© The Author(s), published by EDP Sciences, 2025

1 Introduction

The recent research studies of optical pulses focus mainly on the dynamical features of propagating solitons in different optical fiber systems and other photonic media [1–4]. In addition, there are different fields of sciences that exploit soliton phenomena in applications such as quantum electronics [5–7], solid state physics [8, 9], Bose-Einstein condensates [10–16] and others. Understanding the physical properties of optical solitons can help to overcome the data over-demand that contribute in many applications in science, technology, and industry. Based on the continuous studies, scientists and engineers have reached an effective innovative tool known as birefringent fiber that possesses an optical property of light polarization enhancing higher transmission speeds. Birefringence is a mechanism in which the light passes through certain materials and is split into two rays having different index of refraction. Due to its prominent significancy in various scientific applications, this phenomena has attained wide attention in the fields of fiber-optic communication system, fiber lasers, and fiber sensors [17]. In optical communication systems, for instance, birefringence fiber can play a vital role in transmitting light signals for long distance with minimal degradation, resulting in more reliable communication. Hence, a lot of studies in literature have intensively discussed the pulse propagation in birefringent fiber.

The dynamic features of optical solitons are dominated by the nonlinear Schrödinger equation (NLSE) and its analogous types of models with various forms of nonlinear refractive index. For example, Menyuk [18, 19] examined the birefringence impact on soliton propagation in single-mode optical fibers. Carrying out numerical computation to a model of coupled NLSE, the analysis of pulse evolution shows that the Kerr effect stabilizes solitons against spreading caused by birefringence. Furthermore, the study concluded that the birefringence must be linear and the standard single-mode birefringent fibers are much suitable for soliton communication system. Recently, Liu et al. [20] investigated the optical nondegenerate solitons in a birefringent fibers with 35° elliptical angle. Making use of the bilinear forms, the nondegenerate bright one- and two-soliton solutions are constructed by solving the coupled NLSE. By virtue of the asymptotic analysis, two degenerate solitons are found to be elastic interactions under certain conditions while two nondegenerate solitons are elastic interactions but it can be made inelastic by adjusting the soliton phase. In an inhomogeneous birefringent nonlinear dispersive medium, Zaabat et al. [21] scrutinized wave-speed management of soliton pulses in emergence of variable dispersion and nonlinearity parameters and external potential. New types of periodic nonlinear waves expressed in the form of Jacobi elliptic functions are extracted which degenerate, as the elliptic modulus approaching unity, to soliton pulse solutions including bright-dipole and bright-dark solutions. It is noted that the wave speed of solitons can be controlled by varying dispersion parameter. In addition, the influence of fractional order derivatives on optical solitons transmitted through birefringence fibers are also reported by different groups of authors. Applying the semi-inverse and fractional variational method, Fu et al. [22] discussed the space-time fractional NLSE with different types of nonlinearity such as Kerr, power, parabolic, dual-power, and log law. Their study results in miscellaneous soliton structures including bright, dark, and singular solitons which display diverse behaviors because of fractional order effects. Considering conformable fractional derivative, Li et al. [23] dealt with Fokas-Lenells equation with the aid of the plane dynamics system. The optical soliton solutions, qualitative analysis and chaotic behaviors of the model are illustrated thoroughly.

Moreover, the effects of the phase chirp on transmitted solitons cannot be ignored as this chirp can rehash the physical aspects of propagating pulses. Mahmood [24] studied the behavior of chirped solitons in a single-mode birefringent fiber described by a coupled NLSE. It is detected that the chirp has a significant role on controlling the threshold amplitude for soliton trapping without causing excessive pulse broadening. Besides, Triki et al. [25] investigated ultrashort light pulses in a birefringent optical fiber under the effect of several physical features. The chirped solitons in the form of dark-dark and bright-bright soliton pairs are induced in the presence of all fiber parameters. The consequence of study indicates that the frequency chirp corresponding to these solitons depends on the intensity of the pulse. Using numerical simulation scheme, Xiao et al. [26, 27] discussed chirped bright and dark vector quasi-solitons in birefringent fiber system characterized by the coupled Ginzburg-Landau equation. The analyzed behaviors displays that both bright and dark vector solitons can be transmitted stably in the birefringent fiber system. For more details about the analytical studies of soliton propagation in birefringent fiber, the reader is referred to the references [28–40].

As mentioned above, the dynamics of light pulses transmitted through nonlinear optical media are described by the NLSE-type equations and relevant models. Kaup-Newell equation (KNE) is one of the vital models that describes wave propagations in optical fiber and plasma physics [41–44]. This model is found to characterize the sub-pico-second pulses in mono-mode optical fibers. The dimensionless form of KNE is given by [45, 46]$${\mathrm{{\rm Y}}}_t+\mathrm{ia}{\mathrm{{\rm Y}}}_{\mathrm{xx}}+b\left(\right|\mathrm{{\rm Y}}{|}^2\mathrm{{\rm Y}}{)}_x=0,$$(1)where Υ(x, t) represents a complex-valued function referring to the wave profile. The parameter a denotes the group velocity dispersion while the parameter b describes the effect of nonlinearity in the medium. Equation (1) has been scrutinized in the past to derive the exact soliton solitons. Two group of authors [47, 48] investigated the chirped optical solitons using different integration methodologies. Numerous types of structures are obtained such as bright, dark and singular solitons. The chirped solitons are extracted with their corresponding chirp which is expressed nonlinearly in terms of soliton intensity. With the aid of a novel mathematical technique, new chirped optical soliton solutions are retrieved which can be useful in the fields of optical fibers and plasma physics [49]. Further to this, the perturbed model of KNE has been studied in the past to detect soliton pulses, see [50, 51].

The KNE can be applied in birefringent fiber without four-wave mixing to study the dynamics of optical solitons and the physical properties of the medium. Thus, the model of KNE is addressed as$$\begin{array}{l}{p}_t+i{a}_1{p}_{\mathrm{xx}}+b_1\left(\right|p{|}^{2}p{)}_x+c_1\left(\right|q{|}^{2}q{)}_x=0,\\ q_t+i{a}_2{q}_{\mathrm{xx}}+b_2\left(\right|q{|}^{2}q{)}_x+c_2\left(\right|p{|}^{2}p{)}_x=0,\end{array}$$(2)where a_j(j = 1, 2) represents the coefficients of group velocity dispersion while b_j and c_j(j = 1, 2) stand for the nonlinear influence. The previous studies in literature that scrutinized the coupled equations (2) have investigated the chirped-free solitons only. For example, three groups of authors [52–54] studies sub pico-second optical pulses of equations (2) by applying distinct mathematical tools. Different wave structures are extracted including bright, dark and singular solitons. Exploiting three integration approaches, the system of KNE (2) is revisited by Rehman et al. [55] to investigate the sub-pico-second optical solitons. The detected wave forms include bright, dark, singular and bright-singular combo solitons as well as periodic singular waves. Recently, Li [56] discussed the dynamical behavior of the coupled KNE by means of bifurcation theory of planar dynamical system. Using this technique, the phase portrait and optical soliton solutions are created.

Our present attention is concentrated on investigating the chirped bright and dark optical pulses of fractional KNE in birefringent fiber in the sense of conformable derivatives [57]. The study is carried out by utilizing the Jacobi elliptic equation method expressed in a form of a first-order nonlinear ordinary differential equation (ODE) with three-degree terms. The modulation instability of the addressed model is examined. The arrangement of this work is as follows. The next section elucidates the properties of conformable fractional derivative. Section 3 describes the governing model of fractional order derivative and its mathematical analysis. In Section 4, abundant types of chirped bright and dark are constructed by using the proposed strategy. Section 5 displays the diagnosis process of the modulation instability through employing the linear stability analysis method. The discussion of obtained results and optical pulse behaviors are illustrated in Section 6. Finally, the conclusion of work is given in Section 7.

2 Conformable fractional derivative

Fractional calculus can be defined in various ways according to the characteristics and conditions of fractional derivative that must be satisfied. Among the introduced definitions in literature are Riemann and Liouville, Caputo-Katugampola, Grünwald-Letnikov, Marchaud and Others [58]. Conformable fractional derivative [59] is one of the most applied techniques in the recent research activities involving fractional calculus, see [60]. The definition of conformable fractional derivative is described as follows.

3 Governing model and mathematical analysis

The space-time fractional Kaup-Newell equation in birefringent fiber is described by$$\begin{array}{ll}& D_t^{\alpha }p+i{a}_1{D}_x^{2\alpha }p+b_1{D}_x^{\alpha }\left(\right|p{|}^{2}p)+c_1{D}_x^{\alpha }(\left|q{|}^{2}q\right)=0,\\ & D_t^{\alpha }q+i{a}_2{D}_x^{2\alpha }q+b_2{D}_x^{\alpha }\left(\right|q{|}^{2}q)+c_2{D}_x^{\alpha }(\left|p{|}^{2}p\right)=0.\end{array}$$(5)

This equation characterizes the propagation of very short (sub-picosecond) light pulses in birefringent fiber which incorporates fractional dispersion. To the best of our knowledge, equation (5) has not been discussed before. Consider the traveling wave transformation introduced as$$\begin{array}{ll}& p(x,t)={\psi }_1\left(\xi \right)e^{i\left({\varphi }_1\left(\xi \right)-{\omega }_1\frac{t^{\alpha }}{\alpha }\right)},\\ & q(x,t)={\psi }_2\left(\xi \right)e^{i\left({\varphi }_2\left(\xi \right)-{\omega }_2\frac{t^{\alpha }}{\alpha }\right)},\end{array}$$(6)where α denotes fractional-order derivative and ξ represents the wave coordinate of spatial variable x and temporal variable t. The variable ξ is given by$$\xi =\frac{x^{\alpha }}{\alpha }-\nu \frac{t^{\alpha }}{\alpha }, 0<\alpha \le 1.$$(7)

The two functions ψ₁(ξ) and ψ₂(ξ) stand for the amplitudes of the solitons while the functions ϕ₁(ξ) and ϕ₂(ξ) account for nonlinear phase shift. The parameters ω₁, ω_2, and ν are real constants denoting the wave number and the soliton velocity.

Applying the transformation (6), the coupled equation (5) breaks up into a real part given by$$\begin{array}{ll}& -\nu {\psi }_1^{\text{'}}-a_1{\psi }_1{\varphi }_1^{\mathrm{″}}-2a_1{\psi }_1^{\mathrm{\prime } }{\varphi }_1^{\text{'}}+3b_1{\psi }_1^2{\psi }_1^{\text{'}}+3c_1{\psi }_2^2{\psi }_2^{\mathrm{\prime }}=0,\\ & -\nu {\psi }_2^{\text{'}}-a_2{\psi }_2{\varphi }_2^{\mathrm{″}}-2a_2{\psi }_2^{\mathrm{\prime }}{\varphi }_2^{\text{'}}+3b_2{\psi }_2^2{\psi }_2^{\text{'}}+3c_2{\psi }_1^2{\psi }_1^{\mathrm{\prime }}=0,\end{array}$$(8)and an imaginary part given by$$\begin{array}{ll}& a_1{\psi }_1^{\mathrm{″}}-{\omega }_1{\psi }_1-\nu {\psi }_1{\varphi }_1^{\text{'}}-a_1{\psi }_1{\varphi }_1^{\mathrm{\prime }2}+b_1{\varphi }_1^{\mathrm{\prime }}{\psi }_1^3+c_1{\varphi }_2^{\mathrm{\prime }}{\psi }_2^3=0,\\ & a_2{\psi }_2^{″}-{\omega }_2{\psi }_2-\nu {\psi }_2{\varphi }_2^{\text{'}}-a_2{\psi }_2{\varphi }_2^{\mathrm{\prime }2}+b_2{\varphi }_2^{\mathrm{\prime }}{\psi }_2^3+c_2{\varphi }_1^{\mathrm{\prime }}{\psi }_1^3=0.\end{array}$$(9)

To handle the set of obtained equations analytically, the relation between ψ₁ and ψ₂ is proposed as$${\psi }_2=\lambda {\psi }_1,$$(10)where λ ≠ 1 is a real constant. As a result, the systems (8) and (9) are converted respectively to$$\begin{array}{ll}& -\nu {\psi }_1^{\text{'}}-a_1\left({\psi }_1{\varphi }_1^{″}+2{\psi }_1^{\mathrm{\prime }}{\varphi }_1^{\mathrm{\prime }}\right)+3(b_1+c_1{\lambda }^3){\psi }_1^2{\psi }_1^{\mathrm{\prime }}=0,\\ & -\lambda \nu {\psi }_1^{\mathrm{\prime }}-a_2\lambda \left({\psi }_1{\varphi }_2^{\mathrm{″}}+2{\psi }_1^{\mathrm{\prime }}{\varphi }_2^{\mathrm{\prime }}\right)+3(b_2{\lambda }^3+c_2){\psi }_1^2{\psi }_1^{\mathrm{\prime }}=0,\end{array}$$(11)and$$\begin{array}{ll}& a_1{\psi }_1^{\mathrm{″}}-{\omega }_1{\psi }_1-\nu {\psi }_1{\varphi }_1^{\mathrm{\prime }}-a_1{\psi }_1{\varphi }_1^{\mathrm{\prime }2}+(b_1{\varphi }_1^{\mathrm{\prime }}+c_1{\lambda }^3{\varphi }_2^{\mathrm{\prime }}){\psi }_1^3=0,\\ & a_2\lambda {\psi }_1^{\mathrm{″}}-{\omega }_2\lambda {\psi }_1-\nu \lambda {\psi }_1{\varphi }_2^{\mathrm{\prime }}-a_2\lambda {\psi }_1{\varphi }_2^{\mathrm{\prime }2}+(b_2{\lambda }^3{\varphi }_2^{\mathrm{\prime }}+c_2{\varphi }_1^{\mathrm{\prime }}){\psi }_1^3=0.\end{array}$$(12)

One can deduce that both equations in (11) can be integrated and resulting in the first integral as$$\begin{array}{ll}& {\varphi }_1^{\mathrm{\prime }}=-\frac{\nu }{2a_1}+\frac{3(b_1+c_1{\lambda }^3)}{4a_1}{\psi }_1^2,\\ & {\varphi }_2^{\mathrm{\prime }}=-\frac{\nu }{2a_2}+\frac{3(b_2{\lambda }^3+c_2)}{4a_2\lambda }{\psi }_1^2,\end{array}$$(13)where the integration constant is set zero. Consequently, the chirping expression of the first field component has the form$$\delta {\omega }_1=\frac{\nu }{2a_1}-\frac{3(b_1+c_1{\lambda }^3)}{4a_1}{\psi }_1^2,$$(14)whereas the chirping expression of the second field component is identified as$$\delta {\omega }_2=\frac{\nu }{2a_2}-\frac{3(b_2{\lambda }^3+c_2)}{4a_2\lambda }{\psi }_1^2.$$(15)

Substituting (13) into equation (12), we obtain$$\begin{array}{l}{\psi }_1^{\mathrm{″}}-{\sigma }_1{\psi }_1-{\gamma }_1{\psi }_1^3+{\beta }_1{\psi }_1^5=0,\\ {\psi }_1^{\mathrm{″}}-{\sigma }_2{\psi }_1-{\gamma }_2{\psi }_1^3+{\beta }_2{\psi }_1^5=0.\end{array}$$(16)

The parameters σ_j, γ_j, β_j; (j = 1, 2) in the system of equations (16) are defined as$$\begin{array}{ll}{\sigma }_1=\frac{4a_1{\omega }_1-{\nu }^2}{4a_1^2},{\gamma }_1=\frac{\nu (a_2{b}_1+a_1{c}_1{\lambda }^3)}{2a_2{a}_1^2}, {\beta }_1=\frac{3a_2(b_1+c_1{\lambda }^3)(b_1-3c_1{\lambda }^3)+12a_1{c}_1{\lambda }^2(b_2{\lambda }^3+c_2)}{16a_2{a}_1^2},& \\ {\sigma }_2=\frac{4a_2{\omega }_2-{\nu }^2}{4a_2^2},{\gamma }_2=\frac{\nu (a_1{b}_2{\lambda }^3+a_2{c}_2)}{2\lambda a_1{a}_2^2},{\beta }_2=\frac{3a_1(b_2{\lambda }^3+c_2)(b_2{\lambda }^3-3c_2)+12a_2{c}_2\lambda (b_1+c_1{\lambda }^3)}{16a_1{a}_2^2{\lambda }^2},& \end{array}$$(17)where both a₁ and a₂ are not equal to zero. The coupled equations (16) are consistent under the conditions$$\frac{{\sigma }_1}{{\sigma }_2}=\frac{{\gamma }_1}{{\gamma }_2}=\frac{{\beta }_1}{{\beta }_2}.$$(18)

Accordingly, the system of equations (16) turns into$${\psi }_1^{\mathrm{″}}-\sigma {\psi }_1-\gamma {\psi }_1^3+\beta {\psi }_1^5=0,$$(19)where the parameters β, γ and σ are given by$$\sigma ={\sigma }_1={\sigma }_2, \gamma ={\gamma }_1={\gamma }_2, \beta ={\beta }_1={\beta }_2.$$(20)

Since equation (19) can be integrated, then it is reduced to$$6{\psi }_1^{\mathrm{\prime }2}-6\sigma {\psi }_1^2-3\gamma {\psi }_1^4+2\beta {\psi }_1^6+12\mu =0,$$(21)where μ is the integration constant. For convenience, the form of equation (21) can be modified by introducing the variable transformation given as$${\psi }_1={\mathrm{\Psi }}^{\frac{1}{2}}.$$(22)

Subsequently, one can reach the equation of the form$$3{\mathrm{\Psi }}^{\mathrm{\prime }2}+24\mu \mathrm{\Psi }-12\sigma {\mathrm{\Psi }}^2-6\gamma {\mathrm{\Psi }}^3+4\beta {\mathrm{\Psi }}^4=0.$$(23)

Based on the relations (6), (10) and (22), the solutions of equation (23) reveal the amplitude structures of soliton waves that propagate in birefringent optical fiber. Further to this, one can recover the general form of solutions for the coupled equations (5) as$$\begin{array}{ll}& p(x,t)={\mathrm{\Psi }}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ & q(x,t)=\lambda {\mathrm{\Psi }}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha }))},\end{array}$$(24)where the phase variable ϕ₁(ξ) and ϕ₂(ξ) can be obtained by integrating equations (13).

Thus, our aim in the next section is to solve equation (23) analytically by the proposed method so as to derive soliton solutions of bright and dark structures.

4 Chirped soliton solutions

The solution of equation (23) is derived by means of the Jacobi elliptic equation that has a form of a first-order nonlinear ODE with three-degree terms. Before implementing this technique, we put forward the transformation given by$$\begin{array}{cc}\mathrm{\Psi }\left(\xi \right)=\mathrm{\Gamma }\left(\zeta \right), & \zeta =ϵ\xi ,\end{array}$$(25)by which equation (23) is rearranged to$$3{ϵ}^2{\mathrm{\Gamma }}^{\mathrm{\prime }2}+24\mu \mathrm{\Gamma }-12\sigma {\mathrm{\Gamma }}^2-6\gamma {\mathrm{\Gamma }}^3+4\beta {\mathrm{\Gamma }}^4=0.$$(26)where ϵ is a constant to be determined. It is worth mentioning that equation (26) can be expressed in terms of the elliptic Jacobi sine [61–64]. However, we are interested in creating various solutions to equation (26) by considering its solution in the form$$\mathrm{\Gamma }\left(\zeta \right)=d_0+d_1\mathrm{\Lambda }\left(\zeta \right)+d_2\sqrt{\mathrm{\Lambda }\left(\zeta \right)}+d_3\frac{{\mathrm{\Lambda }}^2\left(\zeta \right)}{1+\mathrm{\Lambda }\left(\zeta \right)},$$(27)where d_j, (j = 0, 1, 2, 3) are constants to be identified. The function Λ(ζ) satisfies the following ODE given by [65]$$\left(\mathrm{\Lambda }\text{'}\right(\zeta ){)}^2=f\mathrm{\Lambda }(\zeta )+g\mathrm{\Lambda }(\zeta {)}^2+h\mathrm{\Lambda }(\zeta {)}^3,$$(28)which is an equivalent form of equation [66]$$\left({\mathrm{\Xi }}^{\mathrm{\prime }}\right(\zeta ){)}^2=\frac{1}{4}\left[f+g\mathrm{\Xi }(\zeta {)}^2+h\mathrm{\Xi }(\zeta {)}^4\right],$$(29)where Λ(ζ) = Ξ(ζ)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(ζ, m), cn(ζ, m), dn(ζ, 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(ζ, m) = sin(ξ), cn(ζ, m) = cos(ξ), dn(ζ, m) = 1 as m tends to 0 while it is found that sn(ζ, m) = tanh(ξ), cn(ζ, m) = sech(ξ), dn(ζ, m) = sech(ξ) 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 Λ^l(ζ), (l = 0, 1, …, 8). Equating the coefficients of Λ^l(ζ) 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. $$d_0=d_2=d_3=0, d_1=\frac{2\sigma h}{\gamma g}, ϵ=2\sqrt{\frac{\sigma }{g}},$$(30)under the constraint conditions$$\begin{array}{cc}\mu \gamma g^2+fh{\sigma }^2=0,& \beta =0,\end{array}$$(31)where σg > 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 + m²), h = 4m², Λ(ζ) = sn²(ζ, m), equation (5) secures JEF solution of the form$$\begin{array}{l}p(x,t)=\sqrt{-\frac{2\sigma m^2}{\gamma (m^2+1)}}\mathrm{sn}\left(\sqrt{-\frac{\sigma }{m^2+1}}\xi \right)e^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda \sqrt{-\frac{2\sigma m^2}{\gamma (m^2+1)}}\mathrm{sn}\left(\sqrt{-\frac{\sigma }{m^2+1}}\xi \right)e^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(32)provided that σ < 0 and γ > 0. The conditions (31) reduce to$$\begin{array}{cc}\mu \gamma (m^2+1{)}^2+{\sigma }^2{m}^2=0,& \beta =0.\end{array}$$(33)

As m approaches 1, solution (32) collapses to a soliton solution given by$$\begin{array}{l}p(x,t)=\sqrt{-\frac{\sigma }{\gamma }}\tanh \left(\sqrt{-\frac{\sigma }{2}}\xi \right)e^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda \sqrt{-\frac{\sigma }{\gamma }}\tanh \left(\sqrt{-\frac{\sigma }{2}}\xi \right)e^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(34)which represent dark soliton as shown in Figure 1, where σ < 0 and γ > 0. From (33), we arrive at$$\begin{array}{cc}4\mu \gamma +{\sigma }^2=0,& \beta =0.\end{array}$$(35)

Figure 1 Intensity profile of dark soliton given by solutions (34), (42) and (52) with parameter values a1 = a2 = b1 = b2 = c1 = c2 = ω1 = 0.5, ν = 1.25, λ = 1.5. (a) 3D plot of dark soliton; (b) |p|2 with distinct values of α; (c) |q|2 with distinct values of α; (d) Chirping profile.

Case I2. If f = 4(1 − m²), g = −4(1 − 2m²), h = −4m², Λ(ζ) = cn²(ζ, m), equation (5) possesses JEF solution of the form$$\begin{array}{l}p(x,t)=\sqrt{-\frac{2\sigma m^2}{\gamma (2m^2-1)}}\mathrm{cn}\left(\sqrt{\frac{\sigma }{2m^2-1}}\xi \right)e^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda \sqrt{-\frac{2\sigma m^2}{\gamma (2m^2-1)}}\mathrm{cn}\left(\sqrt{\frac{\sigma }{2m^2-1}}\xi \right)e^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })}.\end{array}$$(36)where σ > 0, γ < 0 as m² > 1/2 and σ < 0, γ < 0 as m² < 1/2. The conditions (31) reduce to$$\begin{array}{cc}\mu \gamma (2m^2-1{)}^2+{\sigma }^2{m}^2(m^2-1)=0, & \beta =0.\end{array}$$(37)

As m approaches 1, solution (36) co llapses to a soliton solution given by$$\begin{array}{l}p(x,t)=\sqrt{-\frac{2\sigma }{\gamma }}\mathrm{sech}\left(\sqrt{\sigma }\xi \right)e^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda \sqrt{-\frac{2\sigma }{\gamma }}\mathrm{sech}\left(\sqrt{\sigma }\xi \right)e^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(38)which describes bright soliton as depicted in Figure 2, where σ > 0 and γ < 0. From (37), we arrive at$$\begin{array}{cc}\mu =0, & \beta =0.\end{array}$$(39)

Figure 2 Intensity profile of bright soliton given by solutions (38), (48) and (56) with parameter values a1 = 0.93, a2 = b2 = −0.5, b1 = c1 = c2 = 0.5, ω1 = 1.5, ν = 0.75, λ = 1.5. (a) 3D plot of bright soliton; (b) |p|2 with distinct values of α; (c) |q|2 with distinct values of α; (d) Chirping profile.

Case I3. If f = 1, g = 2(1 − 2m²), h = 1, $\mathrm{\Lambda }\left(\zeta \right)=\frac{{\mathrm{sn}}^2(\zeta ,m)}{(1\pm \mathrm{cn}(\zeta ,m){)}^2}$, equation (5) admits JEF solution of the form$$\begin{array}{l}p(x,t)={\left[-\frac{\sigma }{\gamma (2m^2-1)}\frac{1-\mathrm{cn}\left(\sqrt{-\frac{2\sigma }{2m^2-1}}\xi \right)}{1+\mathrm{cn}\left(\sqrt{-\frac{2\sigma }{2m^2-1}}\xi \right)}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda {\left[-\frac{\sigma }{\gamma (2m^2-1)}\frac{1-\mathrm{cn}\left(\sqrt{-\frac{2\sigma }{2m^2-1}}\xi \right)}{1+\mathrm{cn}\left(\sqrt{-\frac{2\sigma }{2m^2-1}}\xi \right)}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })}.\end{array}$$(40)where σ < 0, γ > 0 as m² > 1/2 and σ > 0, γ > 0 as m² < 1/2. The conditions (31) reduce to$$\begin{array}{cc}4\mu \gamma (2m^2-1{)}^2+{\sigma }^2=0, & \beta =0.\end{array}$$(41)

As m approaches 1, solution (40) convertes to a soliton solution given by$$\begin{array}{l}p(x,t)={\left[-\frac{\sigma }{\gamma }\frac{1-\mathrm{sech}\left(\sqrt{-2\sigma }\xi \right)}{1+\mathrm{sech}\left(\sqrt{-2\sigma }\xi \right)}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda {\left[-\frac{\sigma }{\gamma }\frac{1-\mathrm{sech}\left(\sqrt{-2\sigma }\xi \right)}{1+\mathrm{sech}\left(\sqrt{-2\sigma }\xi \right)}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(42)which characterizes dark soliton as plotted in Figure 1, where σ < 0 and γ > 0. From (41), we arrive at$$\begin{array}{cc}4\mu \gamma +{\sigma }^2=0, & \beta =0.\end{array}$$(43)

Set II. $$d_0=-\frac{2\sigma (g+\tau )}{\gamma (g+3\tau )}, d_1=-\frac{4\sigma h}{\gamma (g+3\tau )}, d_2=d_3=0, ϵ=2\sqrt{-\frac{2\sigma }{g+3\tau }},$$(44)under the constraint conditions$$\begin{array}{cc}\mu \gamma (g+3\tau {)}^2+2\tau {\sigma }^2(g+\tau )=0,& \beta =0,\end{array}$$(45)where $\tau =\sqrt{g^2-4fh}$ and σ(g + 3τ) < 0. Applying these outcomes to (27) and using (24) and (25), the following cases of solutions to equation (5) are extracted.

Case II1. If f = 4, g = −4(1 + m²), h = 4m², Λ(ζ) = sn²(ζ, m), equation (5) acquires JEF solution of the form$$\begin{array}{l}p(x,t)=\sqrt{\frac{2\sigma }{\gamma (m^2-2)}}\mathrm{dn}\left(\sqrt{-\frac{\sigma }{m^2-2}}\xi \right)e^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda \sqrt{\frac{2\sigma }{\gamma (m^2-2)}}\mathrm{dn}\left(\sqrt{-\frac{\sigma }{m^2-2}}\xi \right)e^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })}.\end{array}$$(46)where σ > 0 and γ < 0. The conditions (45) reduce to$$\begin{array}{cc}\mu \gamma (m^2-2{)}^2+{\sigma }^2(m^2-1)=0, & \beta =0.\end{array}$$(47)

As m approaches 1, solution (46) changes to a soliton solution given by$$\begin{array}{l}p(x,t)=\sqrt{-\frac{2\sigma }{\gamma }}\mathrm{sech}\left(\sqrt{\sigma }\xi \right)e^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda \sqrt{-\frac{2\sigma }{\gamma }}\mathrm{sech}\left(\sqrt{\sigma }\xi \right)e^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(48)which illustrates bright soliton as exhibited in Figure 2, where σ > 0 and γ < 0. From (47), we arrive at$$\begin{array}{cc}\mu =0, & \beta =0.\end{array}$$(49)

Case II2. If f = 4(1 − m²), g = −4(1 − 2m²), h = −4m², Λ(ζ) = cn²(ζ, m), equation (5) has JEF solution of the form$$\begin{array}{l}p(x,t)=m\sqrt{-\frac{2\sigma }{\gamma (m^2+1)}}\mathrm{sn}\left(\sqrt{-\frac{\sigma }{m^2+1}}\xi \right)e^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda m\sqrt{-\frac{2\sigma }{\gamma (m^2+1)}}\mathrm{sn}\left(\sqrt{-\frac{\sigma }{m^2+1}}\xi \right)e^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(50)provided that σ < 0 and γ > 0. The conditions (45) reduce to$$\begin{array}{cc}\mu \gamma (m^2+1{)}^2+{\sigma }^2{m}^2=0, & \beta =0.\end{array}$$(51)

As m approaches 1, solution (50) reduces to a soliton solution given by$$\begin{array}{l}p(x,t)=\sqrt{-\frac{\sigma }{\gamma }}\tanh \left(\sqrt{-\frac{\sigma }{2}}\xi \right)e^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda \sqrt{-\frac{\sigma }{\gamma }}\tanh \left(\sqrt{-\frac{\sigma }{2}}\xi \right)e^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(52)which presents dark soliton as shown in Figure 1, where σ < 0 and γ > 0. From (51), we arrive at$$\begin{array}{cc}4\mu \gamma +{\sigma }^2=0, & \beta =0.\end{array}$$(53)

Case II3. If f = 1, g = 2(1 − 2m²), h = 1, $\mathrm{\Lambda }\left(\zeta \right)=\frac{{\mathrm{sn}}^2(\zeta ,m)}{(1\pm \mathrm{cn}(\zeta ,m){)}^2}$, equation (5) gains JEF solution of the form$$\begin{array}{l}p(x,t)={\left[\frac{4\sigma }{\gamma (m^2+n^2-6\mathrm{mn})}\frac{1-(m^2-\mathrm{mn})\left\{1+\mathrm{cn}\left(2\sqrt{\frac{\sigma }{m^2+n^2-6\mathrm{mn}}}\xi \right)\right\}}{1+\mathrm{cn}\left(2\sqrt{\frac{\sigma }{m^2+n^2-6\mathrm{mn}}}\xi \right)}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda {\left[\frac{4\sigma }{\gamma (m^2+n^2-6\mathrm{mn})}\frac{1-(m^2-\mathrm{mn})\left\{1+\mathrm{cn}\left(2\sqrt{\frac{\sigma }{m^2+n^2-6\mathrm{mn}}}\xi \right)\right\}}{1+\mathrm{cn}\left(2\sqrt{\frac{\sigma }{m^2+n^2-6\mathrm{mn}}}\xi \right)}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(54)where $n=\sqrt{m^2-1}$. The conditions (45) reduce to$$\begin{array}{cc}\mu \gamma (m^2+n^2-6\mathrm{mn}{)}^2-4{\sigma }^2\mathrm{mn}(m-n{)}^2=0, & \beta =0.\end{array}$$(55)

As m approaches 1, solution (54) degenerates to a soliton solution given by$$\begin{array}{l}p(x,t)={\left[-\frac{4\sigma }{\gamma }\frac{\mathrm{sech}\left(2\sqrt{\sigma }\xi \right)}{1+\mathrm{sech}\left(2\sqrt{\sigma }\xi \right)}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda {\left[-\frac{4\sigma }{\gamma }\frac{\mathrm{sech}\left(2\sqrt{\sigma }\xi \right)}{1+\mathrm{sech}\left(2\sqrt{\sigma }\xi \right)}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(56)which describes bright soliton as shown in Figure 2, where σ > 0 and γ < 0. From (55), we arrive at$$\begin{array}{cc}\mu =0, & \beta =0.\end{array}$$(57)

Set III. $$d_0=d_2=0, d_1=\frac{2\sigma (3f-2g+h)}{\gamma (g-3f)}, d_3=-\frac{2\sigma (3f-2g+h)}{\gamma (g-3f)}, ϵ=2\sqrt{\frac{\sigma }{g-3f}},$$(58)under the constraint conditions$$\begin{array}{c}\mu \gamma (g-3f{)}^2+f{\sigma }^2(3f-2g+h)=0,\\ 4\beta \sigma (3f-2g+h{)}^2-3{\gamma }^2(g-3f\left)\right(f-g+h)=0,\end{array}$$(59)where σ(g − 3f) > 0. Applying these outcomes to (27) and using (24) and (25), the following cases of solutions to equation (5) are extracted.

Case III1. If f = 4, g = −4(1 + m²), h = 4m², Λ(ζ) = sn²(ζ, m), equation (5) attains JEF solution of the form$$\begin{array}{l}p(x,t)={\left[-\frac{2\sigma (3m^2+5)}{\gamma (m^2+4)}\frac{\mathrm{sn}{\left(\sqrt{-\frac{\sigma }{m^2+4}}\xi \right)}^2}{1+\mathrm{sn}{\left(\sqrt{-\frac{\sigma }{m^2+4}}\xi \right)}^2}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda {\left[-\frac{2\sigma (3m^2+5)}{\gamma (m^2+4)}\frac{\mathrm{sn}{\left(\sqrt{-\frac{\sigma }{m^2+4}}\xi \right)}^2}{1+\mathrm{sn}{\left(\sqrt{-\frac{\sigma }{m^2+4}}\xi \right)}^2}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(60)provided that σ < 0 and γ > 0. The conditions (59) reduce to$$\begin{array}{cc}\mu \gamma (m^2+4{)}^2+{\sigma }^2(3m^2+5)=0,& 2\beta \sigma (3m^2+5{)}^2+3{\gamma }^2(m^2+4\left)\right(m^2+1)=0.\end{array}$$(61)

As m approaches 1, solution (60) transforms to a soliton solution given by$$\begin{array}{l}p(x,t)={\left[-\frac{16\sigma }{5\gamma }\frac{\tanh {\left(\sqrt{-\frac{\sigma }{5}}\xi \right)}^2}{1+\tanh {\left(\sqrt{-\frac{\sigma }{5}}\xi \right)}^2}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda {\left[-\frac{16\sigma }{5\gamma }\frac{\tanh {\left(\sqrt{-\frac{\sigma }{5}}\xi \right)}^2}{1+\tanh {\left(\sqrt{-\frac{\sigma }{5}}\xi \right)}^2}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(62)which represents dark soliton as exhibited in Figure 3, where σ < 0 and γ > 0. From (61), we arrive at$$\begin{array}{cc}25\mu \gamma +8{\sigma }^2=0, & 64\beta \sigma +15{\gamma }^2=0.\end{array}$$(63)

Figure 3 Intensity profile of dark soliton given by solutions (62), (70) and (96) with the same parameter values as in Figure 1 except b1 = −0.5, ω1 = 0.128, ν = 0.75. (a) 3D plot of dark soliton; (b) |p|2 with distinct values of α; (c) |q|2 with distinct values of α; (d) Chirping profile.

Case III2. If f = 4(1 − m²), g = −4(1 − 2m²), h = −4m², Λ(ζ) = cn²(ζ, m), equation (5) possesses JEF solution of the form$$\begin{array}{l}p(x,t)={\left[-\frac{2\sigma (8m^2-5)}{\gamma (5m^2-4)}\frac{\mathrm{cn}{\left(\sqrt{\frac{\sigma }{5m^2-4}}\xi \right)}^2}{1+\mathrm{cn}{\left(\sqrt{-\frac{\sigma }{5m^2-4}}\xi \right)}^2}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda {\left[-\frac{2\sigma (8m^2-5)}{\gamma (5m^2-4)}\frac{\mathrm{cn}{\left(\sqrt{\frac{\sigma }{5m^2-4}}\xi \right)}^2}{1+\mathrm{cn}{\left(\sqrt{\frac{\sigma }{5m^2-4}}\xi \right)}^2}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })}.\end{array}$$(64)

The conditions (59) become$$\begin{array}{cc}\mu \gamma (5m^2-4{)}^2+{\sigma }^2(m^2-1\left)\right(8m^2-5)=0, & 2\beta \sigma (8m^2-5{)}^2+3{\gamma }^2(2m^2-1\left)\right(5m^2-4)=0.\end{array}$$(65)

As m approaches 1, solution (64) converts to a soliton solution given by$$\begin{array}{l}p(x,t)={\left[-\frac{6\sigma }{\gamma }\frac{\mathrm{sech}{\left(\sqrt{\sigma }\xi \right)}^2}{1+\mathrm{sech}{\left(\sqrt{\sigma }\xi \right)}^2}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_1\right(\xi )-{\omega }_1\frac{t^{\alpha }}{\alpha })},\\ q(x,t)=\lambda {\left[-\frac{6\sigma }{\gamma }\frac{\mathrm{sech}{\left(\sqrt{\sigma }\xi \right)}^2}{1+\mathrm{sech}{\left(\sqrt{\sigma }\xi \right)}^2}\right]}^{\frac{1}{2}}{e}^{i\left({\varphi }_2\right(\xi )-{\omega }_2\frac{t^{\alpha }}{\alpha })},\end{array}$$(66)which characterizes bright soliton as demonstrated in Figure 4, where σ > 0 and γ < 0. From (65), we obtain$$\begin{array}{cc}\mu =0, & 6\beta \sigma +{\gamma }^2=0.\end{array}$$(67)