)-(22) are all possible solutions to Eq. (1) in the case
a6 6= 0. It is significant to observe that the obtained solutions exist under the parametric condition 3h(t) + 2p(t) = 0. The later means that self-steepening and selffrequency shift coefficients are not independent and the existing soliton solutions are
obtained in the framework of this relationship.
3. CONCLUSIONS
In this paper, we have investigated a higher-order nonlinear Schrödinger equation with time-dependent coefficients, modeling the propagation of ultrashort (femtosecond) optical pulses in nonlinear optical fibers. The model used combines cubic
and quintic nonlinearities, as well as the self-steepening and self-frequency shift effects. The trial equation method is used to construct families of bright, dark, and
singular soliton solutions for the nonlinear dynamical model. Conditions for the existence of propagating envelope solutions have also been reported. The work reveals
the power of the trial equation method in handling nonlinear evolution equations
with variable coefficients. The trial equation method is a useful systematic method
capable of solving nonlinear evolution equations by using linear methods.
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