J. Eur. Opt. Society-Rapid Publ. 21, 11( 2025) 117
Figure 7. Intensity profile of dark soliton given by solutions( 90) and( 100) with the same parameter values as in Figure 4.( a) 3D plot of dark soliton;( b) | p | 2 with distinct values of a;( c) | q | 2 with distinct values of a;( d) Chirping profile.
Figure 8. Intensity profile of bright soliton given by solutions( 91) and( 101) with the same parameter values as in Figure 4.( a) 3D plot of bright soliton;( b) | p | 2 with distinct values of a;( c) | q | 2 with distinct values of a;( d) Chirping profile.
provided that r < 0 and c > 0. The conditions( 87) become
lcð4m 2 þ 1Þ 2 þ 8r 2 m 2 ¼ 0; 64brm 2 þ 3c 2 ð4m 2 þ 1Þ ¼0: ð94Þ
As m approaches 1, solution( 93) with(+) sign turns into a soliton solution given by
h n pffiffiffiffiffiffi
oi1 pðx; tÞ ¼ � 8r 1 þ sech 2 � r 2 5c 5 n e ið / t
1ðnÞ�x a 1 a Þ; h n pffiffiffiffiffiffi
oi1 qðx; tÞ ¼k � 8r 1 þ sech 2 � r 2 5c 5 n e ið / t
2ðnÞ�x a 2 a Þ:
ð95Þ
which represents bright soliton as demonstrated in Figure 5, while solution( 93) with( �) sign becomes a soliton solution given by h n pffiffiffiffiffiffi
oi1 pðx; tÞ ¼ � 8r 1 � sech 2 � r 2 5c 5 n e ið / t
1ðnÞ�x a 1 a Þ; h n pffiffiffiffiffiffi
oi1 qðx; tÞ ¼k � 8r 1 � sech 2 � r 2 5c 5 n e ið / t
2ðnÞ�x a 2 a Þ:
ð96Þ
which depictes dark soliton as presented in Figure 3, where r < 0 and c > 0. From( 94), wefind
25lc þ 8r 2 ¼ 0; 64br þ 15c 2 ¼ 0: ð97Þ
Case V3. If f = 1, g = 2( 1 � 2m 2), h = 1, KðfÞ ¼ sn2 ðf; mÞ
, equation( 5) attains JEF solution of the ð1cnðf; mÞÞ form 2
See Equation( 98) at the bottom of this page p where n ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 2 � 1. The conditions( 87) turn to
2
8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
9
31
2
2rðm � nÞ
>< r u1 � cn 2 n
>= 6 m pðx; tÞ ¼ � ðm � nÞ 2 �3mnþn t 2 cðm 2 � 3mn þ n 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7
4
5 e ið / t 1ðnÞ�x a 1 a Þ; >: r
1 þ cn 2 n >; m 2 �3mnþn 2
2
8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
931
2
2rðm � nÞ
>< r
1 � cn 2 n >=
6 u m qðx; tÞ ¼k � ðm � nÞ 2 �3mnþn t 2 cðm 2 � 3mn þ n 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7
4
5 e ið / t 2ðnÞ�x a 2 a Þ; >: r
1 þ cn 2 n >; m 2 �3mnþn 2 ð98Þ