J. Eur. Opt. Society-Rapid Publ. 21, 11( 2025) 111
which is an equivalent form of equation [ 66 ] ðN 0 ðfÞÞ 2 ¼ 1
4 f þ gNðfÞ2 þ hNðfÞ 4; ð29Þ
where K( f)= N( f) 2 and f, g, h are constants. The last equation is widely employed in literature and is known as the Jacobi elliptic equation because it permits solutions in terms of the Jacobi elliptic functions( JEFs) sn( f, m), cn( f, m), dn( f, m), and so on [ 67 ]. Therefore, equation( 28) has similarly different JEFs solutions [ 65, 68, 69 ]. The parameter m is the modulus of JEFs such that 0 < m < 1. It is well known, when m approaches 0 or 1, JEFs degenerate to trigonometric and hyperbolic functions. For instance, one can reach sn( f, m) = sin( n), cn( f, m) = cos( n), dn( f, m)= 1asm tends to 0 while it is found that sn( f, m) = tanh( n), cn( f, m) = sech( n), dn( f, m) = sech( n) when m approaches 1. As mentioned above, equation( 28) has abundant JEFs solutions, however, we focus only on three form of JEFs as given in Table 1.
Inserting( 27) along with( 28) into equation( 26), we arrive at a polynomial in K l( f),( l = 0,1,..., 8). Equating the coefficients of K l( f) to zero, this yields a system of algebraic equations. Solving this system of equations, we obtain four main sets that detect distinct values for the constants d j,( j = 0,1,..., 5) under specific restrictions and hence each set induces various cases of solutions according to the variety of JEFs.
Set I. d 0 ¼ d 2 ¼ d 3 ¼ 0; d 1 ¼ 2rh rffiffiffi cg; ¼ 2 r
; ð30Þ g under the constraint conditions lcg 2 þ fhr 2 ¼ 0; b ¼ 0; ð31Þ
where rg > 0. Implementing these findings to( 27) and using( 24) and( 25), the following cases of solutions to equation( 5) are derived.
Case I1. If f = 4, g = �4( 1 + m 2), h = 4m 2, K( f)= sn 2( f, m), equation( 5) secures JEF solution of the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðx; tÞ ¼ � 2rm2 sn � r n e ið / t 1ðnÞ�x a Þ 1 cðm 2 þ 1Þ m 2 a; þ 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qðx; tÞ ¼k � 2rm2 sn � r n e ið / t 2ðnÞ�x a 2 Þ cðm 2 þ 1Þ m 2 a; þ 1 ð32Þ provided that r < 0andc > 0. The conditions( 31) reduce to lcðm 2 þ 1Þ 2 þ r 2 m 2 ¼ 0; b ¼ 0: ð33Þ
As m approaches 1, solution( 32) collapses to a soliton solution given by q pðx; tÞ ¼ ffiffiffiffiffiffi pffiffiffiffiffiffi
� r c tanh � r 2 n e ið / t 1ðnÞ�x a 1 a Þ; qffiffiffiffiffiffi pffiffiffiffiffiffi
ð34Þ qðx; tÞ ¼k � r c tanh � r 2 n e ið / t 2ðnÞ�x a 2 a Þ;
Table 1. Three Jacobi elliptic solutions to equation( 28).
f g h K( f)
4 |
�4( 1 + m 2) |
4 m 2 |
sn 2( f, m) |
4( 1 � m 2) |
�4( 1 � 2m 2) |
�4 m 2 |
cn 2( f, m) |
1 2( 1 – 2 m 2) 1
which represent dark soliton as shown in Figure 1, where r < 0 and c > 0. From( 33), we arrive at
4lc þ r 2 ¼ 0; b ¼ 0: ð35Þ
Case I2. Iff = 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 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pðx; tÞ ¼ � 2rm2 r cn n e ið / t
1ðnÞ�x a 1 cð2m 2 � 1Þ 2m 2 a Þ; � 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qðx; tÞ ¼k � 2rm2 r cn n e ið / t
2ðnÞ�x a 2 cð2m 2 � 1Þ 2m 2 a Þ: � 1 ð36Þ
where r > 0, c < 0asm 2 > 1 / 2 and r < 0, c < 0as m 2 < 1 / 2. The conditions( 31) reduce to
lcð2m 2 � 1Þ 2 þ r 2 m 2 ðm 2 � 1Þ ¼0; b ¼ 0: ð37Þ
As m approaches 1, solution( 36) co llapses to a soliton solution given by qffiffiffiffiffiffiffiffi pffiffiffi pðx; tÞ ¼ � 2r sech r c ð nÞe ið / 1 ðnÞ�x t a 1 a Þ; qffiffiffiffiffiffiffiffi pffiffiffi ð38Þ qðx; tÞ ¼k sech ð r nÞe ið / 2 ðnÞ�x t a 2 a Þ;
� 2r c
which describes bright soliton as depicted in Figure 2, where r > 0 and c < 0. From( 37), we arrive at
l ¼ 0; b ¼ 0: ð39Þ Case I3. Iff = 1, g = 2( 1� 2m 2), h = 1, KðfÞ ¼ sn2 ðf; mÞ ð1cnðf; mÞÞ 2, equation( 5) admits JEF solution of the form 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
31 r 1 � cn � 2r 2 n 6
2m pðx; tÞ ¼ � 2 �1 7 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 e ið / t
1ðnÞ�x a 1 cð2m 2 a Þ; � 1Þ
1 þ cn � 2r n
2m 2 �1
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
31 r 1 � cn � 2r 2
6 n
2m qðx; tÞ ¼k � 2 �1 7 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 e ið / t
2ðnÞ�x a 2 cð2m 2 a Þ: � 1Þ
1 þ cn � 2r n
2m 2 �1 sn 2 ðf; mÞ ð1cnðf; mÞÞ 2
ð40Þ
where r < 0, c > 0asm 2 > 1 / 2 and r > 0, c > 0as m 2 < 1 / 2. The conditions( 31) reduce to
4lcð2m 2 � 1Þ 2 þ r 2 ¼ 0; b ¼ 0: ð41Þ
As m approaches 1, solution( 40) convertes to a soliton solution given by