118
J. Eur. Opt. Society-Rapid Publ. 21, 11( 2025)
lcðm 2 � 3mn þ n 2 Þ 2 � 2r 2 mnðm � nÞ 2 ¼ 0; 16brðm � nÞ 2 þ 3c 2 ðm 2 � 3mn þ n 2 Þ¼0: ð99Þ
As m approaches 1, solution( 98) with(+) sign switches to a soliton solution given by
"( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðx; tÞ ¼ � 2r p
1 � sech 2 ffiffiffi)# 1
2 ð r nÞ 1 þ p c 1 þ sech 2 ffiffiffi e ið / t
1ðnÞ�x a 1 a Þ; ð r nÞ "( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðx; tÞ ¼k � 2r p
1 � sech 2 ffiffiffi)# 1
2 ð r nÞ 1 þ p c 1 þ sech 2 ffiffiffi e ið / t 2ðnÞ�x a 2 a Þ; ð r nÞ
ð100Þ
which delineates dark soliton as shown in Figure 7, while solution( 98) with( �) sign mutates to a soliton solution given by
"( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðx; tÞ ¼ � 2r p
1 � sech 2 ffiffiffi)# 1
2 ð r nÞ 1 � p c 1 þ sech 2 ffiffiffi e ið / t
1ðnÞ�x a 1 a Þ; ð r nÞ "( sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðx; tÞ ¼k � 2r p
1 � sech 2 ffiffiffi)# 1
2 ð r nÞ 1 � p c 1 þ sech 2 ffiffiffi e ið / t 2ðnÞ�x a 2 a Þ; ð r nÞ
ð101Þ
which represents bright soliton as displayed in Figure 8, where r > 0 and c < 0. From( 99), we catch
5 Stability analysis
l ¼ 0; 16br þ 3c 2 ¼ 0: ð102Þ
The linear stability analysis technique is discussed here to study the modulation instability of the space-time fractional Kaup-Newell equation( 5) in birefringent fiber. Accordingly, we assume the perturbed steady-state solutions of the form hpffiffiffi i pðx; tÞ ¼ d þ! ðx; tÞ e idt; hpffiffiffi i ð103Þ qðx; tÞ ¼ d þ Xðx; tÞ e idt;
where d is the normalized optical power while( x, t) and X( x, t) are small perturbations such that both and X are d. To examine the effect of perturbations and X, the method of linear stability analysis is employed. Substituting( 103) into system( 5), one can come to the linearized disturbance equations with respect to and X as
Figure 9. The dispersion relation ~ x ¼ ~ xð ~ jÞ between frequency ~ x and wave number ~ j given in( 110).
@ a
! @ t þ i d! þ a @ 2a! a 1
@ x 2a
þ c 1 d 2 @ a X @ x þ @ a X a @ x a
@ a
X @ t þ i dX þ a @ 2a
X a 2 @ x 2a
þ c 2 d 2 @ a!
@ x þ @ a! a @ x a
þ b 1 d 2 @ a!
@ x þ @ a! a @ x a
¼ 0;
þ b 2 d 2 @ a X
@ x þ @ a X a @ x a
¼ 0;
ð104Þ
where * denotes the conjugate. As both equations in( 104) have the same structures, the modulation instability analysis of the perturbations( x, t) and X( x, t) can be examined using the same way. Hence, we deal with the first equation in( 104) to study the perturbation evolution. Suppose that the coupled equations has solution expressed as
! ðx; tÞ ¼qe i ð jxa ~ a � xta ~ aÞ þ ge
�i ðj~ xa a � xta ~ aÞ; ð105Þ
Xðx; tÞ ¼qe i ð jxa ~ a � xta ~ aÞ þ ge
�i ðj~ xa a � xta ~ aÞ;
! ðx; tÞ ¼qe �i ð jxa ~ a � xta ~ aÞ þ ge i ðj~ xa a � xta ~ aÞ; ð106Þ
X ðx; tÞ ¼qe �i ð jxa ~ a � xta ~ aÞ þ ge i ðj~ xa a � xta ~ aÞ;
where x~ and j~ are the frequency of perturbation and normalized wave numbers, respectively. Plugging ansatz( 105) and( 106) into the first equation of( 104) and separating the coefficients of expfi j~ xa � x~ ta
� a a g and
� expf�i j~ xa � x~ ta a a g, this induces a pair of equations in q and g as
qð x~ þ a 1 j~ 2 Þ�dðqþ jðg ~ þ 2qÞðb 1 þ c 1 ÞÞ ¼ 0; gð x~ � a 1 j~ 2 Þþdðg� jðq ~ þ 2gÞðb 1 þ c 1 ÞÞ ¼ 0: ð107Þ
The system( 107) creates the coefficient matrix of q and g as
x~ þ a 1 j~ 2 � dð1 þ 2 jðb ~ 1 þ c 1 ÞÞ |
�d jðb ~ 1 þ c 1 Þ |
q
|
�d jðb ~ 1 þ c 1 Þ |
x~ � a 1 j~ 2 þ dð1 � 2 jðb ~ 1 þ c 1 ÞÞ |
g |
¼ 0
: ð108Þ 0