J. Eur. Opt. Society-Rapid Publ. 21, 11( 2025) 109
derivatives [ 57 ]. The study is carried out by utilizing the Jacobi elliptic equation method expressed in a form of a first-order nonlinear ordinary differential equation( ODE) with three-degree terms. The modulation instability of the addressed model is examined. The arrangement of this work is as follows. The next section elucidates the properties of conformable fractional derivative. Section 3 describes the governing model of fractional order derivative and its mathematical analysis. In Section 4, abundant types of chirped bright and dark are constructed by using the proposed strategy. Section 5 displays the diagnosis process of the modulation instability through employing the linear stability analysis method. The discussion of obtained results and optical pulse behaviors are illustrated in Section 6. Finally, the conclusion of work is given in Section 7.
2 Conformable fractional derivative
Fractional calculus can be defined in various ways according to the characteristics and conditions of fractional derivative that must be satisfied. Among the introduced definitions in literature are Riemann and Liouville, Caputo-Katugampola, Grünwald-Letnikov, Marchaud and Others [ 58 ]. Conformable fractional derivative [ 59 ] is one of the most applied techniques in the recent research activities involving fractional calculus, see [ 60 ]. The definition of conformable fractional derivative is described as follows.
Definition 1. Letf:( 0,1)? R, then the conformable fractional derivative of f of order a is defined as
D a t f ðtÞ ¼lim e! 0 f ðt þ e t 1�a Þ�fðtÞ; ð3Þ e
for all t > 0,0 < a 1. In case that the conformable fractional derivative of f of order a exists, then it is said that f is a-differentiable. The conformable fractional derivative satisfies some properties displayed in the following theorems.
Theorem 1. Leta 2( 0, 1 ] and f = f( t), g = g( t) be a-differentiable at a point t > 0, then
1. D a t ð. Þ¼0;. is a constant. 2. D a t ðt # Þ¼ # t # �a, for all # 2 R. 3. D a t ðaf þ bgÞ ¼aDa t f þ bDa t g, for all a, b 2 R.
4. D a t ðfgÞ ¼gDa t f þ fDa t g.
5. D a f t
¼ gDa t f � fDa t g. g g 2
Additionally, if f is differentiable, then D a df t f ðtÞ ¼t1�a. dt
Theorem 2. Letf:( 0,1)? R, be a function such that f is differentiable and also a-differentiable. Let g be a function defined in the range of f and also differentiable. Then,
D a t ðf gÞðtÞ ¼t1�a g 0 ðÞf t 0 ðgðtÞÞ; ð4Þ
where prime denotes the classical derivatives with respect to t.
3 Governing model and mathematical analysis
The space-time fractional Kaup-Newell equation in birefringent fiber is described by
D a t p þ ia 1D 2a x p þ b 1D a x ðjpj2 pÞþc 1 D a x ðjqj2 qÞ¼0; D a t q þ ia 2D 2a x q þ b 2D a x ðjqj2 qÞþc 2 D a x ðjpj2 pÞ¼0:
This equation characterizes the propagation of very short( sub-picosecond) light pulses in birefringent fiber which incorporates fractional dispersion. To the best of our knowledge, equation( 5) has not been discussed before. Consider the traveling wave transformation introduced as
pðx; tÞ ¼w 1 ðnÞe i / t ð 1ðnÞ�x a 1 aÞ; ð6Þ qðx; tÞ ¼w 2 ðnÞe i / t ð 2ðnÞ�x a 2 aÞ; ð5Þ
where a denotes fractional-order derivative and n represents the wave coordinate of spatial variable x and temporal variable t. The variable n is given by
n ¼ xa ta � m a a; 0 < a 1: ð7Þ
The two functions w 1( n) andw 2( n) stand for the amplitudes of the solitons while the functions / 1( n) and / 2( n) account for nonlinear phase shift. The parameters x 1, x 2, and m are real constants denoting the wave number and the soliton velocity.
Applying the transformation( 6), the coupled equation( 5) breaks up into a real part given by
�mw 0 1 � a 1w 1 / 00
1 � 2a 1w 0 1 / 0 1 þ 3b 1w 2 1 w0 1 þ 3c 1w 2
2 w0 2 ¼ 0;
�mw 0 2 � a 2w 2 / 00
2 � 2a 2w 0 2 / 0 2 þ 3b 2w 2 2 w0 2 þ 3c 2w 2
1 w0 1 ¼ 0; ð8Þ and an imaginary part given by a 1 w 00
1 � x 1w 1 � mw 1 / 0 1 � a 1w 1 / 0 2 1 þ b 1 / 0 1 w3 1 þ c 1 / 0 2 w3 2 ¼ 0; a 2 w 00
2 � x 2w 2 � mw 2 / 0 2 � a 2w 2 / 0 2
2 þ b 2 / 0 2 w3 2 þ c 2 / 0 1 w3 1 ¼ 0: ð9Þ
To handle the set of obtained equations analytically, the relation between w 1 and w 2 is proposed as
w 2 ¼ kw 1; ð10Þ
where k 6¼ 1 is a real constant. As a result, the systems( 8) |
and( 9) are converted respectively to |
|
|
|
�mw 0 1 � a 1 |
w 1 / 00
1 þ 2w0 1 / 0 1
|
þ 3ðb 1 þ c 1 k 3 Þw 2
1 w0 1 ¼ 0;
|
|
�kmw 0 1 � a 2k w 1 / 00
2 þ 2w0 1 / 0 2
|
þ 3ðb 2 k 3 þ c 2 Þw 2
1 w0 1 ¼ 0;
|
and See Equation( 12) at the bottom of the following page
ð11Þ
One can deduce that both equations in( 11) can be integrated and resulting in the firstintegralas