J. Eur. Opt. Society-Rapid Publ. 21, 11( 2025) 119
Figure 10. Nonlinearity effect on the chirped bright soliton given by solution( 66) with various values of parameters b 1, c 1, b 2, c 2.( a) | p | 2 with distinct values of b 1;( b) | p | 2 with distinct values of c 1;( c) | p | 2 with distinct values of b 2;( d) | p | 2 with distinct values of c 2.
See Equation( 108) at the bottom of the previous page
To ensure that the coefficient matrix having nontrivial solution, the determinant has to be vanish. Consequently, we obtain the dispersion relation in the form
ð x~ � 2d jðb ~ 1 þ c 1 ÞÞ 2 � d 2 j~ 2 ðb 1 þ c 1 Þ 2 �ðd�a 1 j~ 2 Þ 2 ¼ 0: ð109Þ
The solution of the dispersion relation( 109) for x~ provides qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x~ ¼ 2d jðb ~ 1 þ c 1 Þ d 2 j~ 2 ðb 1 þ c 1 Þ 2 þðd�a 1 j~ 2 Þ 2: ð110Þ
This expression reveals the situation of the steady-state stability. One can clearly note that d 2 ~ j 2 ðb 1 þ c 1 Þ 2 þ ðd � a 1 ~ j 2 Þ 2 is always 0. This means that ~ x is real for all values of wave numbers ~ j. Hence, the steady state is stable against the wave number perturbation. The dispersion relation that shows the depends of perturbation frequency on the normalized wave number is illustrated in Figure 9.
6 Results and discussion
The utilized scheme has been effective on generating optical solitons. Although this work sheds light only on chirped bright and dark solitons, there are abundant types of those soliton structures that are derived for the KNE in birefringent fiber. The dynamical behaviors of chirped solitons due to the effect of fractional order derivatives, a, are displayed through the graphical representations. The profile of soliton intensity is depicted in 2 and 3 dimensions with three distinct values of a which are 0.7, 0.8, 1.0. It is found that the fractional order derivative affects remarkably the evolution of chirped bright and dark solitons. In addition to this, the corresponding chirp to each soliton solution is also presented with integer order derivative. The parameter values are carefully selected to fulfil the constraint conditions for the existence of each soliton solution.
We observed that solutions( 34),( 42),( 52) have the same behavior which represent a profile of chirped dark soliton as presented in Figure 1 with parameter values a 1 = a 2 = b 1 = b 2 = c 1 = c 2 = x 1 = 0.5, m = 1.25, k = 1.5. Moreover, solutions( 38),( 48),( 56) exhibit the shape of bright soliton wave as shown in Figure 2 with parameter values a 1 = 0.93, a 2 = b 2 = �0.5, b 1 = c 1 = c 2 = 0.5, x 1 = 1.5, m = 0.75, k = 1.5. Additionally, it is noted that solutions( 62),( 70),( 96) describe the variation of dark soliton as plotted in Figure 3 with the same parameter values as in Figure 1 except b 1 = �0.5, x 1 = 0.128, m = 0.75. One can obviously see that Figure 4 illustrates the profile of chirped bright soliton for solution( 66) with the same parametervaluesasinFigure 1 except c 1 = �0.5, x 1 = 1.65, m = 1.75. In Figure 5, the graph delineates bright soliton structure for solutions( 76),( 84),( 95) which is depicted with the same parameter values as in Figure 3. Further to this, Figure 6 demonstrates the shape of dark soliton wave for solution( 80) which is drawn with the same parameter values as in Figure 4. InFigure 7, the plot displays the profile of dark soliton for solutions( 90) and( 100) while the plot in Figure 8 characterizes bright soliton wave for solutions( 91) and( 101), where both graphs are drawn with the same parameter values as in Figure 4.
The effect on nonlinearity parameters b 1, c 1, b 2, c 2 on the soliton propagation is detected by taking the bright soliton given by solution( 66) as example. It is obviously seen from Figure 10 that b 1 causes noteworthy variations on the soliton behaviors, where the increase in b 1 value reduces the amplitude and expands the width of wave. The most negative value of c 1 decreases both of the soliton amplitude and width. Moreover, the increment of b 2 leads to a reduction in the soliton amplitude with almost no change in its width. The parameter c 2 has nearly a negligible influence in the pulse propagation.
Overall, one can see that the chirped bright and dark optical solitons retrieved here experience different evolutions based on the impact of fractional order derivative and diversity of model parameter values.
7 Conclusion
We have studied chirped bright and dark solitons of fractional-order Kaup-Newell equation( KNE) in birefringent fiber. The employed scheme which depends on alternative form of Jacobi elliptic equation is found to be powerful on deriving optical solitons. Various types of chirped bright and dark solitons with their associated chirping are extracted which present distinct wave profiles. The behavior of obtained optical solitons is noted to undergo