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J. Eur. Opt. Society-Rapid Publ. 21, 11( 2025)
Figure 3. Intensity profile of dark soliton given by solutions( 62),( 70) and( 96) with the same parameter values as in Figure 1 except b 1 = �0.5, x 1 = 0.128, m = 0.75.( 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 4. Intensity profile of bright soliton given by solution( 66) with the same parameter values as in Figure 1 except c 1 = �0.5, x 1 = 1.65, m = 1.75.( 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.
25lc þ 8r 2 ¼ 0; 64br þ 15c 2 ¼ 0: ð63Þ
Case III2. 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
2 pffiffiffiffiffiffiffiffiffiffi
3 2 1 2 r
6 pðx; tÞ ¼ � 2rð8m2 � 5Þ cn n
5m 2 �4 cð5m 2 � 4Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
7 4
5 e ið / t 1ðnÞ�x a 1 a Þ; 2
1 þ cn � r n
5m 2 �4
2 pffiffiffiffiffiffiffiffiffiffi
3 2 1 2 r
6 qðx; tÞ ¼k � 2rð8m2 � 5Þ cn n
5m 2 �4 cð5m 2 � 4Þ pffiffiffiffiffiffiffiffiffiffi
7 4
5 e ið / t 2ðnÞ�x a 2 a Þ: 2 r
1 þ cn n
The conditions( 59) become
5m 2 �4
lcð5m 2 � 4Þ 2 þ r 2 ðm 2 � 1Þð8m 2 � 5Þ ¼0; 2brð8m 2 � 5Þ 2 þ 3c 2 ð2m 2 � 1Þð5m 2 � 4Þ ¼0: ð64Þ
ð65Þ
As m approaches 1, solution( 64) converts to a soliton solution given by
" pðx; tÞ ¼ � 6r pffiffiffi
# 1
2 2 sech ð r nÞ pffiffiffi
2 e ið / t 1ðnÞ�x a 1 a Þ; c 1 þ sech ð r nÞ "
qðx; tÞ ¼k � 6r pffiffiffi
# ð66Þ
1
2 2 sech ð r nÞ pffiffiffi
2 e ið / t 2ðnÞ�x a 2 a Þ; c 1 þ sech ð r nÞ which characterizes bright soliton as demonstrated in Figure 4, where r > 0 and c < 0. From( 65), we obtain
l ¼ 0; 6br þ c 2 ¼ 0: ð67Þ
Case III3. If f = 1, g = 2( 1 � 2m 2), h = 1, KðfÞ ¼ sn2 ðf; mÞ
, equation( 5) secures JEF solution of the ð1cnðf; mÞÞ form 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 pðx; tÞ ¼ � 8m2 2
r r 1 � cn 2 � n e ið / t
1ðnÞ�x a 1 cð4m 2 þ 1Þ
4m 2 a Þ; þ 1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 qðx; tÞ ¼k � 8m2 2
r r 1 � cn 2 � n e ið / t
2ðnÞ�x a 2 cð4m 2 þ 1Þ
4m 2 a Þ; þ 1 ð68Þ provided that r < 0 and c > 0. The conditions( 59) turn to
lcð4m 2 þ 1Þ 2 þ 8r 2 m 2 ¼ 0; 64brm 2 þ 3c 2 ð4m 2 þ 1Þ ¼0: ð69Þ
As m approaches 1, solution( 68) changes to 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 o ð70Þ i1 qðx; tÞ ¼k � 8r 1 � sech 2 � r 2 5c 5 n e ið / t
2ðnÞ�x a 2 a Þ;
which represents dark soliton as shown in Figure 3, where r < 0 and c > 0. From( 69), we reach