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Figure 5. Intensity profile of bright soliton given by solutions( 76) and( 84) with the same parameter values as in Figure 3.( 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.
Figure 6. Intensity profile of dark soliton given by solution( 80) 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.
Set V. 4rðg þ sÞ d 0 ¼� cð2g þ 3sÞ; d 1 ¼ d 3 ¼ 0; d 2 ¼� 4r
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi �2hðg þ sÞ
; cð2g þ 3sÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4 � r; ð86Þ
2g þ 3s
under the constraint conditions lcð2g þ 3sÞ 2 þ 4sr 2 ðg þ sÞ ¼0; ð87Þ
32brðg þ sÞþ3c 2 ð2g þ 3sÞ ¼0; p where s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g 2 � 4fh. Applying these outcomes to( 27) and using( 24) and( 25), the following cases of solutions to equation( 5) are extracted.
Case V1. If f = 4, g = �4( 1 + m 2), h = 4m 2, K( f)= sn 2( f, m), equation( 5) captures JEF solution of the form
h n pffiffiffiffiffiffiffiffiffiffiffiffiffi oi1
8r pðx; tÞ ¼ 1 sn 2 � r
2 n e ið / t
1ðnÞ�x a 1 cðm 2 �5Þ m 2 a Þ;
�5 h n pffiffiffiffiffiffiffiffiffiffiffiffiffi oi1
8r qðx; tÞ ¼k 1 sn 2 � r
2 n e ið / t
2ðnÞ�x a 2 cðm 2 �5Þ m 2 a Þ;
�5 ð88Þ provided that r > 0andc < 0. The conditions( 87) reduce to lcðm 2 � 5Þ 2 � 8r 2 ðm 2 � 1Þ ¼0; 64br � 3c 2 ðm 2 � 5Þ ¼0: ð89Þ
As m approaches 1, solution( 88) with(+) sign degenerates to a soliton solution given by
h i1 pffiffiffi pðx; tÞ ¼ � 2r 2
1 þ tanh r c f ð nÞg e ið / t
1ðnÞ�x a 1 a Þ; h ð90Þ i1 pffiffiffi qðx; tÞ ¼k � 2r 2
1 þ tanh r c f ð nÞg e ið / t
2ðnÞ�x a 2 a Þ;
which describes dark soliton as presented in Figure 7, while solution( 88) with( �) sign collapses to a soliton solution given by
h i1 pffiffiffi pðx; tÞ ¼ � 2r 2
1 � tanh r c f ð nÞg e ið / t
1ðnÞ�x a 1 a Þ; h ð91Þ i1 pffiffiffi qðx; tÞ ¼k � 2r 2
1 � tanh r c f ð nÞg e ið / t
2ðnÞ�x a 2 a Þ;
which characterizes bright soliton as plotted in Figure 8, where r > 0 and c < 0. From( 89), we catch
l ¼ 0; 16br þ 3c 2 ¼ 0: ð92Þ
Case V2. If f = 4( 1 � m 2), g = �4( 1 � 2m 2), h = �4m 2, K( f)= cn 2( f, m), equation( 5) possesses JEF solution of the form
h n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oi1 pðx; tÞ ¼ � 8rm2 1 cn 2 � r
2 n e ið / t
1ðnÞ�x a 1 cð4m 2 þ1Þ 4m 2 a Þ;
þ1 h n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi oi1 qðx; tÞ ¼k � 8rm2 1 cn 2 � r
2 n e ið / t
2ðnÞ�x a 2 cð4m 2 a Þ: þ1Þ 4m 2 þ1 ð93Þ