Vol.:(0123456789) J Opt (November 2024) 53(5):4113–4136 https://doi.org/10.1007/s12596-023-01392-7 RESEARCH ARTICLE Optical solitons and conservation laws for the concatenation model in the absence of self‑phase modulation Ahmed H. Arnous1 · Anjan Biswas2,3,4,5  · Abdul H. Kara6 · Yakup Yıldırım7,8 · Carmelia Mariana Balanica Dragomir4 · Asim Asiri3  Received: 1 August 2023 / Accepted: 26 August 2023 / Published online: 5 October 2023 © The Author(s), under exclusive licence to The Optical Society of India 2023 Abstract This paper addresses the concatenation model with the absence of self-phase modulation. Two integration approaches, namely the generalized sine-Gordon equation method and the projective Riccati equation approach yielded a plethora of soliton solutions. The parameter constraints for the existence of such solitons are also presented. The conservation laws are also enumerated that are recovered with the multipliers approach. Keywords Solitons · Sine-Gordon · Riccati · Concatenation · Self-phase modulation Introduction The theory of optical solitons have made remarkable advances during the past half-a-century. The technology has advanced in unfathomable speed. There is still a lot to be achieved. Some of the factors that need improvements are the internet bottleneck effect, mitigating the noise effect and cross-talk during the soliton propagation, addressing the evolution of ghost pulses during soliton propagation and many others. There are a lot of means and measures that are continuously adopted to overcome these effects. One of the new models that has been recently proposed is the conjunction of three well-known equation that dictate the propagation of solitons through optical fibers. These are the nonlinear Schrödinger’s equation (NLSE), Laksh- manan–Porsezian–Daniel (LPD) equation and the Sasa–Sat- suma equation (SSE). This is referred to as the concatenation model that was first conceived in 2014 [1, 2]. Later, a deluge of results have emerged and reported across the board. These range from the bifurcation analysis, numerical study of such solitons that emerged from the model, the Painleve analysis, conservation laws, quiescent solitons as well as studying the soliton dynamics with power-law of nonlinearity [3–14]. Subsequently, the model was extended to study it in birefrin- gent fibers [6]. It is now time to move further on. The current paper takes a look at the concatenation model that comes with the absence of self-phase modulation (SPM). Two integration approaches give way to a full spectrum of solitons that are enumerated in the paper. They are the general- ized sine-Gordon equation approach and the projective Riccati equation method. The parameter constraints for the existence of such solitons also naturally fell out from the two schemes. Finally, the multipliers approach also led to the retrieval of the conservation laws, and the conserved quantities are computed with the usage of the bright solitons that emerged from the * Anjan Biswas biswas.anjan@gmail.com 1 Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt 2 Department of Mathematics and Physics, Grambling State University, Grambling, LA 71245-2715, USA 3 Mathematical Modeling and Applied Computation (MMAC) Research Group, Center of Modern Mathematical Sciences and their Applications (CMMSA), Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia 4 Department of Applied Sciences, Cross-Border Faculty of Humanities, Economics and Engineering, Dunarea de Jos University of Galati, 111 Domneasca Street, Galati 800201, Romania 5 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa 0204, South Africa 6 School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits, Johannesburg 2050, South Africa 7 Department of Computer Engineering, Biruni University, Istanbul 34010, Turkey 8 Department of Mathematics, Near East University, 99138 Nicosia, Cyprus http://crossmark.crossref.org/dialog/?doi=10.1007/s12596-023-01392-7&domain=pdf http://orcid.org/0000-0002-8131-6044 4114 J Opt (November 2024) 53(5):4113–4136 two integration schemes. The details are exhibited in the rest of the paper after a short ride through the introductory process. Governing Model The concatenation model in the absence of SPM is: Equation (1) is the dimensionless form of the concatena- tion model that will be studied in this paper. Here in Eq. 4, q(x, t) is the complex-valued function that represents the wave profile, where x and t are the two independent vari- ables that represents the spatial and temporal coordinates, respectively. Also i = √ −1 . The first two terms in Eq. (1) is the NLSE with the missing SPM and the coefficient of a being the chromatic dispersion (CD). The first term is from the linear temporal evolution. The coefficient of c1 comes from LPD equation, while the coefficient of c2 comes from SSE. Thus, Eq. (1) is a concatenation of NLSE, LPD model and the SSE. This equation that will be studied in the rest of the paper to fetch its soliton solutions. Assume the solution structure of Eq. (1) as follows: where the wave variable � is given by Here, U(�) represents the amplitude component of the soliton solution and v is the speed of the soliton, while the phase component �(x, t) is defined as where � is the frequency of the solitons, while � represents the wave number, and �0 is the phase constant. Substituting (2) into (1) and then decomposing into real and imaginary parts and From the imaginary part, the soliton speed reaches (1) iqt + aqxx + c1 [ �1qxxxx + �2 ( qx )2 q∗ + �3 ||qx||2q + �4|q|2qxx + �5q 2q∗ xx + �6|q|4q ] + ic2 [ �7qxxx + �8|q|2qx + �9q 2q∗ x ] = 0. (2)q(x, t) = U(�)ei�(x,t), (3)� = k(x − vt). (4)�(x, t) = −�x + �t + �0, (5) − k2 ( a − 6c1� 2�1 + 3c2��7 ) U�� + ( a�2 − c1� 4�1 + c2� 3�7 + � ) U − c1k 4�1U ���� − c1k 2 ( �4 + �5 ) U2U�� − c1k 2 ( �2 + �3 ) UU�2 + � ( c1� ( �2 − �3 + �4 + �5 ) + c2 ( �9 − �8 )) U3 − c1�6U 5 = 0, (6) k ( 2a� − 4c1� 3�1 + 3c2� 2�7 + v ) U� + k3 ( 4c1��1 − c2�7 ) U���+ k ( 2c1� ( �2 + �4 − �5 ) − c2 ( �8 + �9 )) U2U� = 0. while the wave number reads with parametric restriction Equation (5) can be simplified as where An overview of the integration algorithms Consider a governing model where u = u(x, t) denotes a wave profile, while t and x depict the time and space variables in sequence. The relations condense Eq. (12) to (7)v = −2a� + 4c1� 3�1 − 3c2� 2�7, (8)� = c2�7 4c1�1 , (9)2c1� ( �2 + �4 − �5 ) − c2 ( �8 + �9 ) = 0. (10) k2U���� + s6U 2U�� + s5U �� + s4UU�2 + s3U 5 + s2U 3 + s1U = 0. (11) ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ s1 = − a�2 − c1� 4�1 + c2� 3�7 + � c1k 2�1 , s2 = − � � c1� � �2 − �3 + �4 + �5 � + c2 � �9 − �8 �� c1k 2�1 , s3 = �6 k2�1 , s4 = �2 + �3 �1 , s5 = a − 6c1� 2�1 + 3c2��7 c1�1 , s6 = �4 + �5 �1 . (12)F(u, ux, ut, uxt, uxx, ...) = 0, (13)u(x, t) = U(�), � = k(x − �t), (14)P(U,−k�U�, kU�, k2U��, ...) = 0, 4115J Opt (November 2024) 53(5):4113–4136 where k is the wave width, � is the wave variable, and � is the wave velocity. Generalized sine‑Gordon equation method The performance steps for the generalized sine-Gordon equation method are: Step 1  Assume Eq. (14) has the formal solution along with the general sine-Gordon travelling wave reduc- tion equation Step 2  Equation (16) possesses the following cases: Case‑1: � = 0, � = 1 which gives the hyperbolic function solutions or Case‑2: � = 1, � = −m2 which gives the Jacobi’s elliptic function solutions or Case‑3: � = m2, � = −1 which gives the Jacobi’s elliptic function solutions or where i = √ −1. (15) U(�) = A0 + N∑ j=1 cos[G(�)]j−1 ( Aj sin[G(�)] + Bj cos[G(�)] ) , (16)G�(�) = √ � + � sin[G(�)]2. (17)G�(�) = ∓ sin[G(�)], (18)sin[G(�)] = sech[�], and cos[G(�)] = ± tanh[�], (19)sin[G(�)] = ±i csch[�], and cos[G(�)] = ± coth[�]. (20)G�(�) = √ 1 − m2 sin[G(�)]2, (21)sin[G(�)] = sn[�;m], and cos[G(�)] = cn[�;m], (22) sin[G(�)] = 1 m ns[�;m], and cos[G(�)] = − i m ds[�;m]. (23)G�(�) = √ m2 − sin[G(�)]2, (24)sin[G(�)] = m sn[�;m], and cos[G(�)] = dn[�;m], (25)sin[G(�)] = ns[�;m], and cos[G(�)] = −i cs[�;m], Step 3  Substituting Eq. (15) along with Eq. (16) into Eq. (14), we get a polynomial in sin[G(�)] and cos[G(�)] which equal to zero. The obtained coefficients of this polynomial give the needed parameters in Eq. (13) and Eq. (15). Remark 1 The convergence of Jacobi’s elliptic functions toward their hyperbolic function limits as m tends to unity is listed as Projective Riccati equation method The algorithmic process of the projective Riccati equation’s method is as follows: Step 1  Assume Eq. (14) has the formal solution where �(�) and �(�) satisfy the following ODEs: with where r is constant and N a positive integer comes from the balancing principle in Eq. (14). Also, a0, ai and bi(i = 0, 1, ...,N) are constants. Step 2  The solutions of Eq.  (28) are listed as follows: Case‑1:  R(r) = 0 or Case‑2:  R(r) = 24 25 r2 (26) lim m→1 cn[�;m] = lim m→1 dn[�;m] = sech[�], lim m→1 sn[�;m] = tanh[�], lim m→1 cs[�;m] = lim m→1 ds[�;m] = csch[�], lim m→1 ns[�;m] = coth[�]. (27)U(�) = a0 + N∑ i=1 � i−1(�) ( ai�(�) + bi�(�) ) , (28) � �(�) = −�(�)�(�), ��(�) = 1 − �2(�) − r�(�), (29)�(�)2 = 1 − 2r�(�) + R(r)�(�)2, (30)�(�) = 1 2r sech 2 [ � 2 ] , and �(�) = tanh [ � 2 ] , (31)�(�) = − 1 2r csch 2 [ � 2 ] , and �(�) = coth [ � 2 ] . 4116 J Opt (November 2024) 53(5):4113–4136 Case‑3:  R(r) = 5 9 r2 Case‑4:  R(r) = r2 − 1 or Case‑5:  R(r) = r2 + 1 Step 3:  Inserting Eq. (27) along with Eq. (28) and Eq. (29) into Eq. (14), we get a polynomial of �(�) and �(�) which (32)�(�) = 1 r 5 sech[�] 5 sech[�] ± 1 , and �(�) = tanh[�] 1 ± 5 sech[�] . (33) �(�) = 1 r 3 sech[�] 3 sech[�] ± 2 , and �(�) = 2 2 coth[�] ± 3 csch[�] . (34)�(�) = 4 sech[�] 3 tanh[�] + 4r sech[�] + 5 , and �(�) = 5 tanh[�] + 3 3 tanh[�] + 4r sech[�] + 5 , (35)�(�) = sech[�] r sech[�] + 1 , and �(�) = tanh[�] r sech[�] + 1 . (36)�(�) = csch[�] r csch[�] + 1 , and �(�) = coth[�] r csch[�] + 1 . equal to zero. The obtained coefficients of this polynomial give the needed parameters in Eq. (13) and Eq. (27). Optical solitons The current section will apply the two integration algorithms to derive the soliton solutions to the concatenation model that is with no SPM. The subsequent two subsections detail the derivation. The generalized sine‑Gordon equation method Balancing U′′′′ with U5 in Eq. (10) gives N = 1 ; accordingly, the solution takes the form Inserting Eq. (37) together with Eq. (28) and Eq. (29) into Eq. (10), we get a system of algebraic equations (37)U(�) = A0 + A1 sin[G(�)] + B1 cos[G(�)]. (38)A1 ( −A2 1 ( 10B2 1 s3 + � ( s4 + 2s6 )) + A4 1 s3 + 3B2 1 � ( s4 + 2s6 ) + 5B4 1 s3 + 24k2�2 ) = 0, (39)2A0A1B1 ( −10A2 1 s3 + 10B2 1 s3 + � ( s4 + 4s6 )) = 0, (40) A1 ( A2 1 ( −10A2 0 s3 + 10B2 1 s3 + �s4 + �s6 + �s4 + 3�s6 − s2 ) + 2� ( A2 0 s6 − 2k2(5� + 7�) + s5 ) + B2 1 ( 30A2 0 s3 − 3�s4 − 3�s6 − 4�s4 − 5�s6 + 3s2 ) − 2A4 1 s3 ) = 0, (41) 2A0A1B1 ( 10A2 0 s3 + 10A2 1 s3 − �s4 − 2�s6 − �s4 − 3�s6 + 3s2 ) = 0, (42) A1 ( A2 0 ( 10A2 1 s3 − s6(� + �) + 3s2 ) − A2 1 �s6 − A2 1 �s6 + 5A4 0 s3 + A2 1 s2 + A4 1 s3 + B2 1 �s4 + B2 1 �s4 + k2�2 + 6k2�� + 5k2�2 − �s5 − �s5 + s1 ) = 0, (43)B1 ( −A2 1 ( 10B2 1 s3 + 3� ( s4 + 2s6 )) + 5A4 1 s3 + B2 1 � ( s4 + 2s6 ) + B4 1 s3 + 24k2�2 ) = 0, (44)A0 ( −A2 1 ( 30B2 1 s3 + � ( s4 + 4s6 )) + 5A4 1 s3 + B2 1 � ( s4 + 4s6 ) + 5B4 1 s3 ) = 0, (45) B2 1 ( 10A2 0 s3 − ( s4 + s6 ) (� + 2�) + s2 ) + 2� ( A2 0 s6 − 10k2(� + 2�) + s5 ) − 10A4 1 s3 + B1 ( A2 1 ( −30A2 0 s3 + 10B2 1 s3 + 3�s4 + 3�s6 + 5�s4 + 10�s6 − 3s2 )) = 0, 4117J Opt (November 2024) 53(5):4113–4136 Solving these equations together yields the following results. Case‑1: Result‑1: Plugging the obtained parameters in Eq. (49) with Eq. (18) or Eq. (19) into Eq. (37), as a consequence, we get dark and singular solitons with 6s5 − 5s1 > 0 , 2s2 + s4 − −4s6 > 0, and 2s5 ( s2 − s4 − −2s6 ) + s1 ( 2 ( s4 + s6 ) − s2 ) > 0 , a s shown below or Result‑2: (46) A0 ( A2 1 ( −10A2 0 s3 + 30B2 1 s3 + �s4 + 2�s6 + �s4 + 6�s6 − 3s2 ) + B2 1 ( 10A2 0 s3 − ( s4 + 2s6 ) (� + 2�) + 3s2 ) − 10A4 1 s3 ) = 0, (47) B1 ( A2 0 ( 30A2 1 s3 − s6(� + 2�) + 3s2 ) − 2A2 1 �s4 − 3A2 1 �s6 − 2A2 1 �s4 − 4A2 1 �s6 + 5A4 0 s3 + 3A2 1 s2 + 5A4 1 s3 + B2 1 �s4 + B2 1 �s4 + k2�2 + 16k2�� + 16k2�2 − �s5 − 2�s5 + s1 ) = 0, (48)A0 ( −2A2 1 �s6 − 2A2 1 �s6 + A4 0 s3 + A2 0 ( 10A2 1 s3 + s2 ) + 3A2 1 s2 + 5A4 1 s3 + B2 1 �s4 + B2 1 �s4 + s1 ) = 0. (49) A0 = A1 = 0, B1 = ± √ 6s5 − 5s1 2s2 + s4 − 4s6 , k = 1 2 √√√√2s5 ( s2 − s4 − 2s6 ) + s1 ( 2 ( s4 + s6 ) − s2 ) 2 ( 2s2 + s4 − 4s6 ) s3 = ( 2s2 + s4 − 4s6 )( s1 ( 3s2 − s4 + 4s6 ) − 6s2s5 ) ( 5s1 − 6s5 )2 . (50) q(x, t) = ± � 6s5 − 5s1 2s2 + s4 − 4s6 tanh ⎡ ⎢⎢⎣ 1 2 ����2s5 � s2 − s4 − 2s6 � + s1 � 2 � s4 + s6 � − s2 � 2 � 2s2 + s4 − 4s6 � (x − vt) ⎤⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � , (51) q(x, t) = ± � 6s5 − 5s1 2s2 + s4 − 4s6 coth ⎡ ⎢⎢⎣ 1 2 ����2s5 � s2 − s4 − 2s6 � + s1 � 2 � s4 + s6 � − s2 � 2 � 2s2 + s4 − 4s6 � (x − vt) ⎤⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (52) A0 = B1 = 0, A1 = ± √ − 2 ( 10s1 + 9s5 ) s2 + s4 + s6 , k = √ − ( s1 + s5 ) , s3 = ( 3s5 ( 4s2 + s4 − 2s6 ) + 2s1 ( 6s2 + s4 − 4s6 ))( s2 + s4 + s6 ) 2 ( 10s1 + 9s5 )2 . Substituting the parameters acquired in Eq. (52) with Eq. (18) or Eq. (19) into Eq. (37), as a consequence, we get bright soliton with 2 ( 10s1 + 9s5 ) (s2 + s4 + s6) < 0 , s1 + s5 < 0, and singular soliton with 2 ( 10s1 + 9s5 ) (s2 + s4 + s6) > 0 , s1 + s5 < 0 , as presented below 4118 J Opt (November 2024) 53(5):4113–4136 or Result‑3: (53)q(x, t) = ± √ − 2 ( 10s1 + 9s5 ) s2 + s4 + s6 sech [√ − ( s1 + s5 ) (x − vt) ] e i ( − { c2�7 4c1�1 } x+�t+�0 ) , (54)q(x, t) = ± √ 2 ( 10s1 + 9s5 ) s2 + s4 + s6 csch [√ − ( s1 + s5 ) (x − vt) ] e i ( − { c2�7 4c1�1 } x+�t+�0 ) . (55) A0 = 0, A1 = ± √ 20s1 − 6s5 8s2 + s4 − 4s6 , B1 = ± √ 6s5 − 20s1 8s2 + s4 − 4s6 , k = √ 4s1 ( −2s2 + s4 + s6 ) + s5 ( 4s2 − s4 − 2s6 ) 8s2 + s4 − 4s6 , s3 = ( 8s2 + s4 − 4s6 )( s1 ( 12s2 − s4 + 4s6 ) − 6s2s5 ) 4 ( 10s1 − 3s5 )2 . Putting the derived parameters in  Eq. (55) with  Eq. (18) or Eq. (19) into Eq. (37), as a consequence, we get a complexiton, dark soliton and singular soliton solu- tions with 6s5 − −20s1 > 0 , 8s2 + s4 − −4s6 > 0 and 4s1 ( −2s2 + s4 + s6 ) + s5 ( 4s2 − s4 − −2s6 ) > 0 , as given below or and (56) q(x, t) = ± � 6s5 − 20s1 8s2 + s4 − 4s6 � tanh ⎡ ⎢⎢⎣ � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ ± i sech ⎡⎢⎢⎣ � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ � e i � − � c2�7 4c1�1 � x+�t+�0 � , (57) q(x, t) = ± � 6s5 − 20s1 8s2 + s4 − 4s6 tanh ⎡ ⎢⎢⎣ 1 2 � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � , (58) q(x, t) = ± � 6s5 − 20s1 8s2 + s4 − 4s6 coth ⎡ ⎢⎢⎣ 1 2 � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . 4119J Opt (November 2024) 53(5):4113–4136 Case‑2: Result‑1: Inserting the parameters attained in Eq. (59) with Eq. (21) and Eq. (22) into Eq. (37), as a consequence, we get Jacobi’s elliptic-type function solution When the modulus of ellipticity approaches unity, we get a bright soliton solution with 2 ( 10s1 + 9s5 ) (s2 + s4 + s6) < 0, and s1 + s5 < 0 , as given below Also, we have the following Jacobi’s elliptic-type function solution (59) A0 = A1 = 0,B1 = ± √√√√ 2m2 ( 10 ( 2m2 − 1 ) s1 + 3 ( 8m4 − 8m2 + 3 ) s5 ) ( −16m4 + 16m2 − 1 ) s2 + ( 2m2 − 1 )(( 4m4 − 4m2 − 1 ) s4 + ( −16m4 + 16m2 − 1 ) s6 ) , k = √√√√√ s1 (( 2m2 − 1 )( s4 + s6 ) + s2 ) + s5 (( 1 − 2m2 )2 s6 + ( 2m2 − 1 ) s2 + ( 2m4 − 2m2 + 1 ) s4 ) ( −16m4 + 16m2 − 1 ) s2 + ( 2m2 − 1 )(( 4m4 − 4m2 − 1 ) s4 + ( −16m4 + 16m2 − 1 ) s6 ) , s3 = − 1 2 ( 10 ( 2m2 − 1 ) s1 + 3 ( 8m4 − 8m2 + 3 ) s5 )2 × ( 3s5 (( 8m2 − 4 ) s2 + s4 − 2s6 ) + 2s1 (( 2m2 − 1 )( s4 − 4s6 ) + 6s2 )) × (( 1 − 2m2 ( 3 − 4m2 )2) s6 + ( −16m4 + 16m2 − 1 ) s2 + ( 8m6 − 12m4 + 2m2 + 1 ) s4 ) . (60) q(x, t) = ± ���� 2m2 � 10 � 2m2 − 1 � s1 + 3 � 8m4 − 8m2 + 3 � s5 � � −16m4 + 16m2 − 1 � s2 + � 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + � −16m4 + 16m2 − 1 � s6 � cn ⎡⎢⎢⎢⎣ ����� s1 �� 2m2 − 1 �� s4 + s6 � + s2 � + s5 �� 1 − 2m2 �2 s6 + � 2m2 − 1 � s2 + � 2m4 − 2m2 + 1 � s4 � � −16m4 + 16m2 − 1 � s2 + � 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + � −16m4 + 16m2 − 1 � s6 � (x − vt);m ⎤⎥⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (61)q(x, t) = ± √ − 2 ( 10s1 + 9s5 ) s2 + s4 + s6 sech [√ − ( s1 + s5 ) (x − vt) ] e i ( − { c2�7 4c1�1 } x+�t+�0 ) . (62) q(x, t) = ∓ ����− 2 � 10 � 2m2 − 1 � s1 + 3 � 8m4 − 8m2 + 3 � s5 � � −16m4 + 16m2 − 1 � s2 + � 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + � −16m4 + 16m2 − 1 � s6 � ds ⎡⎢⎢⎢⎣ ����� s1 �� 2m2 − 1 �� s4 + s6 � + s2 � + s5 �� 1 − 2m2 �2 s6 + � 2m2 − 1 � s2 + � 2m4 − 2m2 + 1 � s4 � � −16m4 + 16m2 − 1 � s2 + � 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + � −16m4 + 16m2 − 1 � s6 � (x − vt);m ⎤⎥⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . When the modulus of e l l ipt ic i ty approaches unity, we get a singular soliton solution with 4120 J Opt (November 2024) 53(5):4113–4136 2 ( 10s1 + 9s5 ) (s2 + s4 + s6) > 0, and s1 + s5 < 0 , as indi- cated below Result‑2: Plugging the parameters secured in Eq. (64) with Eq. (21) and Eq. (22) into Eq. (37), as a consequence, we get Jacobi’s elliptic-type function solution When the modulus of ellipticity approaches unity, we get a dark soliton solution with 6s5 − 5s1 > 0, 2s2 + s4 − −4s6, and 2s5 ( s2 − s4 − −2s6 ) + s1 ( 2 ( s4 + s6 ) − s2 ) > 0 , a s described below (63)q(x, t) = ± √ 2 ( 10s1 + 9s5 ) s2 + s4 + s6 csch [√ − ( s1 + s5 ) (x − vt) ] e i ( − { c2�7 4c1�1 } x+�t+�0 ) . (64) A0 = B1 = 0,A1 = ± √√√√− 2m2 ( 3 ( 3m4 + 2m2 + 3 ) s5 − 10 ( m2 + 1 ) s1 ) ( m2 + 1 )(( m4 − 6m2 + 1 ) s4 + ( m4 + 14m2 + 1 ) s6 ) − ( m4 + 14m2 + 1 ) s2 , k = √√√√√ s1 ( s2 − ( m2 + 1 )( s4 + s6 )) + s5 (( m4 + 1 ) s4 + ( m2 + 1 )2 s6 − ( m2 + 1 ) s2 ) ( m2 + 1 )(( m4 − 6m2 + 1 ) s4 + ( m4 + 14m2 + 1 ) s6 ) − ( m4 + 14m2 + 1 ) s2 , s3 = − 1 2 ( 10 ( m2 + 1 ) s1 − 3 ( 3m4 + 2m2 + 3 ) s5 )2 × 3s5 (( m2 − 1 )2( s4 − 2s6 ) − 4 ( m2 + 1 ) s2 ) + 2s1 ( 6s2 − ( m2 + 1 )( s4 − 4s6 )) × ( m2 + 1 )(( m4 − 6m2 + 1 ) s4 + ( m4 + 14m2 + 1 ) s6 ) − ( m4 + 14m2 + 1 ) s2. (65) q(x, t) = ± ����− 2m2 � 3 � 3m4 + 2m2 + 3 � s5 − 10 � m2 + 1 � s1 � � m2 + 1 ��� m4 − 6m2 + 1 � s4 + � m4 + 14m2 + 1 � s6 � − � m4 + 14m2 + 1 � s2 sn ⎡⎢⎢⎢⎣ ����� s1 � s2 − � m2 + 1 �� s4 + s6 �� + s5 �� m4 + 1 � s4 + � m2 + 1 �2 s6 − � m2 + 1 � s2 � � m2 + 1 ��� m4 − 6m2 + 1 � s4 + � m4 + 14m2 + 1 � s6 � − � m4 + 14m2 + 1 � s2 (x − vt);m ⎤⎥⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (66) q(x, t) = ± � 6s5 − 5s1 2s2 + s4 − 4s6 tanh ⎡ ⎢⎢⎣ 1 2 ����2s5 � s2 − s4 − 2s6 � + s1 � 2 � s4 + s6 � − s2 � 2 � 2s2 + s4 − 4s6 � (x − vt) ⎤⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . Also, we have the following Jacobi’s elliptic-type function solution 4121J Opt (November 2024) 53(5):4113–4136 When the modulus of ellipticity approaches unity, we get a singular soliton solution with 6s5 − 5s1 > 0, 2s2 + s4 − −4s6, and 2s5 ( s2 − s4 − −2s6 ) + s1 ( 2 ( s4 + s6 ) − s2 ) > 0 , a s shown below Result‑3: (67) q(x, t) = ± ����− 2 � 3 � 3m4 + 2m2 + 3 � s5 − 10 � m2 + 1 � s1 � � m2 + 1 ��� m4 − 6m2 + 1 � s4 + � m4 + 14m2 + 1 � s6 � − � m4 + 14m2 + 1 � s2 ns ⎡ ⎢⎢⎢⎣ ����� s1 � s2 − � m2 + 1 �� s4 + s6 �� + s5 �� m4 + 1 � s4 + � m2 + 1 �2 s6 − � m2 + 1 � s2 � � m2 + 1 ��� m4 − 6m2 + 1 � s4 + � m4 + 14m2 + 1 � s6 � − � m4 + 14m2 + 1 � s2 (x − vt);m ⎤ ⎥⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (68) q(x, t) = ± � 6s5 − 5s1 2s2 + s4 − 4s6 coth ⎡ ⎢⎢⎣ 1 2 ����2s5 � s2 − s4 − 2s6 � + s1 � 2 � s4 + s6 � − s2 � 2 � 2s2 + s4 − 4s6 � (x − vt) ⎤⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (69) A0 =, A1 = ± √√√√− 2m2 ( 10 ( m2 − 2 ) s1 + 3 ( m4 − 6m2 + 6 ) s5 ) ( m2 − 2 )(( m4 + 4m2 − 4 ) s4 − 4 ( m4 − m2 + 1 ) s6 ) − 8 ( m4 − m2 + 1 ) s2 , B1 = ± √√√√ 2m2 ( 10 ( m2 − 2 ) s1 + 3 ( m4 − 6m2 + 6 ) s5 ) ( m2 − 2 )(( m4 + 4m2 − 4 ) s4 − 4 ( m4 − m2 + 1 ) s6 ) − 8 ( m4 − m2 + 1 ) s2 , k = √√√√√4s1 (( m2 − 2 )( s4 + s6 ) + 2s2 ) + s5 ( 2 ( m2 − 2 )2 s6 + 4 ( m2 − 2 ) s2 + ( m4 − 8m2 + 8 ) s4 ) ( m2 − 2 )(( m4 + 4m2 − 4 ) s4 − 4 ( m4 − m2 + 1 ) s6 ) − 8 ( m4 − m2 + 1 ) s2 , s3 = − 1 4 ( 10 ( m2 − 2 ) s1 + 3 ( m4 − 6m2 + 6 ) s5 )2 × 3s5 ( 2 ( m2 − 2 ) s2 − ( m2 − 1 )( s4 − 2s6 )) + s1 (( m2 − 2 )( s4 − 4s6 ) + 12s2 ) × 8 ( m4 − m2 + 1 ) s2 − ( m2 − 2 )(( m4 + 4m2 − 4 ) s4 − 4 ( m4 − m2 + 1 ) s6 ) . Substituting the parameters achieved in Eq. (69) with Eq. (21) and Eq. (22) into Eq. (37), as a consequence, we get Jacobi’s elliptic-type function solution 4122 J Opt (November 2024) 53(5):4113–4136 When the modulus of e l l ipt ic i ty approaches un i ty, we get a complexi ton so l i ton so lu- t i o n w i t h 6s5 − 5s1 > 0, 2s2 + s4 − −4s6, a n d 2s5 ( s2 − s4 − −2s6 ) + s1 ( 2 ( s4 + s6 ) − s2 ) > 0 , as presented below Also, we have the following Jacobi’s elliptic-type function solution (70) q(x, t) = ± ���� 2m2 � 10 � m2 − 2 � s1 + 3 � m4 − 6m2 + 6 � s5 � � m2 − 2 ��� m4 + 4m2 − 4 � s4 − 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 � cn ⎡ ⎢⎢⎢⎣ �����4s1 �� m2 − 2 �� s4 + s6 � + 2s2 � + s5 � 2 � m2 − 2 �2 s6 + 4 � m2 − 2 � s2 + � m4 − 8m2 + 8 � s4 � � m2 − 2 ��� m4 + 4m2 − 4 � s4 − 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 (x − vt);m ⎤ ⎥⎥⎥⎦ + i sn ⎡ ⎢⎢⎢⎣ �����4s1 �� m2 − 2 �� s4 + s6 � + 2s2 � + s5 � 2 � m2 − 2 �2 s6 + 4 � m2 − 2 � s2 + � m4 − 8m2 + 8 � s4 � � m2 − 2 ��� m4 + 4m2 − 4 � s4 − 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 (x − vt);m ⎤ ⎥⎥⎥⎦ � × e i � − � c2�7 4c1�1 � x+�t+�0 � . (71) q(x, t) = � 6s5 − 20s1 8s2 + s4 − 4s6 � − tanh ⎡ ⎢⎢⎣ � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ + i sech ⎡⎢⎢⎣ � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ � e i � − � c2�7 4c1�1 � x+�t+�0 � . (72) q(x, t) = ± ����− 2 � 10 � m2 − 2 � s1 + 3 � m4 − 6m2 + 6 � s5 � � m2 − 2 ��� m4 + 4m2 − 4 � s4 − 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 � ns ⎡⎢⎢⎢⎣ �����4s1 �� m2 − 2 �� s4 + s6 � + 2s2 � + s5 � 2 � m2 − 2 �2 s6 + 4 � m2 − 2 � s2 + � m4 − 8m2 + 8 � s4 � � m2 − 2 ��� m4 + 4m2 − 4 � s4 − 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 (x − vt);m ⎤ ⎥⎥⎥⎦ − ds ⎡⎢⎢⎢⎣ �����4s1 �� m2 − 2 �� s4 + s6 � + 2s2 � + s5 � 2 � m2 − 2 �2 s6 + 4 � m2 − 2 � s2 + � m4 − 8m2 + 8 � s4 � � m2 − 2 ��� m4 + 4m2 − 4 � s4 − 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 (x − vt);m ⎤⎥⎥⎥⎦ � × e i � − � c2�7 4c1�1 � x+�t+�0 � . When the modulus of ellipticity approaches unity, we get dark soliton solution with 6s5 − 5s1 > 0, 2s2 + s4 − −4s6, and 2s5 ( s2 − s4 − −2s6 ) + s1 ( 2 ( s4 + s6 ) − s2 ) > 0 , as given below 4123J Opt (November 2024) 53(5):4113–4136 Case‑3: Result‑1: (73) q(x, t) = ± � 6s5 − 20s1 8s2 + s4 − 4s6 tanh ⎡ ⎢⎢⎣ 1 2 � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤ ⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (74) A0 = A1 = 0,B1 = ± √√√√ 6 ( 3m4 − 8m2 + 8 ) s5 − 20 ( m2 − 2 ) s1( m2 − 2 )(( m4 + 4m2 − 4 ) s4 + ( m4 − 16m2 + 16 ) s6 ) − ( m4 − 16m2 + 16 ) s2 , k = √√√√√ s1 ( s2 − ( m2 − 2 )( s4 + s6 )) + s5 (( m2 − 2 )2 s6 − ( m2 − 2 ) s2 + ( m4 − 2m2 + 2 ) s4 ) ( m2 − 2 )(( m4 + 4m2 − 4 ) s4 + ( m4 − 16m2 + 16 ) s6 ) − ( m4 − 16m2 + 16 ) s2 , s3 = − 1 2 ( 10 ( m2 − 2 ) s1 − 3 ( 3m4 − 8m2 + 8 ) s5 )2 × ( 2s1 ( 6s2 − ( m2 − 2 )( s4 − 4s6 )) + 3s5 ( m4 ( s4 − 2s6 ) − 4 ( m2 − 2 ) s2 )) × (( m2 − 2 )(( m4 + 4m2 − 4 ) s4 + ( m4 − 16m2 + 16 ) s6 ) − ( m4 − 16m2 + 16 ) s2 ) . Putting the obtained parameters in Eq. (74) with Eq. (24) and Eq. (25) into Eq. (37), as a consequence, we get Jacobi’s elliptic-type function solution When the modulus of ellipticity approaches unity, we get a bright soliton solution (75) q(x, t) = ± ���� 6 � 3m4 − 8m2 + 8 � s5 − 20 � m2 − 2 � s1� m2 − 2 ��� m4 + 4m2 − 4 � s4 + � m4 − 16m2 + 16 � s6 � − � m4 − 16m2 + 16 � s2 � dn ⎡⎢⎢⎢⎣ ����� s1 � s2 − � m2 − 2 �� s4 + s6 �� + s5 �� m2 − 2 �2 s6 − � m2 − 2 � s2 + � m4 − 2m2 + 2 � s4 � � m2 − 2 ��� m4 + 4m2 − 4 � s4 + � m4 − 16m2 + 16 � s6 � − � m4 − 16m2 + 16 � s2 (x − vt);m ⎤⎥⎥⎥⎦ � × e i � − � c2�7 4c1�1 � x+�t+�0 � . (76)q(x, t) = ± √ − 2 ( 10s1 + 9s5 ) s2 + s4 + s6 sech [√ − ( s1 + s5 ) (x − vt) ] e i ( − { c2�7 4c1�1 } x+�t+�0 ) . 4124 J Opt (November 2024) 53(5):4113–4136 Also, we have the following Jacobi’s elliptic-type function solution When the modulus of ellipticity approaches unity, we get a singular soliton solution Result‑2: (77) q(x, t) = ∓ ����− 6 � 3m4 − 8m2 + 8 � s5 − 20 � m2 − 2 � s1� m2 − 2 ��� m4 + 4m2 − 4 � s4 + � m4 − 16m2 + 16 � s6 � − � m4 − 16m2 + 16 � s2 � cs ⎡ ⎢⎢⎢⎣ ����� s1 � s2 − � m2 − 2 �� s4 + s6 �� + s5 �� m2 − 2 �2 s6 − � m2 − 2 � s2 + � m4 − 2m2 + 2 � s4 � � m2 − 2 ��� m4 + 4m2 − 4 � s4 + � m4 − 16m2 + 16 � s6 � − � m4 − 16m2 + 16 � s2 (x − vt);m ⎤ ⎥⎥⎥⎦ � × e i � − � c2�7 4c1�1 � x+�t+�0 � . (78)q(x, t) = ∓ √ 2 ( 10s1 + 9s5 ) s2 + s4 + s6 csch [√ − ( s1 + s5 ) (x − vt) ] e i ( − { c2�7 4c1�1 } x+�t+�0 ) . (79) A0 = B1 = 0,A1 = ± √√√√ 20 ( m2 + 1 ) s1 − 6 ( 3m4 + 2m2 + 3 ) s5( m2 + 1 )(( m4 − 6m2 + 1 ) s4 + ( m4 + 14m2 + 1 ) s6 ) − ( m4 + 14m2 + 1 ) s2 , k = √√√√√ s1 ( s2 − ( m2 + 1 )( s4 + s6 )) + s5 (( m4 + 1 ) s4 + ( m2 + 1 )2 s6 − ( m2 + 1 ) s2 ) ( m2 + 1 )(( m4 − 6m2 + 1 ) s4 + ( m4 + 14m2 + 1 ) s6 ) − ( m4 + 14m2 + 1 ) s2 , s3 = − 1 2 ( 10 ( m2 + 1 ) s1 − 3 ( 3m4 + 2m2 + 3 ) s5 )2 × ( 3s5 (( m2 − 1 )2( s4 − 2s6 ) − 4 ( m2 + 1 ) s2 ) + 2s1 ( 6s2 − ( m2 + 1 )( s4 − 4s6 ))) × (( m2 + 1 )(( m4 − 6m2 + 1 ) s4 + ( m4 + 14m2 + 1 ) s6 ) − ( m4 + 14m2 + 1 ) s2 ) . Inserting the parameters acquired in Eq. (79) with Eq. (24) and Eq. (25) into Eq. (37), as a consequence, we get Jacobi’s elliptic-type function solution When the modulus of ellipticity approaches unity, we get dark soliton solution (80) q(x, t) = ± ���� m2 � 20 � m2 + 1 � s1 − 6 � 3m4 + 2m2 + 3 � s5 � � m2 + 1 ��� m4 − 6m2 + 1 � s4 + � m4 + 14m2 + 1 � s6 � − � m4 + 14m2 + 1 � s2 � sn ⎡⎢⎢⎢⎣ ����� s1 � s2 − � m2 + 1 �� s4 + s6 �� + s5 �� m4 + 1 � s4 + � m2 + 1 �2 s6 − � m2 + 1 � s2 � � m2 + 1 ��� m4 − 6m2 + 1 � s4 + � m4 + 14m2 + 1 � s6 � − � m4 + 14m2 + 1 � s2 (x − vt);m ⎤⎥⎥⎥⎦ � × e i � − � c2�7 4c1�1 � x+�t+�0 � . 4125J Opt (November 2024) 53(5):4113–4136 Also, we have the following Jacobi’s elliptic-type function solution (81) q(x, t) = ± � 6s5 − 5s1 2s2 + s4 − 4s6 tanh ⎡ ⎢⎢⎣ 1 2 ���� s5 � −2s2 + 2s4 + 4s6 � + s1 � s2 − 2 � s4 + s6 �� 2 � 4s6 − s4 � − 4s2 (x − vt) ⎤ ⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (82) q(x, t) = ± ���� 20 � m2 + 1 � s1 − 6 � 3m4 + 2m2 + 3 � s5� m2 + 1 ��� m4 − 6m2 + 1 � s4 + � m4 + 14m2 + 1 � s6 � − � m4 + 14m2 + 1 � s2 � ns ⎛ ⎜⎜⎜⎝ ����� s1 � s2 − � m2 + 1 �� s4 + s6 �� + s5 �� m4 + 1 � s4 + � m2 + 1 �2 s6 − � m2 + 1 � s2 � � m2 + 1 ��� m4 − 6m2 + 1 � s4 + � m4 + 14m2 + 1 � s6 � − � m4 + 14m2 + 1 � s2 (x − tv);m ⎞ ⎟⎟⎟⎠ � × e i � − � c2�7 4c1�1 � x+�t+�0 � . When the modulus of ellipticity approaches unity, we get singular soliton solution Result‑3: Plugging the derived parameters in Eq. (84) with Eq. (24) and Eq. (25) into Eq. (37), as a consequence, we get Jacobi’s elliptic-type function solution (83) q(x, t) = ± � 6s5 − 5s1 2s2 + s4 − 4s6 coth ⎡ ⎢⎢⎣ 1 2 ���� s5 � −2s2 + 2s4 + 4s6 � + s1 � s2 − 2 � s4 + s6 �� 2 � 4s6 − s4 � − 4s2 (x − vt) ⎤⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (84) A0 = 0,A1 = ± √√√√ 20 ( 2m2 − 1 ) s1 − 6 ( 6m4 − 6m2 + 1 ) s5( 2m2 − 1 )(( 4m4 − 4m2 − 1 ) s4 + 4 ( m4 − m2 + 1 ) s6 ) − 8 ( m4 − m2 + 1 ) s2 , B1 = ± √√√√ ( 20 − 40m2 ) s1 + 6 ( 6m4 − 6m2 + 1 ) s5( 2m2 − 1 )(( 4m4 − 4m2 − 1 ) s4 + 4 ( m4 − m2 + 1 ) s6 ) − 8 ( m4 − m2 + 1 ) s2 , k = √√√√√4s1 ( 2s2 − ( 2m2 − 1 )( s4 + s6 )) + s5 ( 2 ( 1 − 2m2 )2 s6 + ( 4 − 8m2 ) s2 + ( 8m4 − 8m2 + 1 ) s4 ) ( 2m2 − 1 )(( 4m4 − 4m2 − 1 ) s4 + 4 ( m4 − m2 + 1 ) s6 ) − 8 ( m4 − m2 + 1 ) s2 , s3 = 1 4 ( 10 ( 2m2 − 1 ) s1 − 3 ( 6m4 − 6m2 + 1 ) s5 )2 × ( 3s5 (( 4m2 − 2 ) s2 − m2 ( m2 − 1 )( s4 − 2s6 )) + s1 (( 2m2 − 1 )( s4 − 4s6 ) − 12s2 )) × (( 2m2 − 1 )(( 4m4 − 4m2 − 1 ) s4 + 4 ( m4 − m2 + 1 ) s6 ) − 8 ( m4 − m2 + 1 ) s2 ) . 4126 J Opt (November 2024) 53(5):4113–4136 When the modulus of ellipticity approaches unity, we get complexiton soliton solution Also, we have the following Jacobi’s elliptic-type function solution When the modulus of ellipticity approaches unity, we get dark soliton solution (85) q(x, t) = ± ���� � 20 − 40m2 � s1 + 6 � 6m4 − 6m2 + 1 � s5� 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 � dn ⎡ ⎢⎢⎢⎣ �����4s1 � 2s2 − � 2m2 − 1 �� s4 + s6 �� + s5 � 2 � 1 − 2m2 �2 s6 + � 4 − 8m2 � s2 + � 8m4 − 8m2 + 1 � s4 � � 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 (x − tv);m ⎤ ⎥⎥⎥⎦ + im sn ⎡ ⎢⎢⎢⎣ �����4s1 � 2s2 − � 2m2 − 1 �� s4 + s6 �� + s5 � 2 � 1 − 2m2 �2 s6 + � 4 − 8m2 � s2 + � 8m4 − 8m2 + 1 � s4 � � 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 (x − vt);m ⎤ ⎥⎥⎥⎦ � × e i � − � c2�7 4c1�1 � x+�t+�0 � . (86) q(x, t) = ± � 6s5 − 20s1 8s2 + s4 − 4s6 � − tanh ⎡ ⎢⎢⎣ � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ + i sech ⎡⎢⎢⎣ � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ � e i � − � c2�7 4c1�1 � x+�t+�0 � . (87) q(x, t) = ± ���� � 20 − 40m2 � s1 + 6 � 6m4 − 6m2 + 1 � s5� 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 � ins ⎡⎢⎢⎢⎣ �����4s1 � 2s2 − � 2m2 − 1 �� s4 + s6 �� + s5 � 2 � 1 − 2m2 �2 s6 + � 4 − 8m2 � s2 + � 8m4 − 8m2 + 1 � s4 � � 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 (x − vt);m ⎤⎥⎥⎥⎦ − i cs ⎡⎢⎢⎢⎣ �����4s1 � 2s2 − � 2m2 − 1 �� s4 + s6 �� + s5 � 2 � 1 − 2m2 �2 s6 + � 4 − 8m2 � s2 + � 8m4 − 8m2 + 1 � s4 � � 2m2 − 1 ��� 4m4 − 4m2 − 1 � s4 + 4 � m4 − m2 + 1 � s6 � − 8 � m4 − m2 + 1 � s2 (x − vt);m ⎤⎥⎥⎥⎦ � × e i � − � c2�7 4c1�1 � x+�t+�0 � . (88) q(x, t) = ± � 6s5 − 20s1 8s2 + s4 − 4s6 tanh ⎡ ⎢⎢⎣ 1 2 � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . Projective Riccati equation method Balancing U′′′′ with U5 in Eq. (10) gives N = 1 , accordingly the solution takes the form (89)U(�) = a0 + a1�(�) + b1�(�). 4127J Opt (November 2024) 53(5):4113–4136 Substituting Eq. (89) together with Eq. (28) and Eq. (29) into Eq. (10), we get a system of algebraic equations (90)a1R(r) 2 ( 5b4 1 s3 + 3b2 1 ( s4 + 2s6 ) + 24k2 ) + a3 1 ( 10b2 1 s3 + s4 + 2s6 ) R(r) + a5 1 s3 = 0, (91)b1 ( a2 1 ( 10b2 1 s3 + 3s4 + 6s6 ) R(r) + 5a4 1 s3 + R(r)2 ( b4 1 s3 + b2 1 ( s4 + 2s6 ) + 24k2 )) = 0, (92)a0b 2 1 ( 5b2 1 s3 + s4 + 4s6 ) R(r)2 + a3 1 ( 5a0a1s3 − r ( 20b2 1 s3 + 2s4 + 3s6 )) + a1R(r) ( a0a1 ( 30b2 1 s3 + s4 + 4s6 ) − r ( 20b4 1 s3 + b2 1 ( 8s4 + 17s6 ) + 60k2 )) = 0, (93)− b1R(r) ( r ( 4b4 1 s3 + b2 1 ( 2s4 + 5s6 ) + 36k2 ) − 2a0a1 ( 10b2 1 s3 + s4 + 4s6 )) − b1a 2 1 ( r ( 20b2 1 s3 + 4s4 + 7s6 ) − 20a0a1s3 ) = 0, (94) a3 1 ( 10a2 0 s3 + 10b2 1 s3 + s2 + s4 + s6 ) − 2a0a 2 1 r ( 30b2 1 s3 + s4 + 3s6 ) − 2a0b 2 1 r ( 10b2 1 s3 + s4 + 5s6 ) R(r) + a1 ( 5r2 ( 4b4 1 s3 + b2 1 ( s4 + 2s6 ) + 6k2 )) + R(r) ( b2 1 ( 30a2 0 s3 + 3s2 + 2s4 + 7s6 ) + 2 ( a2 0 s6 + 10k2 + s5 ) + 10b4 1 s3 ) = 0, (95)b1 ( R(r) ( b2 1 ( 10a2 0 s3 + s2 + 2s6 ) + 2 ( a2 0 s6 + 4k2 + s5 ) + 2b4 1 s3 ) − 2a0a1r ( 20b2 1 s3 + s4 + 4s6 )) + b1 ( a2 1 ( 30a2 0 s3 + 10b2 1 s3 + 3s2 + s4 + 2s6 ) + r2 ( 4b4 1 s3 + b2 1 ( s4 + 2s6 ) + 6k2 )) = 0, (96) a0b 2 1 ( R(r) ( 10a2 0 s3 + 10b2 1 s3 + 3s2 + 4s6 ) + r2 ( 20b2 1 s3 + s4 + 4s6 )) + a0a 2 1 ( 10a2 0 s3 + 30b2 1 s3 + 3s2 + s4 + 2s6 ) − a1r ( b2 1 ( 60a2 0 s3 + 6s2 + 2s4 + 7s6 ) + 3 ( a2 0 s6 + 5k2 + s5 ) + 20b4 1 s3 ) = 0, (97)− b1 ( a2 0 r ( 20b2 1 s3 + s6 ) − 2a1a0 ( 10b2 1 s3 + 3s2 + s6 ) − 20a1a 3 0 s3 + r ( 4b4 1 s3 + b2 1 ( 2s2 + s6 ) + k2 + s5 )) = 0, (98) a1 ( a2 0 ( 30b2 1 s3 + 3s2 + s6 ) + 5a4 0 s3 + 3b2 1 s2 + 5b4 1 s3 + b2 1 s6 + k2 + s1 + s5 ) − 2a0b 2 1 r ( 10a2 0 s3 + 10b2 1 s3 + 3s2 + s6 ) = 0, (99) b1 ( a2 0 ( 10b2 1 s3 + 3s2 ) + 5a4 0 s3 + b2 1 ( b2 1 s3 + s2 ) + s1 ) = 0, Solving these equations together yields the following results: Case‑1: R(r) = 0 (100) a0 ( a2 0 ( 10b2 1 s3 + s2 ) + a4 0 s3 + 3b2 1 s2 + 5b4 1 s3 + s1 ) = 0. (101) a0 = a1 = 0, b1 = ± √ 6s5 − 20s1 8s2 + s4 − 4s6 , k = √ 4s1 ( −2s2 + s4 + s6 ) + s5 ( 4s2 − s4 − 2s6 ) 8s2 + s4 − 4s6 , s3 = ( 8s2 + s4 − 4s6 )( s1 ( 12s2 − s4 + 4s6 ) − 6s2s5 ) 4 ( 10s1 − 3s5 )2 . 4128 J Opt (November 2024) 53(5):4113–4136 Putting the parameters attained in Eq. (101) with Eq. (30) or Eq. (31) into Eq. (89), as a consequence, we get dark and singular solitons with 6s 5 − 20s 1 > 0 , 8s 2 + s 4 − 4s 6 > 0 and 4s 1 ( −2s 2 + s 4 + s 6 ) + s 5 ( 4s 2 − s 4 − 2s 6 ) > 0 , as indicated below Case‑2: R(r) = 24 25 r2 Result‑1: (102) q(x, t) = ± � 6s5 − 20s1 8s2 + s4 − 4s6 tanh ⎡ ⎢⎢⎣ 1 2 � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤ ⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � , (103) q(x, t) = ± � 6s5 − 20s1 8s2 + s4 − 4s6 coth ⎡ ⎢⎢⎣ 1 2 � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 (x − vt) ⎤ ⎥⎥⎦ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (104) a0 = 0, a1 = 4 5 √ 3 2 b1r, b1 = √ 6s5 − 20s1 8s2 + s4 − 4s6 , k = √ 4s1 ( −2s2 + s4 + s6 ) + s5 ( 4s2 − s4 − 2s6 ) 8s2 + s4 − 4s6 , s3 = ( 8s2 + s4 − 4s6 )( s1 ( 12s2 − s4 + 4s6 ) − 6s2s5 ) 4 ( 10s1 − 3s5 )2 . Inserting the parameters secured in Eq. (104) with Eq. (32) into Eq. (89), as a consequence, we get a straddled singu- lar–singular soliton with 6s 5 − 20s 1 > 0 , 8s 2 + s 4 − 4s 6 > 0 a n d 4s 1 ( −2s 2 + s 4 + s 6 ) + s 5 ( 4s 2 − s 4 − 2s 6 ) > 0 , a s described below Result‑2: (105) q(x, t) = � 6s5 − 20s1 8s2 + s4 − 4s6 × ⎧⎪⎪⎨⎪⎪⎩ ±2 √ 6csch �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + 1 coth �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � ± 5csch �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � ⎫⎪⎪⎬⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (106) a0 = b1 = 0, a1 = 12 √ 2 5 r � s5 2s4 + 3s6 , k = � − s5 5 , s3 = − 2s2 4 + 11s6s4 + 12s2 6 60s5 , s2 = 1 16 � 6s4 + 17s6 � , s1 = − 4s5 5 . 4129J Opt (November 2024) 53(5):4113–4136 Plugging the parameters achieved in Eq. (106) with Eq. (32) into Eq. (89), as a consequence, we get a straddled bright–bright soliton with s5 < 0 and 2s4 + 3s6 < 0 , as shown below Result‑3: (107) q(x, t) = ⎧ ⎪⎪⎨⎪⎪⎩ 12 � 2s5 2s4 + 3s6 sech �� − s5 5 (x − vt) � 5sech �� − s5 5 (x − vt) � ± 1 ⎫ ⎪⎪⎬⎪⎪⎭ e i � − � c2�7 4c1�1 � x+�t+�0 � . Substituting the obtained parameters in Eq. (108) with Eq. (32) into Eq. (89), as a consequence, we get a strad- dled bright–dark soliton with s5 > 0, 4s4 + 7s6 > 0, and 2s4 + 3s6 < 0 , as presented below Case‑3: R(r) = 5 9 r2 Result‑1: Putting the parameters acquired in Eq. (110) with Eq. (33) into Eq. (89), as a consequence, we get a straddled singular–sin- gular soliton with 6s5 − −20s1 > 0 , 8s2 + s4 − −4s6 > 0 and 4s1 ( −2s2 + s4 + s6 ) + s5 ( 4s2 − s4 − −2s6 ) > 0 , as given below (108) a 0 = a 1 = 0, b 1 = 2 √ 6 � s 5 4s 4 + 7s 6 , k = � 2 5 � − s 5 � 2s 4 + 3s 6 � 4s 4 + 7s 6 , s 3 = − 4s 2 4 + 23s 6 s 4 + 28s 2 6 120s 5 , s 2 = s 4 3 + 47s 6 48 , s 1 = − s 5 � 32s 4 + 43s 6 � 40s 4 + 70s 6 . (109)q(x, t) = ⎧⎪⎪⎨⎪⎪⎩ 2 � 6s5 4s4+7s6 tanh �� 2 5 � − s5(2s4+3s6) 4s4+7s6 (x − vt) � 1 ± 5sech �� 2 5 � − s5(2s4+3s6) 4s4+7s6 (x − vt) � ⎫⎪⎪⎬⎪⎪⎭ e i � − � c2�7 4c1�1 � x+�t+�0 � . (110) a0 = 0, a1 = √ 5 3 b1r, b1 = � 6s5 − 20s1 8s2 + s4 − 4s6 , k = � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 , s3 = � 8s2 + s4 − 4s6 �� s1 � 12s2 − s4 + 4s6 � − 6s2s5 � 4 � 10s1 − 3s5 �2 . (111) q(x, t) = � 6s5 − 20s1 8s2 + s4 − 4s6 × ⎧⎪⎪⎨⎪⎪⎩ √ 5csch �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � ± 2 3csch �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � ± 2 coth �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � ⎫⎪⎪⎬⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � . 4130 J Opt (November 2024) 53(5):4113–4136 Result‑2: Inserting the derived parameters in Eq. (112) with Eq. (33) into Eq. (89), as a consequence, we get a straddled bright–bright soliton with s5 < 0, and 2s4 + 3s6 < 0 , as indicated below (112) a0 = 0, a1 = 2 √ 5 3 r √ s5 2s4 + 3s6 , b1 = 0, k = √ − s5 5 , s1 = − 4s5 5 , s2 = 1 15 ( 17s4 + 33s6 ) , s3 = − 2s2 4 + 11s6s4 + 12s2 6 60s5 . (113)q(x, t) = ⎧ ⎪⎪⎨⎪⎪⎩ 2 � 5 3 � s5 2s4 + 3s6 � 3sech �� − s5 5 (x − vt) �� 3sech �� − s5 5 (x − vt) � ± 2 ⎫ ⎪⎪⎬⎪⎪⎭ e i � − � c2�7 4c1�1 � x+�t+�0 � . Result‑3: Plugging the parameters attained in Eq. (114) with Eq. (33) into Eq. (89), as a consequence, we get a straddled (114) a0 = a1 = 0, b1 = 2 √ 15s5 10s4 + 63s6 , k = √ − s5 ( 2s4 + 3s6 ) 10s4 + 63s6 , s1 = 4s5 ( 3s6 − 2s4 ) 10s4 + 63s6 , s2 = s4 3 + 3s6 5 , s3 = − 10s2 4 + 103s6s4 + 252s2 6 300s5 . singular–singular soliton with s5 > 0, 4s4 + 7s6 > 0, and 2s4 + 3s6 < 0 , as described below Case‑4: R(r) = r2 − 1 Result‑1: Substituting the parameters secured in Eq. (116) with Eq. (34) or Eq. (35) into Eq. (89), as a consequence, we get straddled bright–dark solitons and straddled singular–singu- lar solitons with 6s5 − −20s1 > 0 , 8s2 + s4 − −4s6 > 0 and 4s1 ( −2s2 + s4 + s6 ) + s5 ( 4s2 − s4 − −2s6 ) > 0 , as shown below (115) q(x, t) = ⎧ ⎪⎪⎨⎪⎪⎩ 4 � 15s5 10s4+63s6 2 coth �� − s5(2s4+3s6) 10s4+63s6 (x − vt) � ± 3csch �� − s5(2s4+3s6) 10s4+63s6 (x − vt) � ⎫⎪⎪⎬⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (116) a0 = 0, a1 = b1 √ r2 − 1, b1 = � 6s5 − 20s1 8s2 + s4 − 4s6 , k = � 4s1 � −2s2 + s4 + s6 � + s5 � 4s2 − s4 − 2s6 � 8s2 + s4 − 4s6 , s3 = � 8s2 + s4 − 4s6 �� s1 � 12s2 − s4 + 4s6 � − 6s2s5 � 4 � 10s1 − 3s5 �2 . 4131J Opt (November 2024) 53(5):4113–4136 or Result‑2: Putting the parameters achieved in Eq. (119) with Eq. (34) or Eq. (35) into Eq. (89), as a consequence, we get straddled (117) q(x, t) = � 6s5 − 20s1 8s2 + s4 − 4s6 × ⎧ ⎪⎪⎨⎪⎪⎩ 4 √ r2 − 1sech �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + 5 tanh �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + 3 4r sech �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + 3 tanh �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + 5 ⎫ ⎪⎪⎬⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � , (118) q(x, t) = � 6s5 − 20s1 8s2 + s4 − 4s6 × ⎧⎪⎪⎨⎪⎪⎩ √ r2 − 1csch �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + 1 rcsch �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + coth �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − tv) � ⎫⎪⎪⎬⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (119) a1 = 2 √ 3 ( r2 − 1 ) s5 2s4 + 3s6 , a0 = b1 = 0, k = √ − s5 5 , s3 = − 2s2 4 + 11s6s4 + 12s2 6 60s5 , s2 = 2 ( r2 + 2 ) s4 + 3 ( 2r2 + 1 ) s6 6 ( r2 − 1 ) , s1 = − 1 5 ( 4s5 ) . bright–dark solitons and straddled bright–bright solitons with s5 < 0, r > 1, and 2s4 + 3s6 < 0 , as presented below or (120)q(x, t) = ⎧ ⎪⎪⎪⎨⎪⎪⎪⎩ 8 � 3 � r2 − 1 � s5 2s4 + 3s6 sech �� − s5 5 (x − vt) � 4rsech �� − s5 5 (x − vt) � + 3 tanh �� − s5 5 (x − vt) � + 5 ⎫ ⎪⎪⎪⎬⎪⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � , 4132 J Opt (November 2024) 53(5):4113–4136 Result‑3: Inserting the obtained parameters in Eq. (122) with Eq. (34) or Eq. (35) into Eq. (89), as a consequence, we get straddled bright–dark solitons with (r2 − 1)s5 > 0, 2(r2 − −1)s4 + 3(r2 + 3)s6 > 0, and 2s4 + 3s6 < 0 , as given below or (121)q(x, t) = ⎧ ⎪⎪⎪⎨⎪⎪⎪⎩ 2 � 3 � r2 − 1 � s5 2s4 + 3s6 sech �� − s5 5 (x − vt) � rsech �� − s5 5 (x − vt) � + 1 ⎫ ⎪⎪⎪⎬⎪⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (122) a0 = a1 = 0, b1 = 2 √√√√ 3 ( r2 − 1 ) s5 2 ( r2 − 1 ) s4 + 3 ( r2 + 3 ) s6 , k = √√√√− ( r2 − 1 ) s5 ( 2s4 + 3s6 ) 10 ( r2 − 1 ) s4 + 15 ( r2 + 3 ) s6 , s3 = − ( s4 + 4s6 )( 2 ( r2 − 1 ) s4 + 3 ( r2 + 3 ) s6 ) 60 ( r2 − 1 ) s5 , s2 = ( 2r2 − 3 ) s6 2 ( r2 − 1 ) + s4 3 , s1 = − 2s5 ( 4 ( r2 − 1 ) s4 + 3 ( 2r2 − 7 ) s6 ) 5 ( 2 ( r2 − 1 ) s4 + 3 ( r2 + 3 ) s6 ) . (123) q(x, t) =2 ���� 3 � r2 − 1 � s5 2 � r2 − 1 � s4 + 3 � r2 + 3 � s6 ⎧⎪⎪⎨⎪⎪⎩ 5 tanh �� − (r2−1)s5(2s4+3s6) 10(r2−1)s4+15(r2+3)s6 (x − vt) � + 3 3 tanh �� − (r2−1)s5(2s4+3s6) 10(r2−1)s4+15(r2+3)s6 (x − vt) � + 4rsech �� − (r2−1)s5(2s4+3s6) 10(r2−1)s4+15(r2+3)s6 (x − vt) � + 5 ⎫⎪⎪⎬⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � , (124) q(x, t) = ⎧ ⎪⎪⎪⎨⎪⎪⎪⎩ 2 ���� 3 � r2 − 1 � s5 2 � r2 − 1 � s4 + 3 � r2 + 3 � s6 tanh ⎡⎢⎢⎣ ����− � r2 − 1 � s5 � 2s4 + 3s6 � 10 � r2 − 1 � s4 + 15 � r2 + 3 � s6 (x − vt) ⎤⎥⎥⎦ rsech ⎡⎢⎢⎣ ����− � r2 − 1 � s5 � 2s4 + 3s6 � 10 � r2 − 1 � s4 + 15 � r2 + 3 � s6 (x − vt) ⎤⎥⎥⎦ + 1 ⎫⎪⎪⎪⎬⎪⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � . Case‑5: R(r) = r2 + 1 4133J Opt (November 2024) 53(5):4113–4136 Result‑1: Plugging the parameters acquired in Eq. (125) with Eq. (36) into Eq. (89), as a consequence, we get a straddled bright–dark soliton with 6s 5 − 20s 1 > 0 , 8s 2 + s 4 − 4s 6 > 0 and 4s 1 ( −2s 2 + s 4 + s 6 ) + v 5 ( 4s 2 − s 4 − 2s 6 ) > 0 , as indi- cated below Result‑2: (125) a0 = 0, a1 = √ − 2 ( r2 + 1 )( 10s1 − 3s5 ) 8s2 + s4 − 4s6 , b1 = √ 6s5 − 20s1 8s2 + s4 − 4s6 , k = √ 4s1 ( −2s2 + s4 + s6 ) + s5 ( 4s2 − s4 − 2s6 ) 8s2 + s4 − 4s6 , s3 = ( 8s2 + s4 − 4s6 )( s1 ( 12s2 − s4 + 4s6 ) − 6s2s5 ) 4 ( 10s1 − 3s5 )2 . (126) q(x, t) = � 6s5 − 20s1 8s2 + s4 − 4s6 × ⎧⎪⎪⎨⎪⎪⎩ √ r2 + 1sech �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + 1 rsech �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � + tanh �� 4s1(−2s2+s4+s6)+s5(4s2−s4−2s6) 8s2+s4−4s6 (x − vt) � ⎫⎪⎪⎬⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � . (127) a0 = a1 = 0, b1 = 2 √√√√ 3 ( r2 + 1 ) s5 2 ( r2 + 1 ) s4 + 3 ( r2 − 3 ) s6 , k = √√√√− ( r2 + 1 ) s5 ( 2s4 + 3s6 ) 10 ( r2 + 1 ) s4 + 15 ( r2 − 3 ) s6 , s3 = − ( s4 + 4s6 )( 2 ( r2 + 1 ) s4 + 3 ( r2 − 3 ) s6 ) 60 ( r2 + 1 ) s5 , s2 = ( 2r2 + 3 ) s6 2 ( r2 + 1 ) + s4 3 , s1 = − 2s5 ( 4 ( r2 + 1 ) s4 + 3 ( 2r2 + 7 ) s6 ) 5 ( 2 ( r2 + 1 ) s4 + 3 ( r2 − 3 ) s6 ) . Substituting the derived parameters in Eq. (127) with Eq. (36) into Eq. (89), as a consequence, we get straddled a singular–singular soliton with s5 > 0, 2(r2 + 1)s4 + 3(r2 − −3)s6 > 0, and 2s4 + 3s6 < 0 , as described below (128) q(x, t) = ⎧ ⎪⎪⎪⎨⎪⎪⎪⎩ 2 ���� 3 � r2 + 1 � s5 2 � r2 + 1 � s4 + 3 � r2 − 3 � s6 coth ⎡⎢⎢⎣ ����− � r2 + 1 � s5 � 2s4 + 3s6 � 10 � r2 + 1 � s4 + 15 � r2 − 3 � s6 (x − vt) ⎤⎥⎥⎦ rcsch ⎡⎢⎢⎣ ����− � r2 + 1 � s5 � 2s4 + 3s6 � 10 � r2 + 1 � s4 + 15 � r2 − 3 � s6 (x − vt) ⎤⎥⎥⎦ + 1 ⎫⎪⎪⎪⎬⎪⎪⎪⎭ × e i � − � c2�7 4c1�1 � x+�t+�0 � . 4134 J Opt (November 2024) 53(5):4113–4136 Result‑3: Putting the parameters attained in Eq. (129) with Eq. (36) into Eq. (89), as a consequence, we get a straddled singu- lar–singular soliton with s5 < 0, and 2s4 + 3s6 < 0 , as shown below Conservation laws For the conserved flow that renders a closed form of the respective PDE, we let q = u + iv and split the PDE into a real system whose conserved vectors (Tt, Tx) satisfy the (129) a1 = 2 √ 3 ( r2 + 1 ) s5 2s4 + 3s6 , a0 = b1 = 0, k = √ − s5 5 , s3 = − 2s2 4 + 11s6s4 + 12s2 6 60s5 , s2 = 2 ( r2 − 2 ) s4 + 3 ( 2r2 − 1 ) s6 6 ( r2 + 1 ) , s1 = − 4s5 5 . (130)q(x, t) = ⎧ ⎪⎪⎪⎨⎪⎪⎪⎩ 2 � 3 � r2 + 1 � s5 2s4 + 3s6 csch �� − s5 5 (x − vt) � rcsch �� − s5 5 (x − vt) � + 1 ⎫ ⎪⎪⎪⎬⎪⎪⎪⎭ e i � − � c2�7 4c1�1 � x+�t+�0 � . DtT t + DxT x) = 0 along the solutions of the PDEs. We pre- sent the final conserved densities for special cases of (1) as Φt . Subject to the condition we have the following: (a) Power (P) density: In addition to the above condition, if we have �4 = �2 + �5, (131)Φt P = 1 2 |q|2. (132)�3 = 2�5, and �9 = 0, we obtain (b) Linear momentum (M) density: and (c) Hamiltonian (H) density: Now, the bright 1-soliton solution is written as where A and B are the amplitude and inverse width of the soliton, respectively. From the phase component, the param- eter � is the frequency of the solitons, while � is the wave number and �0 represents the phase constant. Therefore, the conserved quantities are: (133)Φt M = 1 2 ℑ ( q∗qx ) , (134) Φt H = 1 2 aℜ ( qq∗ xx ) + c1 [ 1 2 �1ℜ ( qq∗ xxxx ) + 1 4 �2 {( ℜ ( qq∗ x ))2 + ( ℑ ( q∗qx ))2 + |q|2ℜ( qq∗ xx ) + |q|2ℜ( qxx ) + |q|2ℑ( qxx ) + 4ℜ(q)ℑ(q)ℜ ( qx ) ℑ ( qx )} + 1 2 �5 { |q|2ℜ( qq∗ xx ) + |q|2||qx||2 } + 1 6 |q|6 ] + c2 [ − 1 2 �7ℑ ( q∗qxxx ) + 1 4 �8|q|2ℑ ( q∗qx )] . (135)q(x, t) = A sech [B(x − vt)]ei(−�x+�t+�0), (136)P =∫ ∞ −∞ Φt P dx = A B , (137)M =∫ ∞ −∞ Φt M dx = − �A B , 4135J Opt (November 2024) 53(5):4113–4136 and where the notations and are implemented. Conclusions This paper studied the concatenation model that is conserved with the absence of SPM. Nevertheless, the model supported soliton solutions, and they are retrieved by the aid of two inte- gration algorithms. A full spectrum of soliton solutions along with the parameter constraints are identified. These constraints guarantee the existence of such soliton solutions. Finally, the multiplier approach revealed the three conserved quantities for the model. The Hamiltonian came with quadratures. The results are nevertheless encouraging that would lead to several futures avenues to walk upon. The conservation laws would lead to the quasi-monochromatic dynamics of such solitons. The phenomenon of optical soliton cooling would be looked upon. The numerical simulations of such solitons by the Laplace–Adomian decomposition would also be considered. The quiescent solitons for the model are also another inquisi- tive issue. The results would sequentially emerge with time. These results would be recovered and aligned with the pre- existing works, with time [15–30]. References 1. A. Ankiewicz, N. Akhmediev, Higher-order integrable evolution equation and its soliton solutions. Phys Lett A 378, 358–361 (2014) 2. A. Ankiewicz, Y. Wang, S. Wabnitz, N. Akhmediev, Extended non- linear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions. Phys Rev E 89, 012907 (2014) 3. A.H. Arnous, A. Biswas, A.H. Kara, Y. Yildirim, L. Moraru, C. Iticescu, S. Moldovanu, A.A. Alghamdi, Optical solitons and con- servation laws for the concatenation model with spatio-temporal dispersion (Internet traffic regulation). J Eur Opt Soc Rapid Publ 19(2), 35 (2023) (138) H = ∫ ∞ −∞ Φt H dx = aA2 3B ( B2 + 3�2 ) + c1 [ �1A 2 15B ( 7B4 + 30�2B2 + 15�4 ) + �2 { 2A4 15B ( 2B2 + 5�2 ) + A3 4B ∫ ∞ −∞ {( B2 − �2 ) sech 3� − 2B2 sech 5� } cos�dx − �A3 2 ∫ ∞ −∞ sech 3� tanh � sin�dx + A4 4B ∫ ∞ −∞ {( B2 − �2 ) sech 4� − B2 sech 6� } sin 2 2�dx + �A4 4 ∫ ∞ −∞ sech � tanh � sin 4�dx } + �5 { 2A4 15B ( 3B2 + 5�2 ) + 4A6 45B }] − c2 [ �7�A 2 B ( B2 + �2 ) + �8�A 4 3B ] , (139)� = B(x − vt), (140)� = −�x + �t + �0 4. A.H. Arnous, A. Biswas, A.H. Kara, Y. Yildirim, L. Moraru, C. Iticescu, S. Moldovanu, A.A. Alghamdi, Optical solitons and con- servation laws for the concatenation model: Power-law nonlinear- ity. Appear Ain Shams Eng J (2023). https:// doi. org/ 10. 1016/j. asej. 2023. 102381 5. A. Biswas, J. Vega-Guzman, A.H. Kara, S. Khan, H. Triki, O. Gonzalez-Gaxiola, L. Moraru, P.L. Georgescu, Optical solitons and conservation laws for the concatenation model: undetermined coefficients and multipliers approach. Universe. 9(1), Article 15 (2023) 6. A. Biswas, J. Vega-Guzman, Y. Yildirim, L. Moraru, C. Iticescu, A.A. Alghamdi, Optical solitons for the concatenation model with differential group delay: undetermined coefficients. Mathematics. 11(9), 2012 (2023) 7. A. Biswas, J.M. Vega-Guzman, Y. Yildirim, S.P. Moshokoa, M. Aphane, A.A. Alghamdi, Optical solitons for the concatenation model with power-law nonlinearity: undetermined coefficients. Ukr J Phys Opt 24(3), 185–192 (2023) 8. O. González-Gaxiola, A. Biswas, J.R.D. Chavez, A. Asiri, Bright and dark optical solitons for the concatenation model by the Laplace-Adomian decomposition scheme. Ukr J Phys Opt 24(3), 222–234 (2023) 9. N.A. Kudryashov, A. Biswas, A.G. Borodina, Y. Yildirim, H.M. Alshehri, Painleve analysis and optical solitons for a concatenated model. Optik. 272, 170255 (2023) 10. A. Kukkar, S. Kumar, S. Malik, A. Biswas, Y. Yildirim, S.P. Moshokoa, S. Khan, A.A. Alghamdi, Optical solitons for the concatenation model with Kudryashov’s approaches. Ukr J Phys Opt 24(2), 155–160 (2023) 11. L. Tang, A. Biswas, Y. Yildirim, A.A. Alghamdi, Bifurcation analysis and optical solitons for the concatenation model. Phys Lett A 480, 128943 (2023) 12. H. Triki, Y. Sun, Q. Zhou, A. Biswas, Y. Yildirim, H.M. Alshehri, Dark solitary pulses and moving fronts in an optical medium with the higher-order dispersive and nonlinear effects. Chaos Solitons Fractals 164, 112622 (2022) 13. M.-Y. Wang, A. Biswas, Y. Yıldırım, L. Moraru, S. Moldovanu, H.M. Alshehri, Optical solitons for a concatenation model by trial equation approach. Electronics 12(1), Article 19 (2023) 14. Y. Yildirim, A. Biswas, L. Moraru, A.A. Alghamdi, Quiescent optical solitons for the concatenation model with nonlinear chro- matic dispersion. Mathematics. 11(7), Article 1709 (2023) 15. T. Han, Z. Li, C. Li, L. Zhao, Bifurcations, stationary optical soli- tons and exact solutions for complex Ginzburg-Landau equation with nonlinear chromatic dispersion in non-Kerr law media. J Opt (2023). https:// doi. org/ 10. 1007/ s12596- 022- 01041-5 16. Z. Li, E. Zhu, Optical soliton solutions of stochastic Schrödinger- Hirota equation in birefringent fibers with spatiotemporal disper- sion and parabolic law nonlinearity. J Opt (2023). https:// doi. org/ 10. 1007/ s12596- 023- 01287-7 https://doi.org/10.1016/j.asej.2023.102381 https://doi.org/10.1016/j.asej.2023.102381 https://doi.org/10.1007/s12596-022-01041-5 https://doi.org/10.1007/s12596-023-01287-7 https://doi.org/10.1007/s12596-023-01287-7 4136 J Opt (November 2024) 53(5):4113–4136 17. S. Nandy, V. Lakshminarayanan, Adomian decomposition of sca- lar and coupled nonlinear Schrödinger equations and dark and bright solitary wave solutions. J Opt 44, 397–404 (2015) 18. L. Tang, Bifurcations and optical solitons for the coupled non- linear Schrödinger equation in optical fiber Bragg gratings. J Opt (2023). https:// doi. org/ 10. 1007/ s12596- 022- 00963-4 19. L. Tang, Phase portraits and multiple optical solitons perturbation in optical fibers with the nonlinear Fokas-Lenells equation. J Opt (2023). https:// doi. org/ 10. 1007/ s12596- 023- 01097-x 20. S. Wang, Novel soliton solutions of CNLSEs with Hirota bilinear method. J Opt (2023). https:// doi. org/ 10. 1007/ s12596- 022- 01065-x 21. S.A. AlQahtani, M.S. Al-Rakhami, M.A. Reham Shohib, M.E.M. Alngar, P. Pathak, Dispersive optical solitons with Schrödinger- Hirota equation using the Φ6-model expansion approach. Opt Quantum Electron 55(8), 701 (2023) 22. E.M. Zayed, R.M. Shohib, Solitons and other solutions to the improved perturbed nonlinear Schrodinger equation with the dual-power law nonlinearity using different techniques. Optik 171, 27–43 (2018) 23. E.M. Zayed, R.M. Shohib, Optical solitons to the generalized non- linear Schrödinger equations for pulse propagation using several different techniques. Optik 187, 81–91 (2019) 24. E.M. Zayed, R.M. Shohib, A.G. Al-Nowehy, On solving the (3+ 1)-dimensional NLEQZK equation and the (3+ 1)-dimensional NLmZK equation using the extended simplest equation method. Comput Math Appl 78(10), 3390–3407 (2019) 25. E.M. Zayed, R.M. Shohib, A.G. Al-Nowehy, Solitons and other solutions for higher-order NLS equation and quantum ZK equa- tion using the extended simplest equation method. Comput Math Appl 76(9), 2286–2303 (2018) 26. E.M. Zayed, R.M. Shohib, Optical solitons and other solutions to the dual-mode nonlinear Schrödinger equation with Kerr law and dual power law nonlinearities. Optik 208, 163998 (2020) 27. E.M. Zayed, M.E. Alngar, R.M. Shohib, Cubic-quartic embedded solitons with � (2) and � (3) nonlinear susceptibilities having mul- tiplicative white noise via Itô calculus. Chaos Solitons Fractals 168, 113186 (2023) 28. E.M. Zayed, R.M. Shohib, M.E. Alngar, Dispersive optical soli- tons in birefringent fibers for stochastic Schrödinger-Hirota equa- tion with parabolic law nonlinearity and spatiotemporal dispersion having multiplicative white noise. Optik 278, 170736 (2023) 29. E.M. Zayed, M.E. Alngar, R.M. Shohib, Dispersive Optical Soli- tons to Stochastic Resonant NLSE with Both Spatio-Temporal and Inter-Modal Dispersions Having Multiplicative White Noise. Mathematics 10(17), 3197 (2022) 30. E.M. Zayed, R.M. Shohib, M.E. Alngar, Cubic-quartic optical solitons in Bragg gratings fibers for NLSE having parabolic non- local law nonlinearity using two integration schemes. Opt Quan- tum Electron 53(8), 452 (2021) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. https://doi.org/10.1007/s12596-022-00963-4 https://doi.org/10.1007/s12596-023-01097-x https://doi.org/10.1007/s12596-022-01065-x https://doi.org/10.1007/s12596-022-01065-x Optical solitons and conservation laws for the concatenation model in the absence of self-phase modulation Abstract Introduction Governing Model An overview of the integration algorithms Generalized sine-Gordon equation method Projective Riccati equation method Optical solitons The generalized sine-Gordon equation method Projective Riccati equation method Conservation laws Conclusions References