Optical solitons for the resonant nonlinear Schrödinger's equation with time-dependent coefficients by the first integral method

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ARTICLE IN PRESSJLEO-54222; No. of Pages 10

Optik xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Optik

jo ur nal homepage: www.elsev ier .de / i j leo

ptical solitons for the resonant nonlinear Schrödinger’s equationith time-dependent coefficients by the first integral method

. Eslamia, M. Mirzazadehb, B. Fathi Vajargahc, Anjan Biswasd,e,∗

Department of Mathematics, Faculty of Sciences, University of Mazandaran, Babolsar, IranDepartment of Mathematics, Faculty of Sciences, University of Guilan, Rasht, IranDepartment of Statistics, Faculty of Sciences, University of Guilan, Rasht, IranDepartment of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USAFaculty of Science, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

r t i c l e i n f o

rticle history:eceived 21 June 2013ccepted 15 December 2013vailable online xxx

eywords:irst integral methodivision theoremolitonsesonant nonlinear Schrödinger’s equationith time-dependent coefficients

a b s t r a c t

In this paper, the resonant nonlinear Schrödinger’s equation is studied with three forms of nonlinearity.This equation is also considered with time-dependent coefficients. The first integral method is used tocarry out the integration. Exact soliton solutions of this equation are found. These solutions are con-structed through the established first integrals. The power of this manageable method is confirmed.

© 2014 Elsevier GmbH. All rights reserved.

. Introduction

Optical solitons is the most fascinating area of research in the field of nonlinear optics. This area of study has been put to good useor the past few decades. Today there are several and ever increasing number of papers that are being published in this area of research1–40]. The governing equation that is handled in this context is the nonlinear Schrödinger’s equation (NLSE) which is a nonlinear partialifferential equation (PDE) of S-type. It is, however, only for the case of Kerr law nonlinearity, NLSE falls in the category of S-type PDEs. Thisaper will address the resonant NLSE (R-NLSE) with Kerr, parabolic as well as dual-power laws of nonlinearity. The integration architecturehat will be impleted is the first integral method that is widely applied to solve several nonlinear evolution equations (NLEEs).

The construction of exact and analytical traveling wave solutions of nonlinear PDEs is one of the most important and essential tasks inonlinear science, since these solutions will very well describe various natural phenomena, such as vibrations, solitons, and propagationith a finite speed. The rapid developments of nonlinear sciences, a wide range of straightforward and effective methods have been

ntroduced to obtain traveling wave solutions of nonlinear PDEs such as, solitary wave ansatze method [21–25], tanh method [26,27],ultiple exp-function method [28], simplest equation method [29–32], Hirota’s direct method [33,34], transformed rational functionethod [35] and others.Feng [1] introduced a reliable and effective method called the first integral method to look for travelling wave solutions of nonlinear PDEs.

he basic idea of this method is to construct a first integral with polynomial coefficients of an explicit form to an equivalent autonomouslanar system by using the division theorem. Recently, this useful method is widely used in many papers such as in [2–10] and the referenceherein.

It is well known that the NLSE is widely used in basic models of nonlinear waves in many areas of physics, apart from optics. It arises from

Please cite this article in press as: M. Eslami, et al., Optical solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients by the first integral method, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.013

he study of nonlinear wave propagation in dispersive and inhomogeneous media, such as plasma phenomena and nonuniform dielectricedia. It is a generic equation describing the evolution of the slowly varying amplitude of a nonlinear wave train in weakly nonlinear,

trongly dispersive, and hyperbolic systems [36].

∗ Corresponding author at: Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA.E-mail addresses: [email protected] (M. Eslami), [email protected] (M. Mirzazadeh), [email protected] (A. Biswas).

030-4026/$ – see front matter © 2014 Elsevier GmbH. All rights reserved.ttp://dx.doi.org/10.1016/j.ijleo.2014.01.013

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dx.doi.org/10.1016/j.ijleo.2014.01.013

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M. Eslami et al. / Optik xxx (2014) xxx–xxx

The aim of this paper is to extract the exact special solutions of the R-NLSE with time-dependent coefficients [12] by using the firstntegral method. The paper is arranged as follows. In Section 2, we describe briefly the first integral method. In Section 3, we apply this

ethod to the R-NLSE with time-dependent coefficients.

. First integral method

Tascan et al. [4] summarized the main steps for using the first integral method, as follows:Step 1: Suppose a nonlinear PDE

P(u, ut, ux, utt, uxt, uxx, . . .) = 0, (1)

an be converted to an ordinary differential equation (ODE)

Q (U, −ωU ′, kU ′, ω2U ′′, −kωU ′′, k2U ′′, . . .) = 0, (2)

sing a traveling wave variable u(x, t) = U(�), � = kx − ωt, where the prime denotes the derivation with respect to �. If all terms containerivatives, then Eq. (2) is integrated where integration constants are considered zeros.

Step 2: Suppose that the solution of ODE (2) can be written as follows:

u(x, t) = U(�) = f (�). (3)

Step 3: We introduce a new independent variable

X(�) = f (�), Y(�) = f ′(�), (4)

hich leads a system of

X ′(�) = Y(�), (5)

Y ′(�) = F(X(�), Y(�)).

Step 4: According to the qualitative theory of ordinary differential equations [11], if we can find the integrals to (5) under the sameonditions, then the general solutions to (5) can be solved directly. However, in general, it is really difficult for us to realize this even forne first integral, because for a given plane autonomous system, there is neither a systematic theory that can tell us how to find its firstntegrals nor is there a logical way for telling us what these first integrals are. We shall apply the Division Theorem to obtain one firstntegral to (5) which reduces (2) to a first-order integrable ordinary differential equation. An exact solution to (1) is then obtained byolving this equation. Now, let us recall the Division Theorem:

Division Theorem: Suppose that P(w, �) and Q (w, �) are polynomials in C[w, �]; and P(w, �) is irreducible in C[w, �]. If Q (w, �) vanishest all zero points of P(w, �), then there exists a polynomial G(w, �) in C[w, �] such that

Q (w, �) = P(w, �)G(w, �).

. R-NLSE with time-dependent coefficients

The dimensionless form of the R-NLSE that is going to be analyzed in this paper is given by [12–15]

i t + ˛(t) xx + ˇ(t)F(| |2

) + �(t)

(| |xx| |

) = i�(t) . (6)

n Eq. (6), the dependent variable represents the complex wave profile, while x and t are the independent variables that respectivelyepresent the spatial and temporal variables. Also, ˛(t), ˇ(t), �(t) and �(t) are all functions of t. Then, on the left hand side of (6), the firsterm represents the evolution term, while the coefficient of ˛(t) represents the group velocity dispersion, the coefficient of ˇ(t) is theonlinear term and the coefficient of �(t) is the resonant term. On the right-hand side, the coefficient of �(t) is the linear attenuation.hus, there are two nonlinear terms in (6) which are the coefficients of ˇ(t) and �(t). The coefficient of �(t) is sometimes referred to ashe quantum potential or Bohm potential [16–18]. This term commonly appears in the study of chiral solitons in quantum Hall effect. The-NLSE appears in quantum mechanics and in the study of Madelung fluid [18].

Finally, the function F is areal-valued algebraic function with the condition F(|q|2)q : C −→ C . Considering the complex plane C as awo-dimensional linear space R2, the function F(|q|2)q is k times continuously differentiable, so that

F(|q|2)q ∈⋃∞

m,n=1Ck((−n, n) × (−m, m); R2). (7)

he R-NLSE will be considered for the following three forms of nonlinearity as discussed in the following three subsections.

.1. Kerr law

The Kerr law nonlinearity is the case when F(s) = s. This kind of nonlinearity typically arises in the context of water waves or nonlinearber optics when the refractive index of the light is proportional to the intensity. For Kerr-law nonlinearity, the considered generalized


NLS equation with time-varying coefficients is given by

i t + ˛(t) xx + ˇ(t)| |2 + �(t)

(| |xx| |

) = i�(t) . (8)

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hen �(t) = 0, we have

i t + ˛(t) xx + ˇ(t)| |2 + �(t)

(| |xx| |

) = 0. (9)

ow, we assume that Eq. (9) admits a solution of the form

(x, t) = U(�)ei(−�x+ω(t)t), � = x − v(t)t, (10)

here v(t) is the soliton velocity, � is the wave number of the soliton, while ω(t) is the frequency of the soliton velocity, all of them are toe determined. Now, from Eq. (10), we have

t =(i(tdω(t)dt

+ ω(t))U −

(tdv(t)dt

+ v(t))U ′

)ei(−�x+ω(t)t), (11)

xx = (U ′′ − 2i�U ′ − �2U)ei(−�x+ω(t)t), (12)(| |xx| |

) = U ′′ei(−�x+ω(t)t), (13)

here the primes denote derivatives with respect to � . Substituting Eqs. (10)–(13) into (9), and equating the real and imaginary parts,ields the following pair of relations:

tdv(t)dt

+ v(t) + 2�˛(t) = 0, (14)

(˛(t) + �(t))U ′′ −(tdω(t)dt

+ ω(t) + �2˛(t))U + ˇ(t)U3 = 0. (15)

ntegrating Eq. (14) with respect to t yields

v(t) = −2�t

∫ t

0

˛(t′)dt′. (16)

e remark from (16) that the pulse velocity is only affected by the varying dispersion coefficient ˛(t) . Using (3) and (4), Eq. (15) can beewritten as the two-dimensional autonomous system

dX

d�= Y,

dY

d�= t(dω(t)/dt) + ω(t) + �2˛(t)

˛(t) + �(t)X − ˇ(t)

˛(t) + �(t)X3.

(17)

ow, we apply the above division theorem to look for the first integral of system (17). Suppose that X(�) and Y(�) are nontrivial solutionso (17), and Q (X, Y) =

∑mi=0ai(X)Yi is an irreducible polynomial in the complex domain C such that

Q (X(�), Y(�)) =m∑i=0

ai(X(�))Yi(�) = 0, (18)

here ai(X)(i = 0, 1, . . ., m) are polynomials of X and am(X) /= 0 . Eq. (18) is a first integral of system (17). We note that (dQ/d�) is a polynomialf X and Y, and Q(X(�), Y(�)) = 0 implies that dQ/d�|(17) = 0 . According to the division theorem, there exists a polynomial T(X, Y) = g(X) + h(X)Yn the complex domain C such that

dQ

d�= dQ

dX

dX

d�+ dQ

dY

dY

d�= (g(X) + h(X)Y)

m∑i=0

ai(X)Yi. (19)

n this example, we assume that m = 1 in Eq. (18). Taking Eqs. (17) and (19) into account, we get

1∑i=0

a′i(X)Yi+1 +

(t(dω(t)/dt) + ω(t) + k2˛(t)

˛(t) + �(t)X − ˇ(t)

˛(t) + �(t)X3

) 1∑i=0

iai(X)Yi−1 = (g(X) + h(X)Y)1∑i=0

ai(X)Yi, (20)

here the primes denote derivatives with respect to X. Equating the coefficients of Yi(i = 2, 1, 0) in Eq. (20) leads to the system

a′1(X) = h(X)a1(X), (21)

a′0(X) = g(X)a1(X) + h(X)a0(X), (22)

a1(X)

[t(dω(t)/dt) + ω(t) + k2˛(t)

˛(t) + �(t)X − ˇ(t)

˛(t) + �(t)X3

]= g(X)a0(X). (23)


ince ai(X)(i = 0, 1) are polynomials, then from (21) we deduce that a1(X) is constant and h(X) = 0 . For simplicity, take a1(X) = 1. Balancinghe degrees of g(X) and a0(X), we conclude that deg(g(X)) = 1 only. Suppose g(X) = A1X + B0, then we find a0(X) .

a0(X) = A0 + B0X + A1

2X2, (24)

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here A0 is arbitrary integration constant. Substituting a0(X) and g(X) into (23) and setting all the coefficients of powers X to be zero, thene obtain a system of nonlinear algebraic equations and by solving it, we obtain

B0 = 0, A1 =√

−2ˇ(t)

˛(t) + �(t), ω(t) = 1

t

∫ t

0

{−�2˛(t′) + A0

√−2ˇ(t′(˛(t′ + �(t′)}dt′ (25)

B0 = 0, A1 = −√

−2ˇ(t)

˛(t) + �(t), ω(t) = 1

t

∫ t

0

{−�2˛(t′ − A0

√−2ˇ(t′(˛(t′ + �(t′))}dt′, (26)

here A0 and � are arbitrary constants. Using the conditions (25) in Eq. (18), we obtain

Y(�) = −A0 −√

− ˇ(t)2(˛(t) + �(t))

X2(�). (27)

ombining (27) with (17), we obtain the exact solution to Eq. (15) and then exact solutions for the variable coefficients RNLS equation withower law nonlinearity (9) can be written as:

Type 1: When A0

√− ˇ(t)

2(˛(t)+�(t)) < 0, we have

(1) Dark soliton solution

1(x, t) = −

√−A0

√−2ˇ(t)(˛(t) + �(t))

ˇ(t)ei(−�x−((1/t)

∫ t

0{�2˛(t′)−A0

√−2ˇ(t′)(˛(t′)+�(t′))}dt′)t)

× tanh

⎛⎝

√−A0

√−2ˇ(t)(˛(t) + �(t))

2(˛(t) + �(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

)⎞⎠ , (28)

here �0 is an arbitrary constant.(2) Singular soliton solution

2(x, t) = −

√−A0

√−2ˇ(t)(˛(t) + �(t))

ˇ(t)ei(−�x−((1/t)

∫ t

0{�2˛(t′)−A0

√−2ˇ(t′)(˛(t′)+�(t′))}dt′)t)

× coth

⎛⎝

√−A0

√−2ˇ(t)(˛(t) + �(t))

2(˛(t) + �(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

)⎞⎠ , (29)

here �0 is an arbitrary constant.

Type 2: When A0

√− ˇ(t)

2(˛(t)+�(t)) > 0, we can obtain the following periodic solutions

3(x, t) =

√A0

√−2ˇ(t)(˛(t) + �(t))

ˇ(t)ei(−�x−((1/t)

∫ t

0{�2˛(t′)−A0

√−2ˇ(t′)(˛(t′)+�(t′))}dt′)t)

× tan

⎛⎝

√A0

√−2ˇ(t)(˛(t) + �(t))

2(˛(t) + �(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

)⎞⎠ , (30)

4(x, t) = −

√A0

√−2ˇ(t)(˛(t) + �(t))

ˇ(t)ei(−�x−((1/t)

∫ t

0{�2˛(t′)−A0

√−2ˇ(t′)(˛(t′)+�(t′))}dt′)t)

× cot

⎛⎝

√A0

√−2ˇ(t)(˛(t) + �(t))

2(˛(t) + �(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

)⎞⎠ , (31)

here �0 is an arbitrary constant.Type 3: When A0 = 0, we can obtain the following rational solution

5(x, t) =√

−2(˛(t) + �(t))ˇ(t)

ei(−�x−((�2/t)

∫ t

0˛(t′)dt′)t) × 1

x + ((2�/t)∫ t

0˛(t′)dt′)t + �0

, (32)


here �0 is an arbitrary constant. Similarly, in the case of (26), from (18), we obtain

Y(�) = −A0 +√

− ˇ(t)2(˛(t) + �(t))

X2(�), (33)

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nd then the exact solutions of the variable coefficients RNLS equation with power law nonlinearity (9) can be written as:

Type 1: When A0

√− ˇ(t)

2(˛(t)+�(t)) > 0, we have

(1) Dark soliton solution

6(x, t) = −

√A0

√−2ˇ(t)(˛(t) + �(t))

ˇ(t)ei(−�x−((1/t)

∫ t

0{�2˛(t′)+A0

√−2ˇ(t′)(˛(t′)+�(t′))}dt′)t)

× tanh

⎛⎝

√A0

√−2ˇ(t)(˛(t) + �(t))

2(˛(t) + �(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

)⎞⎠ , (34)

here �0 is an arbitrary constant.(2) Singular soliton solution

7(x, t) = −

√A0

√−2ˇ(t)(˛(t) + �(t))

ˇ(t)ei(−�x−((1/t)

∫ t

0{�2˛(t′)+A0

√−2ˇ(t′)(˛(t′)+�(t′))}dt′)t)

× coth

⎛⎝

√A0

√−2ˇ(t)(˛(t) + �(t))

2(˛(t) + �(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

)⎞⎠ , (35)


Type 2: When A0

√− ˇ(t)

2(˛(t)+�(t)) < 0, we can obtain the following periodic solutions

8(x, t) =

√−A0

√−2ˇ(t)(˛(t) + �(t))

ˇ(t)ei(−�x−((1/t)

∫ t

0{�2˛(t′)+A0

√−2ˇ(t′)(˛(t′)+�(t′))}dt′)t)

× tan

⎛⎝

√−A0

√−2ˇ(t)(˛(t) + �(t))

2(˛(t) + �(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

)⎞⎠ , (36)

9(x, t) = −

√−A0

√−2ˇ(t)(˛(t) + �(t))

ˇ(t)ei(−�x−((1/t)

∫ t

0{�2˛(t′)+A0

√−2ˇ(t′)(˛(t′)+�(t′))}dt′)t)

× cot

⎛⎝

√−A0

√−2ˇ(t)(˛(t) + �(t))

2(˛(t) + �(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

)⎞⎠ , (37)

Type 3: When A0 = 0, we can obtain the following rational solution

10(x, t) = −√

−2(˛(t) + �(t))ˇ(t)

ei(−�x−((�2/t)

∫ t

0˛(t′)dt′)t) × 1

x + ((2�/t)∫ t

0˛(t′)dt′)t + �0

, (38)


.2. Parabolic law

For parabolic-law nonlinearity, F(s) = bs + cs2, where b and c are in general constants. Such a kind of nonlinearity appears also in fiberptics [19,20]. In this case, the R-NLSE is

i t + a(t) xx +{b(t)| |2 + c(t)| |4

} + d(t)

(| |xx| |

) = i�(t) . (39)

ere, in (39), the first term represents the evolution term, the second term is the GVD term, while the third and fourth terms in parenthesisogether represents nonlinearity, where a, b, c, d and � are all time-dependent coefficients. The term on the right-hand side of (39) representshe linear attenuation term with a varying coefficient �(t) .


When �(t) = 0, we have

i t + a(t) xx +{b(t)| |2 + c(t)| |4

} + d(t)

(| |xx| |

) = 0. (40)

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or solving (40), we use the wave transformation

(x, t) = U(�)ei(−�x+ω(t)t), � = x − v(t)t, (41)

here v(t) is the soliton velocity, � is the wave number of the soliton, while ω(t) is the frequency of the soliton velocity, all of them are toe determined. Substituting (41) into (40), and equating the real and imaginary parts, yields the following pair of relations:

tdv(t)dt

+ v(t) + 2�a(t) = 0, (42)

(a(t) + d(t))U ′′ −(tdω(t)dt

+ ω(t) + �2a(t))U + b(t)U3 + c(t)U5 = 0. (43)


v(t) = −2�t

∫ t

0

a(t′)dt′. (44)

y using transformation U = V(1/2), Eq. (43) becomes

(a(t) + d(t))(2VV ′′ − (V ′)2) − 4(tdω(t)dt

+ ω(t) + �2a(t))V2 + 4b(t)V3 + 4c(t)V4 = 0. (45)

sing (3) and (4), we obtain the equivalent planar system

dX

d�= Y,

dY

d�= Y2

2X+ 1

(a(t) + d(t))X

[2(tdω(t)dt

+ ω(t) + �2a(t))X2 − 2b(t)X3 − 2c(t)X4]. (46)

ow, we make the transformation d� = Xd� in Eq. (46) to avoid the singular line X = 0 temporarily. Thus, system (46) becomes

dX

d�= XY,

dY

d�= 1

2Y2 + 1

a(t) + d(t)

[2(tdω(t)dt

+ ω(t) + k2a(t))X2 − 2b(t)X3 − 2c(t)X4

]. (47)

uppose that m = 1 in (18). From now on, we shall omit some details because the procedure is the same. Then, by equating the coefficientsf Yi(i = 2, 1, 0) on both sides of (19), we have

Xa′1(X) = (h(X) − 1

2)a1(X), (48)

Xa′0(X) = g(X)a1(X) + h(X)a0(X), (49)

a1(X)[

1a(t) + d(t)

(2(tdω(t)dt

+ ω(t) + �2a(t))X2 − 2b(t)X3 − 2c(t)X4

)]= g(X)a0(X). (50)

s a1(X) and h(X) are polynomials, from Eq. (48), we deduce that h(X) = 12 and a1(X) must be a constant. For simplicity, we can take a1(X) = 1 .

alancing the degrees of g(X) and a0(X), we conclude that deg(g(X)) = 2 and deg(a0(X)) = 2 only. Suppose

g(X) = A0 + A1X + A2X2, a0(X) = B0 + B1X + B2X

2, A2 /= 0, B2 /= 0, (51)

here A0, A1, A2, B0, B1, B2 are all constants to be determined. Substituting (51) into Eq. (49), we obtain

A0 = −12B0, A1 = 1

2B1, A2 = 3

2B2. (52)

ubstituting a0(X), a1(X) and g(X) into (50) and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinearlgebraic equations and by solving it, we obtain

B0 = 0, B1 = −b(t)2

√− 3c(t)(a(t) + d(t))

, B2 = 2

√− c(t)

3(a(t) + d(t)), (53)

ω(t) = −1t

∫ t

0

{ 3b2(t′)16c(t′)

+ �2a(t′)}dt′,

nd

B0 = 0, B1 = b(t)2

√− 3c(t)(a(t) + d(t))

, B2 = −2

√− c(t)

3(a(t) + d(t)), (54)


ω(t) = −1t

∫ t

0

{3b2(t′)16c(t′)

+ �2a(t′)

}dt′.

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sing the conditions (53) and (54) in (18), we obtain

Y = ±b(t)2

√− 3c(t)(a(t) + d(t))

X ∓ 2

√− c(t)

3(a(t) + d(t))X2. (55)

ombining Eq. (46) with Eq. (55) and changing to the original variables, we find exact solutions to Eq. (40) as:Type 1: When 1/(c(t)(a(t) + d(t))) < 0, we can obtain the following envelope solitary wave solutions(1) Dark soliton solution

1,2(x, t) = ±12

√3b(t)2c(t)

ei(−�x−((1/t)

∫ t

0{(3b2(t′)/16c(t′))+�2a(t′)}dt′)t)

×[

1 − tanh

(∓b(t)

4

√− 3c(t)(a(t) + d(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

))](1/2)

, (56)

here �0 is an arbitrary constant.(2) Singular soliton solutions

3,4(x, t) = ±12

√3b(t)2c(t)

ei(−�x−((1/t)

∫ t

0{(3b2(t′)/16c(t′))+�2a(t′)}dt′)t)

×[

1 − coth

(∓b(t)

4

√− 3c(t)(a(t) + d(t))

(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

))](1/2)

, (57)

here �0 is an arbitrary constant.Type 2: When b(t) = 0, we can obtain the following rational solutions

5,6(x, t) = ±√

4 − 3(a(t) + d(t))4c(t)

ei(−�x−((�2/t)

∫ t

0a(t′)dt′)t) ×

[1

x + ((2�/t)∫ t

0˛(t′)dt′)t + �0

](1/2)

, (58)


.3. Dual-power law

The dual-power law nonlinearity appears in LiNbO3 crystals and it is formulated as F(s) = bsn + cs2n, where b and c are in general constantslthough in this paper they are taken to be time-dependent. This law is a generalization of the parabolic law nonlinearity. In fact, setting

= 1, the dual-power law collapses to parabolic law nonlinearity. In this case, the R-NLSE is

i t + a(t) xx +{b(t)| |2n + c(t)| |4n

} + d(t)

(| |xx| |

) = 0. (59)

ithout loss of generality, we assume that the solution (x, t) to Eq. (59) takes the form

(x, t) = U(�)ei(−�x+ω(t)t), � = x − v(t)t. (60)

ubstituting (60) into (59), and equating the real and imaginary parts, yields the following pair of relations:

tdv(t)dt

+ v(t) + 2�a(t) = 0, (61)

(a(t) + d(t))U ′′ −(tdω(t)dt

+ ω(t) + �2a(t))U + b(t)U2n+1 + c(t)U4n+1 = 0. (62)


v(t) = −2�t

∫ t

0

a(t′)dt′. (63)

ue to the difficulty in obtaining the first integral of Eq. (62), we propose a transformation denoted by U = V1

2n . Then Eq. (62) is convertedo

(a(t) + d(t))(2nVV ′′ + (1 − 2n)(V ′)2) − 4n2(tdω(t)dt

+ ω(t) + �2a(t))V2 + 4n2b(t)V3 + 4n2c(t)V4 = 0. (64)

sing (3) and (4), we obtain the equivalent planar system

dX

d�= Y,

dY

d�=

(1 − 1

2n

)Y2

X+ 1

(a(t) + d(t))X

[2n

(tdω(t)dt

+ ω(t) + �2a(t))X2 − 2nb(t)X3 − 2nc(t)X4

]. (65)


aking the following transformation d� = Xd�, then system (65) becomes

dX

d�= XY,

dY

d�=

(1 − 1

2n

)Y2 + 1

a(t) + d(t)

[2n

(tdω(t)dt

+ ω(t) + �2a(t))X2 − 2nb(t)X3 − 2nc(t)X4

]. (66)

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uppose that m = 1 in (18). From now on, we shall omit some details because the procedure is the same. Then, by equating the coefficientsf Yi(i = 2, 1, 0) on both sides of (19), we have

Xa′1(X) =

(h(X) −

(1 − 1

2n

))a1(X), (67)

Xa′0(X) = g(X)a1(X) + h(X)a0(X), (68)

a1(X)[

1a(t) + d(t)

(2n

(tdω(t)dt

+ ω(t) + �2a(t))X2 − 2nb(t)X3 − 2nc(t)X4

)]= g(X)a0(X). (69)

rom Eq. (67), we obtain a1(X) = c0 exp(∫ (

(h(X) − (1 − (1/2n)))/X)), where c0 is an integration constant. As a1(X) and h(X) are polyno-

ials, we deduce that h(X) = 1 − (1/2n) and a1(X) must be a constant. For simplicity, we can take a1(X) = 1. Balancing the degrees of g(X) and0(X), we conclude that deg(g(X)) = 2 and deg(a0(X)) = 2 only. Suppose that

g(X) = A0 + A1X + A2X2, a0(X) = B0 + B1X + B2X

2, A2 /= 0, B2 /= 0, (70)

here A0, A1, A2, B0, B1, B2 are all constants to be determined. Substituting (70) into Eq. (69), we obtain

A0 =(

12n

− 1)B0, A1 = 1

2nB1, A2 =

(1

2n+ 1

)B2. (71)

ubstituting a0(X), a1(X) and g(X) into (69) and setting all the coefficients of powers X to be zero, then we obtain a system of nonlinearlgebraic equations and by solving it, we obtain

B0 = 0, B1 = − nb(t)n + 1

√− 2n + 1c(t)(a(t) + d(t))

, B2 = 2n

√− c(t)

(2n + 1)(a(t) + d(t)), (72)

ω(t) = −1t

∫ t

0

{(2n + 1)b2(t′)

4(n + 1)2c(t′)+ �2a(t′)

}dt′.

nd

B0 = 0, B1 = nb(t)n + 1

√− 2n + 1c(t)(a(t) + d(t))

, B2 = −2n

√− c(t)

(2n + 1)(a(t) + d(t)), (73)

ω(t) = −1∫ t {

(2n + 1)b2(t′)2

+ �2a(t′)

}dt′.


t0 4(n + 1) c(t′)

sing the conditions (72) and (73) in (18), we obtain

Y = ± nb(t)n + 1

√− 2n + 1c(t)(a(t) + d(t))

X ∓ 2n

√− c(t)

(2n + 1)(a(t) + d(t))X2. (74)

ombining Eq. (74) with Eq. (65) and changing to the original variables, we find exact solutions to Eq. (59) as:Type 1: When 1/(c(t)(a(t) + d(t))) < 0, we can obtain the following envelope solitary wave solutions(1) Dark soliton solutions

1,2(x, t) = ei(−�x−((1/t)

∫ t

0{((2n+1)b2(t′)/4(n+1)2c(t′))+�2a(t′)}dt′)t)

×{

± (2n + 1)b(t)4(n + 1)c(t)

[1 − tanh

(± nb(t)

2(n + 1)

√− 2n + 1c(t)(a(t) + d(t))

×(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

))]}(1/2n)

, (75)


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(2) Singular soliton solutions

3,4(x, t) = ei(−�x−((1/t)

∫ t

0{((2n+1)b2(t′)/4(n+1)2c(t′))+�2a(t′)}dt′)t)

×{

± (2n + 1)b(t)4(n + 1)c(t)

[1 − coth

(± nb(t)

2(n + 1)

√− 2n + 1c(t)(a(t) + d(t))

×(x +

(2�t

∫ t

0

˛(t′)dt′)t + �0

))]}(1/2n)

, (76)

here �0 is an arbitrary constant.Type 2: When b(t) = 0, we can obtain the following rational solutions

5,6(x, t) = ei(−�x−((�2/t)

∫ t

0a(t′)dt′)t) ×

[±√

− (2n + 1)(a(t) + d(t))4n2c(t)

1

x + ((2�/t)∫ t

0˛(t′)dt′)t + �0

](1/2n)

, (77)


. Conclusions

NLSE with time-dependent coefficients have been applied in modeling of many physical, engineering, chemistry, biology, etc [12–20,36].aghizadeh et al. [10] have proposed the first integral method to obtain the exact soliton solutions of the NLSE. In this paper, we extendedhe first integral method to construct the exact solutions of the R-NLSE with time-dependent coefficients in three forms of nonlinear-ty such as Kerr law, Parabolic law and Dual-power law nonlinearity. It needs to be noted that there are other laws of nonlinearityhat are studied in the context of optics. They are power law, log law, saturable law threshold law and many others. Unfortunatelyhese laws do not permit retrieval of solutions by the first integral method. These obtained solutions may be important for the expla-ation of some practical physical problems. The results show that this method is efficient in finding the exact soliton solutions of someonlinear partial differential equations with time-dependent coefficients. We predict that the first integral method can be extendedo solve many systems of nonlinear partial differential equations with time-dependent coefficients in mathematical and physical sci-nces.

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Optical solitons for the resonant nonlinear Schrödinger's equation with time-dependent coefficients by the first integral method