110 J. Eur. Opt. Society-Rapid Publ. 21, 11( 2025)
/ 0 1 ¼� m
2a 1 þ 3ðb 1 þ c 1 k 3 Þ
4a 1 w 2
1;
/ 0 2 ¼� m þ 3ðb ð13Þ
2k 3 þ c 2 Þ w 2 1
2a 2 4a 2 k;
where the integration constant is set zero. Consequently, the chirping expression of the first field component has the form dx 1 ¼ m � 3ðb 1 þ c 1 k 3 Þ w 2 1
2a 1 4a; ð14Þ 1
whereas the chirping expression of the second field component is identified as
dx 2 ¼ m � 3ðb 2k 3 þ c 2 Þ w 2 1
2a 2 4a 2 k: ð15Þ
Substituting( 13) into equation( 12), we obtain w 00 1 � r 1w 1 � c 1 w 3
1 þ b 1w 5
1 ¼ 0; w 00 1 � r 2w 1 � c 2 w 3
1 þ b 2w 5
1 ¼ 0: ð16Þ
The parameters r j, c j, b j;( j = 1, 2) in the system of equations( 16) are defined as
See Equation( 17) at the bottom of this page
where both a 1 and a 2 are not equal to zero. The coupled equations( 16) are consistent under the conditions
r 1 r 2
¼ c 1 c 2
¼ b 1 b 2
:
Accordingly, the system of equations( 16) turns into ð18Þ w 00 1 � rw 1 � cw 3
1 þ bw5 1 ¼ 0; ð19Þ where the parameters b, c and r are given by r ¼ r 1 ¼ r 2; c ¼ c 1 ¼ c 2; b ¼ b 1 ¼ b 2: ð20Þ Since equation( 19) can be integrated, then it is reduced to
6w 0 2 1
� 6rw2
1
� 3cw4
1 þ 2bw6 1 þ 12l ¼ 0; ð21Þ
where l is the integration constant. For convenience, the form of equation( 21) can be modified by introducing the variable transformation given as w 1 ¼ W 1
2: ð22Þ Subsequently, one can reach the equation of the form 3W 02 þ 24lW � 12rW 2 � 6cW 3 þ 4bW 4 ¼ 0: ð23Þ
Based on the relations( 6),( 10) and( 22), the solutions of equation( 23) reveal the amplitude structures of soliton waves that propagate in birefringent optical fiber. Further to this, one can recover the general form of solutions for the coupled equations( 5) as
pðx; tÞ ¼W 1
2 e ið / 1 ðnÞ�x 1 t a a Þ; qðx; tÞ ¼kW 1
2 e ið / 2 ðnÞ�x 2 t a a ÞÞ; ð24Þ
where the phase variable / 1( n) and / 2( n) can be obtained by integrating equations( 13).
Thus, our aim in the next section is to solve equation( 23) analytically by the proposed method so as to derive soliton solutions of bright and dark structures.
4 Chirped soliton solutions
The solution of equation( 23) is derived by means of the Jacobi elliptic equation that has a form of a first-order nonlinear ODE with three-degree terms. Before implementing this technique, we put forward the transformation given by
WðnÞ ¼ CðfÞ; f ¼ n; ð25Þ by which equation( 23) is rearranged to
3 2 C 02 þ 24lC � 12rC 2 � 6cC 3 þ 4bC 4 ¼ 0: ð26Þ
where is a constant to be determined. It is worth mentioning that equation( 26) can be expressed in terms of the elliptic Jacobi sine [ 61 – 64 ]. However, we are interested in creating various solutions to equation( 26) by considering its solution in the form pffiffiffiffiffiffiffiffiffi K 2 ðfÞ
CðfÞ ¼d 0 þ d 1 KðfÞþd 2 KðfÞ þ d 3
1 þ KðfÞ; ð27Þ
where d j,( j = 0, 1, 2, 3) are constants to be identified. The function K( f) satisfies the following ODE given by [ 65 ]
ðK 0 ðfÞÞ 2 ¼ f KðfÞþgKðfÞ 2 þ hKðfÞ 3; ð28Þ a 1 w 00
1 � x 1w 1 � mw 1 / 0 1 � a 1w 1 / 02 1 þðb 1 / 0 1 þ c 1k 3 / 0 2 Þw3 1 ¼ 0; a 2 kw 00
1 � x 2kw 1 � mkw 1 / 0 2 � a 2kw 1 / 02 2 þðb 2k 3 / 0 2 þ c 2 / 0 1 Þw3 1 ¼ 0: ð12Þ r 1 ¼ 4a 1x 1 � m 2
4a 2 1
r 2 ¼ 4a 2x 2 � m 2
4a 2 2
; c 1 ¼ mða 2b 1 þ a 1 c 1 k 3 Þ 2a 2 a 2
1
; c 2 ¼ mða 1b 2 k 3 þ a 2 c 2 Þ 2ka 1 a 2
2
; b 1 ¼ 3a 2ðb 1 þ c 1 k 3 Þðb 1 � 3c 1 k 3 Þþ12a 1 c 1 k 2 ðb 2 k 3 þ c 2 Þ;
16a 2 a 2
1
; b 2 ¼ 3a 1ðb 2 k 3 þ c 2 Þðb 2 k 3 � 3c 2 Þþ12a 2 c 2 kðb 1 þ c 1 k 3 Þ;
16a 1 a 2
2k 2 ð17Þ