Introduction

One of the models that was proposed a few decades ago is the Sasa–Satsuma equation that is in fact the perturbed version of the familiar nonlinear Schrödinger’s equation with Kerr law of self–phase modulation [1,2,3,4,5,6,7,8,9]. This model has been extensively studied in a wide range of optoelectronic devices and a pletora of results have been recovered and reported over the decades [10,11,12,13,14,15,16,17,18,19]. The current paper is going to extend the study further along. The study of optical solitons with this model will now be extended to address optical dromions and that too with polarization–mode dispersion (PMD) [20,21,22,23,24,25,26,27,28,29]. To top it off, the model will include spatio–temporal dispersions (STD), along the orthogonal directions, in addition to the usual chromatic dispersion (CD) [30,31,32,33,34,35,36,37,38,39]. The refractive index change is taken in the form of parabolic law of nonlinearity [40,41,42,43,44,45,46,47,48,49]. Additionally, in order to give a flavorful taste from a practical standpoint, the effect of stochasticity, in the form of white noise, is also included along both components of the model with PMD [50,51,52,53,54,55,56,57,58,59]. This is the governing model that will be addressed in order to analytically recover the mathematical structure of the dromions and their classifications will also be carried out. The two mathematical algorithms that are adopted in the paper to recover these dromions are the generalized projective Riccati equations method and the extended auxiliary equation approach [60,61,62,63,64,65]. These two schemes collectively yield a range of structures for optical dromions that are enumerated and exhibited in the rest of the paper.

Governing model

The governing model that describes the propagation of optical dromions with PMD in presence of white noise is structured in its dimensionless form as

$$\begin{aligned}{} & {} iq_{t}+a\left( q_{xx}+q_{yy}\right) +bq_{xt}+cq_{yt}\nonumber \\{} & {} +\left( d\left| q\right| ^{2}+e\left| q\right| ^{4}\right) q+\sigma \left[ q-i\left( bq_{x}+cq_{y}\right) \right] \frac{dW(t)}{dt}\nonumber \\{} & {} =i\left[ \lambda q_{x}+\mu \left( \left| q\right| ^{2}q\right) _{x}\right. \nonumber \\{} & {} \left. +\theta \left( \left| q\right| ^{2}\right) _{x}q+\upsilon \left| q\right| ^{2}q_{x}\right] ,\ \ \ \ \ \ \ \ \ \end{aligned}$$
(1)

where the wave profile is represented by the complex-valued function q(x, t), and \(i=\sqrt{-1}\). Linear temporal evolution is the first term in Eq. (1); coefficients of CD and STD in directions x and y are represented by the constants a, b, and c respectively. Then, d represents the self-phase modulation (SPM) coefficient, e denotes the quintic nonlinearity coefficient, \(\sigma\) denotes the noise strength coefficient, and W(t) represents the standard Wiener process, with dW(t)/dt representing the white noise where W(t) represents the Weiner process.

For PMD, Eq. (1) splits into two components as:

$$\begin{aligned}{} & {} iu_{t}+a_{1}\left( u_{xx}+u_{yy}\right) +b_{1}u_{xt}+c_{1}u_{yt}\nonumber \\{} & {} +\left( d_{1}\left| u\right| ^{2}+e_{1}\left| v\right| ^{2}\right) u+\left( f_{1}\left| u\right| ^{4}\right. \nonumber \\{} & {} \left. +g_{1}\left| u\right| ^{2}\left| v\right| ^{2}+h_{1}\left| v\right| ^{4}\right) u\nonumber \\{} & {} +\sigma _{1}\left[ u-i\left( b_{1}u_{x}+c_{1}u_{y}\right) \right] \frac{ dW_{1}(t)}{dt}\nonumber \\{} & {} =i\left[ \lambda _{1}u_{x}+\mu _{1}\left( \left| u\right| ^{2}u\right) _{x}+\theta _{1}\left( \left| u\right| ^{2}\right) _{x}u\right. \nonumber \\{} & {} \left. +\upsilon _{1}\left| u\right| ^{2}u_{x}\right] , \end{aligned}$$
(2)

and

$$\begin{aligned}{} & {} iv_{t}+a_{2}\left( v_{xx}+v_{yy}\right) +b_{2}v_{xt}+c_{2}v_{yt}\nonumber \\{} & {} +\left( d_{2}\left| v\right| ^{2}+e_{2}\left| u\right| ^{2}\right) v\nonumber \\{} & {} +\left( f_{2}\left| v\right| ^{4}+g_{2}\left| u\right| ^{2}\left| v\right| ^{2}+h_{2}\left| u\right| ^{4}\right) v\nonumber \\{} & {} +\sigma _{2}\left[ v-i\left( b_{2}v_{x}+c_{2}v_{y}\right) \right] \frac{ dW_{2}(t)}{dt}\nonumber \\{} & {} =i\left[ \lambda _{2}v_{x}+\mu _{2}\left( \left| v\right| ^{2}v\right) _{x}\right. \nonumber \\{} & {} \left. +\theta _{2}\left( \left| v\right| ^{2}\right) _{x}v+\upsilon _{2}\left| v\right| ^{2}v_{x}\right] , \end{aligned}$$
(3)

where the wave profiles are represented by the complex-valued functions \(u\left( x,t\right)\) and \(v\left( x,t\right)\). The coefficients of CD and STD in the x and y directions are denoted by the constants \(a_{j},~b_{j}\) and \(c_{j}\). The coefficients for SPM and cross-phase modulation (XPM) are denoted by the constants \(d_{j}\) and \(e_{j}\), respectively, \((j=1,2)\). The coefficients of nonlinear dispersion terms are denoted by the constants \(f_{j},~g_{j}\) and \(h_{j},\) \((j=1,2)\). The standard Wiener processes, \(W_{j}(t),~(j=1,2)\) and the coefficients of noise strength, \(\sigma _{j},\) \((j=1,2),\) such that \(dW_{j}(t)/dt,~(j=1,2)\) are the white noises. In conclusion, the coefficients corresponding to the nonlinear dispersions terms, self-steepening (SS) terms, and intermodal dispersion (IMD) terms are represented by the constants \(\lambda _{j},~\mu _{j},~\theta _{j}\) and \(\upsilon _{j},\) \((j=1,2)\), respectively.

This article aims to derive the dromion solutions for Eqs.(2) and (3) by using the two mathematical techniques as indicated.

Mathematical analysis

The wave profiles are assumed to have the following forms in order to solve Eqs. (2) and (3):

$$\begin{aligned} u\left( x,y,t\right)= & {} \phi _{1}\left( \xi \right) \exp \left[ i\left( \psi _{1}\left( x,t\right) +\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right) \right] , \text { \ \ } \end{aligned}$$
(4)
$$\begin{aligned} v\left( x,y,t\right)= & {} \phi _{2}\left( \xi \right) \exp \left[ i\left( \psi _{2}\left( x,t\right) \right. \right. \nonumber \\{} & {} \left. \left. +\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right) \right] , \end{aligned}$$
(5)
$$\begin{aligned} \xi= & {} x+y-\rho t,~\psi _{j}\left( x,t\right) =-\kappa _{j1}x-\kappa _{j2}y+\omega _{j}t,\ j=1,2, \end{aligned}$$
(6)

where \(\kappa _{j1}\), \(\kappa _{j2}\), \(\omega _{j}\), ( \(j=1,2)\) and \(\rho\) are real-valued constants that are nonzero. The phase component yields the frequencies of the dromions along the x- and y-directions, \(\kappa _{j1}\) and \(\kappa _{j2},(\) \(j=1,2)\), respectively, while \(\omega _{j}\), ( \(j=1,2)\) are the wave number and \(\rho\) is the velocity dromion. The amplitude portions and phase components of the dromions are represented by the real functions \(\phi _{j}(\xi )\) and \(\psi _{j}\left( x,t\right)\) for \(j=1,2\), respectively. The real parts can be obtained by inserting (4) and (5) into Eqs. (2) and (3):

$$\begin{aligned}{} & {} \left[ 2\,a_{1}-\rho \,\left( b_{1}-\,c_{1}\right) \right] \phi _{1}^{\prime \prime }\left( \xi \right) \nonumber \\{} & {} +\left[ \left( \omega _{1}-\sigma _{1}^{2}{}\right) \left( b_{1}\kappa _{11}+c_{1}\kappa _{12}-1\right) \right. \nonumber \\{} & {} \left. -a_{1}\left( \kappa _{11}^{2}{}+\kappa _{12}^{2}{}\right) -\lambda _{1}\kappa _{11}\right] \phi _{1}\left( \xi \right) \nonumber \\{} & {} +\left( h_{1}\phi _{2}^{2}\left( \xi \right) +e_{1}\right) \phi _{1}\left( \xi \right) \phi _{2}^{2}\left( \xi \right) +\left[ g_{1}\phi _{2}^{2}\left( \xi \right) \right. \nonumber \\{} & {} \left. -\left( \mu _{1}+\upsilon _{1}\right) \kappa _{11}+d_{1}\right] \phi _{1}^{3}\left( \xi \right) +f_{1}\phi _{1}^{5}\left( \xi \right) =0, \end{aligned}$$
(7)

and

$$\begin{aligned}{} & {} \left[ 2\,a_{2}-\rho \,\left( b_{2}-\,c_{2}\right) \right] \phi _{2}^{\prime \prime }\left( \xi \right) \nonumber \\{} & {} +\left[ \left( \omega _{2}-\sigma _{2}^{2}{}\right) \left( b_{2}\kappa _{21}+c_{2}\kappa _{22}-1\right) \right. \nonumber \\{} & {} \left. -a_{2}\left( \kappa _{21}^{2}{}+\kappa _{22}^{2}{}\right) -\lambda _{2}\kappa _{21}\right] \phi _{2}\left( \xi \right) \nonumber \\{} & {} +\left( h_{2}\phi _{1}^{2}\left( \xi \right) +e_{2}\right) \phi _{2}\left( \xi \right) \phi _{1}^{2}\left( \xi \right) \nonumber \\{} & {} +\left[ g_{2}\phi _{1}^{2}\left( \xi \right) \right. \nonumber \\{} & {} \left. -\left( \mu _{2}+\upsilon _{2}\right) \kappa _{21}+d_{2}\right] \phi _{2}^{3}\left( \xi \right) +f_{2}\phi _{2}^{5}\left( \xi \right) =0, \end{aligned}$$
(8)

while the imaginary parts are:

$$\begin{aligned}{} & {} [\rho \left( 1-b_{1}\kappa _{11}-c_{1}\kappa _{12}\right) \nonumber \\{} & {} +2a_{1}\left( \,\kappa _{11}+\,\kappa _{12}\right) -\left( b_{1}+c_{1}\right) \left( \omega _{1}-\sigma _{1}^{2}{}\right) +\lambda _{1}]\phi _{1}^{\prime }\left( \xi \right) \nonumber \\{} & {} +\left( \upsilon _{1}+3\,\mu _{1}+2\,\theta _{1}\right) \phi _{1}^{2}\left( \xi \right) \phi _{1}^{\prime }\left( \xi \right) =0, \end{aligned}$$
(9)

and

$$\begin{aligned}{} & {} \left[ \rho \left( 1-b_{2}\kappa _{21}-c_{2}\kappa _{22}\right) +2\,a_{2}\left( \kappa _{21}+\kappa _{22}\right) \right. \nonumber \\{} & {} \left. -\left( b_{2}+c_{2}\right) \left( \omega _{2}-\sigma _{2}^{2}{}\right) +\lambda _{2}\right] \phi _{2}^{\prime }\left( \xi \right) \nonumber \\{} & {} +\left( \upsilon _{2}+3\,\mu _{2}+2\,\theta _{2}\right) \phi _{2}^{2}\left( \xi \right) \phi _{2}^{\prime }\left( \xi \right) =0. \end{aligned}$$
(10)

Applying the linearly independent principle to (9) and (10) yields the wave numbers \(\omega _{1}\) and \(\omega _{2}:\)

$$\begin{aligned} \omega _{j}=\frac{\left( 1-b_{j}\kappa _{j1}-c_{j}\kappa _{j2}\right) \rho +2\left( \,\kappa _{j1}+\,\kappa _{j2}\right) a_{j}+\sigma _{j}^{2}\left( b_{j}+c_{j}\right) +\lambda _{j}}{b_{j}+c_{j}}, \end{aligned}$$
(11)

and

$$\begin{aligned} \upsilon _{j}+3\,\mu _{j}+2\,\theta _{j}=0, \end{aligned}$$
(12)

provided \(b_{j}+c_{j}\ne 0,~(j=1,2).\)

Now, let us set

$$\begin{aligned} \phi _{2}(\xi )=\varkappa \phi _{1}(\xi ), \end{aligned}$$
(13)

where \(\varkappa\) is a nonzero constant, such that \(\varkappa \ne 1.\) Eqs. (7) and (8) can be reduced to:

$$\begin{aligned}{} & {} \left[ 2a_{1}-\rho \left( b_{1}-c_{1}\right) \right] \phi _{1}^{\prime \prime }\left( \xi \right) \nonumber \\{} & {} +\left[ \left( \omega _{1}-\sigma _{1}^{2}\right) \left( b_{1}\kappa _{11}+c_{1}\kappa _{12}-1\right) -a_{1}\left( \kappa _{11}^{2}+\kappa _{12}^{2}\right) -\lambda _{1}\kappa _{11}\right] \phi _{1}\left( \xi \right) \nonumber \\{} & {} \left[ d_{1}-\left( \mu _{1}+\upsilon _{1}\right) \kappa _{11}+\chi ^{2}e_{1} \right] \phi _{1}^{3}\left( \xi \right) \nonumber \\{} & {} +\left( f_{1}+\chi ^{2}g_{1}+\chi ^{4}h_{1}\right) \phi _{1}^{5}\left( \xi \right) =0, \end{aligned}$$
(14)

and

$$\begin{aligned}{} & {} \left[ 2\,a_{2}-\rho \left( b_{2}-c_{2}\right) \right] \phi _{1}^{\prime \prime }\left( \xi \right) \nonumber \\{} & {} +\left[ \left( \omega _{2}-\sigma _{2}^{2}\right) \left( b_{2}\kappa _{21}+c_{2}\kappa _{22}-1\right) -a_{2}\left( \kappa _{21}^{2}+\kappa _{22}^{2}\right) \right. \nonumber \\{} & {} \left. -\lambda _{2}\kappa _{21}\right] \phi _{1}\left( \xi \right) \nonumber \\{} & {} +\left[ \chi ^{2}\left( d_{2}-\left( \mu _{2}+\upsilon _{2}\right) \kappa _{21}\right) +e_{2}\right] \phi _{1}^{3}\left( \xi \right) \nonumber \\{} & {} +\left( \chi ^{4}f_{2}+\chi ^{2}g_{2}+h_{2}\right) \phi _{1}^{5}\left( \xi \right) =0. \end{aligned}$$
(15)

The constraints conditions below make Eqs.(14) and (15) equal:

$$\begin{aligned}{} & {} 2\,a_{1}-\rho \,\left( b_{1}-\,c_{1}\right) =2\,a_{2}-\rho \,\left( b_{2}-\,c_{2}\right) , \end{aligned}$$
(16)
$$\begin{aligned}{} & {} \left( \omega _{1}-\sigma _{1}^{2}{}\right) \left( b_{1}\kappa _{11}+c_{1}\kappa _{12}-1\right) \nonumber \\{} & {} -a_{1}\left( \kappa _{11}^{2}{}+\kappa _{12}^{2}{}\right) -\lambda _{1}\kappa _{11}\nonumber \\{} & {} =\left( \omega _{2}-\sigma _{2}^{2}{}\right) \left( b_{2}\kappa _{21}+c_{2}\kappa _{22}-1\right) \nonumber \\{} & {} -a_{2}\left( \kappa _{21}^{2}{}+\kappa _{22}^{2}{}\right) -\lambda _{2}\kappa _{21}, \end{aligned}$$
(17)
$$\begin{aligned}{} & {} d_{1}-\left( \mu _{1}+\upsilon _{1}\right) \kappa _{11}+\chi ^{2}e_{1}\nonumber \\{} & {} =\chi ^{2}\left( d_{2} -\left( \mu _{2}+\upsilon _{2}\right) \kappa _{21}\right) +e_{2}, \end{aligned}$$
(18)
$$\begin{aligned}{} & {} f_{1}+\chi ^{2}g_{1}+\chi ^{4}h_{1}=\chi ^{4}f_{2}+\chi ^{2}g_{2}+h_{2}. \end{aligned}$$
(19)

From (16), the dromion velocity is yielded as:

$$\begin{aligned} \rho \,=\frac{2\,\left( a_{2}-\,a_{1}\right) }{b_{2}\,+c_{2}-\,(b_{1}+c_{1})}, \end{aligned}$$
(20)

provided \(b_{1}+c_{1}\ne b_{2}\,+c_{2}\) and \(\,a_{1}\ne a_{2}.\)

Equation (14) can be rewritten as:

$$\begin{aligned} \phi _{1}^{\prime \prime }\left( \xi \right) +l\phi _{1}\left( \xi \right) +m\phi _{1}^{3}\left( \xi \right) +n\phi _{1}^{5}\left( \xi \right) =0, \end{aligned}$$
(21)

where l,  m and n are constant coefficients given by:

$$\begin{aligned} l= & {} \frac{\left( \omega _{1}-\sigma _{1}^{2}{}\right) \left( b_{1}\kappa _{11}+c_{1}\kappa _{12}-1\right) -a_{1}\left( \kappa _{11}^{2}{}+\kappa _{12}^{2}{}\right) -\lambda _{1}\kappa _{11}}{2\,a_{1}-\rho \,\left( b_{1}-\,c_{1}\right) }, \end{aligned}$$
(22)
$$\begin{aligned} m= & {} \frac{d_{1}-\left( \mu _{1}+\upsilon _{1}\right) \kappa _{11}+\chi ^{2}e_{1}}{2\,a_{1}-\rho \,\left( b_{1}-\,c_{1}\right) }, \end{aligned}$$
(23)
$$\begin{aligned} n= & {} \frac{f_{1}+\chi ^{2}g_{1}+\chi ^{4}h_{1}}{2\,a_{1}-\rho \,\left( b_{1}-\,c_{1}\right) }, \end{aligned}$$
(24)

provided

$$\begin{aligned} \rho \,\left( b_{1}-\,c_{1}\right) \ne 2\,a_{1}. \end{aligned}$$

In the next two sections, we will use the generalized projective Riccati equations method and the extended auxiliary equation approach to create the dromions and other exact wave solutions of Eqs. (2) and (3).

Generalized projective Riccati equation method

Balancing \(\phi _{1}^{\prime \prime }(\xi )\) and \(\phi _{1}^{5}(\xi )\) in Eq. (21) gives \(N=\) \(\frac{1}{2}\). Therefore, the new wave transformation:

$$\begin{aligned} \phi _{1}(\xi )=[U(\xi )]^{\frac{1}{2}}, \end{aligned}$$
(25)

changes Eq. (21) to the following new nonlinear ordinary differential equations (ODE):

$$\begin{aligned}{} & {} U\left( \xi \right) U^{\prime \prime }\left( \xi \right) -\frac{1}{2} U^{\prime ^{2}}\left( \xi \right) \nonumber \\{} & {} \quad +2U^{2}\left( \xi \right) \left( l+mU\left( \xi \right) +nU^{2}\left( \xi \right) \right) =0, \end{aligned}$$
(26)

where \(U(\xi )\) is a new function of \(\ \xi\), such that \(U(\xi )>0\).

In Eq. (26), balancing the terms \(U(\xi )U^{\prime \prime }(\xi )~\)and \(U^{4}(\xi )\) yields the balance number \(N=1\). The generalized projective Riccati equations method assumes the solution of Eq. (26) has the form:

$$\begin{aligned} U(\xi )=A_{0}+A_{1}F(\xi )+B_{1}G(\xi ), \end{aligned}$$
(27)

where \(A_{0}\), \(A_{1}\) and \(B_{1}\) are constants such that \(A_{1}^{2}+B_{1}^{2}\ne 0\). The functions, \(F(\xi )\) and \(G(\xi )\) satisfy the following system of ODEs:

$$\begin{aligned} F^{\prime }(\xi )= & {} \varepsilon F(\xi )G(\xi ), \end{aligned}$$
(28)
$$\begin{aligned} G^{\prime }(\xi )= & {} R+\varepsilon G^{2}(\xi )-\gamma F(\xi ), \ \ \ \varepsilon =\pm 1, \end{aligned}$$
(29)

where

$$\begin{aligned} G^{2}(\xi )=-\varepsilon \left( R-2\gamma F(\xi )+\frac{\gamma ^{2}+\delta }{ R}F^{2}(\xi )\right) , \end{aligned}$$
(30)

here \(\delta =\pm 1\) while R and \(\gamma\) are nonzero constants.

It is well known that Eqs. (28) and (29) have many explicit solutions. Next, substituting (27) into Eq. (26) and using (28)-(30), in \(F(\xi )\) and \(G(\xi )\), the left side of Eq. (26) turns into a polynomial. This polynomial’s coefficients can be set to zero to produce the following system of algebraic equations:

$$\begin{aligned}&F^{4}:2nA_{1}^{4}-\frac{\varepsilon \left( \gamma ^{2}+\delta \right) }{R}\left( \frac{3}{2}\varepsilon ^{2}A_{1}^{2}\right. \nonumber \\&\left. +12nA_{1}^{2}B_{1}^{2}\right) +\frac{\varepsilon ^{2}\left( \gamma ^{2}+\delta \right) ^{2}}{R^{2}}\left( \frac{3}{2}\varepsilon ^{2}B_{1}^{2}+2nB_{1}^{4}\right) =0,\nonumber \\&F^{3}:2mA_{1}^{3}+2\varepsilon \gamma \left( \frac{3}{2} \varepsilon ^{2}A_{1}^{2}+12nA_{1}^{2}B_{1}^{2}\right) \nonumber \\&-\frac{\varepsilon \left( \gamma ^{2}+\delta \right) }{R}\left( 6mA_{1}B_{1}^{2}-2\gamma \varepsilon B_{1}^{2}\right. \nonumber \\&\left. +2\varepsilon ^{2}A_{0}A_{1}+24nA_{0}A_{1}B_{1}^{2}\right) \nonumber \\&+8nA_{0}A_{1}^{3}-\gamma \varepsilon A_{1}^{2}\nonumber \\&-\frac{4\gamma \varepsilon ^{2}\left( \gamma ^{2}+\delta \right) }{R}\left( \frac{3}{2}\varepsilon ^{2}B_{1}^{2}+2nB_{1}^{4}\right) =0,\nonumber \\&F^{3}G:8nA_{1}^{3}B_{1}-\frac{\varepsilon \left( \gamma ^{2}+\delta \right) }{R}\left( 3\varepsilon ^{2}A_{1}B_{1}+8nA_{1}B_{1}^{3}\right) =0,\nonumber \\&F^{2}:2lA_{1}^{2}-\frac{1}{2}\gamma ^{2}B_{1}^{2}+6mA_{0}A_{1}^{2}+\varepsilon ^{2}\left( 4\gamma ^{2}\right. \nonumber \\&\left. +2\left( \gamma ^{2}+\delta \right) \right) \left( \frac{3}{2}\varepsilon ^{2}B_{1}^{2}+2nB_{1}^{4}\right) \nonumber \\&+12nA_{0}^{2}A_{1}^{2}+R\varepsilon A_{1}^{2}\nonumber \\&-\varepsilon R\left( \frac{3}{2}\varepsilon ^{2}A_{1}^{2}+12nA_{1}^{2}B_{1}^{2}\right) +2\varepsilon \gamma \left( 6mA_{1}B_{1}^{2}-2\gamma \varepsilon B_{1}^{2}\right. \nonumber \\&\left. +2\varepsilon ^{2}A_{0}A_{1}+24nA_{0}A_{1}B_{1}^{2}\right) \nonumber \\&-\frac{\varepsilon \left( \gamma ^{2}+\delta \right) }{R}\left( 2lB_{1}^{2}+6mA_{0}B_{1}^{2}\right. \nonumber \\&\left. +12nA_{0}^{2}B_{1}^{2}+R\varepsilon B_{1}^{2}\right) -\gamma \varepsilon A_{0}A_{1}=0,\nonumber \\&F^{2}G:\frac{\varepsilon \left( \gamma ^{2}+\delta \right) }{R} \left( 2mB_{1}^{3}+8nA_{0}B_{1}^{3}+2\varepsilon ^{2}A_{0}B_{1}\right) \nonumber \\&-2\varepsilon \gamma \left( 3A_{1}\varepsilon ^{2}B_{1}+8nA_{1}B_{1}^{3}\right) -6mA_{1}^{2}B_{1}-24nA_{0}A_{1}^{2}B_{1}\nonumber \\&+3\gamma \varepsilon A_{1}B_{1}=0,\nonumber \\&F:-\varepsilon R\left( 6mA_{1}B_{1}^{2}-2\gamma \varepsilon B_{1}^{2}+2\varepsilon ^{2}A_{0}A_{1}+24nA_{0}A_{1}B_{1}^{2}\right) \nonumber \\&+2\varepsilon \gamma \left( 2lB_{1}^{2}+6mA_{0}B_{1}^{2}+12nA_{0}^{2}B_{1}^{2}\right) \nonumber \\&+6mA_{0}^{2}A_{1}+8nA_{0}^{3}A_{1}+4lA_{0}A_{1}+R\gamma B_{1}^{2}\nonumber \\&-4R\gamma \varepsilon ^{2}\left( \frac{3}{2}\varepsilon ^{2}B_{1}^{2}+2nB_{1}^{4}\right) +R\varepsilon A_{0}A_{1}+2\varepsilon \gamma R\varepsilon B_{1}^{2}=0,\nonumber \\&FG:4lA_{1}B_{1}-\varepsilon R\left( 3A_{1}\varepsilon ^{2}B_{1}+8nA_{1}B_{1}^{3}\right) +2\gamma \left( 2mB_{1}^{3}+8nA_{0}B_{1}^{3}\right. \nonumber \\&\left. +2\varepsilon ^{2}A_{0}B_{1}\right) +24nA_{0}^{2}A_{1}B_{1} \nonumber \\&+2R\varepsilon A_{1}B_{1}+12mA_{0}A_{1}B_{1}-3\gamma \varepsilon A_{0}B_{1}=0,\nonumber \\&G:6mA_{0}^{2}B_{1}+8nA_{0}^{3}B_{1}-R\varepsilon \left( 2mB_{1}^{3}+8nA_{0}B_{1}^{3} +2\varepsilon ^{2}A_{0}B_{1}\right) \nonumber \\&+4lA_{0}B_{1}+2R\varepsilon A_{0}B_{1}=0,\nonumber \\&F^{0}:+2mA_{0}^{3}+2nA_{0}^{4}+R^{2}\varepsilon ^{2}\left( \frac{3 }{2}\varepsilon ^{2}B_{1}^{2}+2nB_{1}^{4}\right) \nonumber \\&-R\varepsilon \left( 2lB_{1}^{2}+6mA_{0}B_{1}^{2}+12nA_{0}^{2}B_{1}^{2}+R\varepsilon B_{1}^{2}\right) \nonumber \\&-\frac{1}{2}R^{2}B_{1}^{2}+2lA_{0}^{2}=0. \end{aligned}$$
(31)

Next, the algebraic equations (31) have two different types of solutions, which will be covered in the following discussion:

Type–1: Substituting \(\varepsilon =-1,\) \(\delta =-1\) in the above algebraic equations (31) and solving them by Maple, we have the results:

Result–1:

$$\begin{aligned} A_{0}= & {} 0,\text { }A_{1}=\frac{\gamma }{m},~B_{1}=0,~R=\nonumber \\{} & {} -\frac{3m^{2}\left( \gamma ^{2}-1\right) }{4n\gamma ^{2}},~l=\frac{3m^{2}\left( \gamma ^{2}-1\right) }{16n\gamma ^{2}}. \end{aligned}$$
(32)

Now, we have the bright dromion solutions:

$$\begin{aligned} u_{1}\left( x,y,t\right)= & {} \left\{ -\dfrac{3m\left( \gamma ^{2}-1\right) }{ 4n\gamma }\left( \dfrac{1}{\gamma +\cosh \left( \sqrt{-\frac{3\left( \gamma ^{2}-1\right) m^{2}}{4n\gamma ^{2}}}\xi \right) }\right) \right\} ^{\frac{1}{ 2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] },\end{aligned}$$
(33)
$$\begin{aligned} v_{1}\left( x,y,t\right)= & {} \varkappa \left\{ -\dfrac{3m\left( \gamma ^{2}-1\right) }{4n\gamma }\left( \dfrac{1}{\gamma +\cosh \left( \sqrt{-\frac{ 3\left( \gamma ^{2}-1\right) m^{2}}{4n\gamma ^{2}}}\xi \right) }\right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(34)

provided \(n\left( \gamma ^{2}-1\right) <0\) and \(m\gamma >0.\)

Result–2:

$$\begin{aligned} A_{0}= & {} -\frac{3m\left( \gamma \pm 1\right) }{4n\left( \gamma \pm 2\right) }, \nonumber \\ A_{1}= & {} -\frac{\gamma \pm 2}{m},~B_{1}=0,~R=-\frac{3m^{2}\left( \gamma ^{2}-1\right) }{4n\left( \gamma \pm 2\right) ^{2}},\nonumber \\ l= & {} \frac{3m^{2}\left( \gamma ^{2}\pm 6\gamma +5\right) }{16n\left( \gamma \pm 2\right) ^{2}}. \end{aligned}$$
(35)

Based on this result, we can infer the bright dromion solutions:

$$\begin{aligned} u_{2}\left( x,y,t\right)= & {} \left\{ -\frac{3m\left( \gamma \pm 1\right) }{ 4n\left( \gamma \pm 2\right) }\left( 1-\dfrac{\left( \gamma \mp 1\right) }{ \gamma +\cosh \left( \sqrt{-\frac{3m^{2}\left( \gamma ^{2}-1\right) }{ 4n\left( \gamma \pm 2\right) ^{2}}}\xi \right) }\right) \right\} ^{\frac{1}{2 }}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(36)
$$\begin{aligned} v_{2}\left( x,y,t\right)= & {} \varkappa \left\{ -\frac{3m\left( \gamma \pm 1\right) }{4n\left( \gamma \pm 2\right) }\left( 1-\dfrac{\left( \gamma \mp 1\right) }{\gamma +\cosh \left( \sqrt{-\frac{3m^{2}\left( \gamma ^{2}-1\right) }{4n\left( \gamma \pm 2\right) ^{2}}}\xi \right) }\right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(37)

provided \(n\left( \gamma ^{2}-1\right) <0\) and \(\frac{m\left( \gamma \pm 1\right) }{n\left( \gamma \pm 2\right) }<0.\)

Result–3:

$$\begin{aligned} A_{0}=-\frac{2l}{m},\text { }A_{1}=0,~B_{1}=\pm \frac{2}{m}\sqrt{-l},~R=-l,~\gamma =0,~n=\frac{3m^{2}}{16l}, \end{aligned}$$
(38)

where \(l<0.\)

Based on this result, we can infer the dark dromion solutions:

$$\begin{aligned} u_{3}\left( x,y,t\right)= & {} \pm \left\{ -\frac{2l}{m}\left( 1\pm \tanh \left( \sqrt{-l}\xi \right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(39)
$$\begin{aligned} v_{3}\left( x,y,t\right)= & {} \pm \varkappa \left\{ -\frac{2l}{m}\left( 1\pm \tanh \left( \sqrt{-l}\xi \right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(40)

provided \(m>0.\)

Result–4:

$$\begin{aligned} A_{0}= & {} -\frac{3m}{8n},\text { }A_{1}=\pm \frac{\sqrt{\gamma ^{2}-1}}{2m},\nonumber \\ B_{1}= & {} \pm \sqrt{-\frac{3}{16n}},~R=-\frac{3m^{2}}{4n},~l=\frac{3m^{2}}{16n}, \end{aligned}$$
(41)

where \(\gamma ^{2}>1\) and \(n<0.\)

This result leads to straddled dark–bright dromion solutions:

$$\begin{aligned} u_{4}\left( x,y,t\right)&=\left\{ -\dfrac{3m}{8n}\left( 1\pm \dfrac{\sqrt{ \gamma ^{2}-1}\text{ sech }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) }{ \gamma \text{ sech }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) +1}\right. \right. \nonumber \\&\left. \left. \pm \dfrac{\tanh \left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) }{\gamma \text{ sech }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) +1}\right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(42)
$$\begin{aligned} v_{4}\left( x,y,t\right)&=\varkappa \left\{ -\dfrac{3m}{8n}\left( 1\pm \dfrac{\sqrt{\gamma ^{2}-1}\text{ sech }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) }{\gamma \text{ sech }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) +1} \right. \right. \nonumber \\&\left. \left. \pm \dfrac{\tanh \left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) }{\gamma \text{ sech }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) +1}\right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(43)

provided \(m>0.\)

Type 2. Substituting \(\varepsilon =-1,\) \(\delta =1\) in the above algebraic equations (31) and solving them by Maple, we get the results:

Result–1:

$$\begin{aligned} A_{0}= & {} 0,\text { }A_{1}=\frac{\gamma }{m},~B_{1}=0,\nonumber \\ R= & {} -\frac{3m^{2}\left( \gamma ^{2}+1\right) }{4n\gamma ^{2}},~l=\frac{3m^{2}\left( \gamma ^{2}+1\right) }{16n\gamma ^{2}}. \end{aligned}$$
(44)

From (44), we construct the singular dromion solutions:

$$\begin{aligned} u_{5}\left( x,y,t\right)= & {} \left\{ -\dfrac{3m\left( \gamma ^{2}+1\right) }{ 4n\gamma }\left( \dfrac{1}{\gamma +\sinh \left( \sqrt{-\frac{3\left( \gamma ^{2}+1\right) m^{2}}{4n\gamma ^{2}}}\xi \right) }\right) \right\} ^{\frac{1}{ 2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(45)
$$\begin{aligned} v_{5}\left( x,y,t\right)= & {} \varkappa \left\{ -\dfrac{3m\left( \gamma ^{2}+1\right) }{4n\gamma }\left( \dfrac{1}{\gamma +\sinh \left( \sqrt{-\frac{ 3\left( \gamma ^{2}+1\right) m^{2}}{4n\gamma ^{2}}}\xi \right) }\right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(46)

provided \(n<0\) and \(m\gamma >0.\)

Result–2:

$$\begin{aligned} A_{0}=-\frac{2l}{m},\text { }A_{1}=0,~B_{1}=\pm \frac{2}{m}\sqrt{-l},~\gamma =0,~R=-l,~n=\frac{3m^{2}}{16l}, \end{aligned}$$
(47)

where \(l<0.\)

Based on this result, we can infer the singular dromion solutions:

$$\begin{aligned} u_{6}\left( x,y,t\right)= & {} \pm \left\{ -\frac{2l}{m}\left( 1\pm \coth \left( \sqrt{-l}\xi \right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(48)
$$\begin{aligned} v_{6}\left( x,y,t\right)= & {} \pm \varkappa \left\{ -\frac{2l}{m}\left( 1\pm \coth \left( \sqrt{-l}\xi \right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(49)

provided \(m>0.\)

Result–3

$$\begin{aligned} A_{0}=-\frac{3m}{8n},\text { }A_{1}=\pm \frac{\sqrt{\gamma ^{2}+1}}{2m},~B_{1}=\pm \sqrt{-\frac{3}{16n}},~R=-\frac{3m^{2}}{4n},~l=\frac{3m^{2}}{16n}, \end{aligned}$$
(50)

where \(n<0.\)

This result arrives the combo singular dromion solutions:

$$\begin{aligned} u_{7}\left( x,y,t\right)&=\left\{ -\dfrac{3m}{8n}\left( 1\pm \dfrac{\sqrt{ \gamma ^{2}+1}\text{ csch }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) }{ \gamma \text{ csch }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) +1}\right. \right. \nonumber \\&\left. \left. \pm \dfrac{\coth \left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) }{\gamma \text{ csch }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) +1}\right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(51)
$$\begin{aligned} v_{7}\left( x,y,t\right)&=\varkappa \left\{ -\dfrac{3m}{8n}\left( 1\pm \dfrac{\sqrt{\gamma ^{2}+1}\text{ csch }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) }{\gamma \text{ csch }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) +1} \right. \right. \nonumber \\&\left. \left. \pm \dfrac{\coth \left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) }{\gamma \text{ csch }\left( \sqrt{-\frac{3m^{2}}{4n}}\xi \right) +1}\right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(52)

provided \(m>0.\)

Extended auxiliary equation approach

The solution to Eq. (26) is assumed to have the following form by the extended auxiliary equation approach:

$$\begin{aligned} U(\xi )=\alpha _{0}+\alpha _{1}H(\xi )+\alpha _{2}H^{2}(\xi ), \end{aligned}$$
(53)

where \(\alpha _{k}~(k=0,1,2)\) are constants such that \(\alpha _{2}\ne 0\). The function \(H(\xi )\) satisfies the following first order nonlinear ODE:

$$\begin{aligned} H^{\prime 2}(\xi )=\sum \limits _{k=0}^{3}C_{2k}H^{2k}(\xi ), \end{aligned}$$
(54)

with the help of the solution

$$\begin{aligned} H(\xi )=\frac{1}{2}\left[ -\frac{C_{4}}{C_{6}}\left( 1\pm h(\xi )\right) \right] ^{\frac{1}{2}}, \end{aligned}$$
(55)

where \(C_{k}\) \((k=0,2,4,6)\) are constants and \(h(\xi )\) is presented via the Jacobi’s elliptic functins (JEFs). Inserting (53) with the usage of (54) into Eq. (26) gives the following system of algebraic equations:

\(H^{8}:2n\alpha _{2}^{4}+6C_{6}\alpha _{2}^{2}=0,\)

\(H^{7}:8n\alpha _{1}\alpha _{2}^{3}+9\alpha _{1}C_{6}\alpha _{2}=0,\)

\(H^{6}:2\,m\alpha _{2}^{3}+4\alpha _{2}^{2}C_{4}+\frac{5}{2}\alpha _{1}^{2}C_{6}+12n\alpha _{1}^{2}\alpha _{2}^{2}+8\alpha _{0}\alpha _{2}C_{6}+8n\alpha _{0}\alpha _{2}^{3}=0,\)

\(H^{5}:3\alpha _{0}\alpha _{1}C_{6}+6\alpha _{1}\alpha _{2}C_{4}+6\,m\alpha _{1}\alpha _{2}^{2}+8n\alpha _{1}^{3}\alpha _{2}+24n\alpha _{0}\alpha _{1}\alpha _{2}^{2}=0,\)

\(H^{4}:2\,l\alpha _{2}^{2}+2n\alpha _{1}^{4}+2\alpha _{2}^{2}C_{2}+\frac{3}{2} \alpha _{1}^{2}C_{4}+12n\alpha _{0}^{2}\alpha _{2}^{2}+6\alpha _{0}\alpha _{2}C_{4}+6\,m\alpha _{0}\alpha _{2}^{2}+6\,m\alpha _{1}^{2}\alpha _{2}+24n\alpha _{0}\alpha _{1}^{2}\alpha _{2}=0,\)

\(H^{3}:2\,m\alpha _{1}^{3}+4\,l\alpha _{1}\alpha _{2}+2\alpha _{0}\alpha _{1}C_{4}+3\alpha _{1}\alpha _{2}C_{2}+8n\alpha _{0}\alpha _{1}^{3}+24n\alpha _{0}^{2}\alpha _{1}\alpha _{2}+12\,m\alpha _{0}\alpha _{1}\alpha _{2}=0,\)

\(H^{2}:2\,l\alpha _{1}^{2}+\frac{1}{2}\alpha _{1}^{2}C_{2}+12n\alpha _{0}^{2}\alpha _{1}^{2}+4\,l\alpha _{0}\alpha _{2}+4\alpha _{0}\alpha _{2}C_{2}+6\,m\alpha _{0}\alpha _{1}^{2}+6\,m\alpha _{0}^{2}\alpha _{2}+8n\alpha _{0}^{3}\alpha _{2}=0,\)

\(H:4\,l\alpha _{0}\alpha _{1}+\alpha _{0}\alpha _{1}C_{2}+6\,m\alpha _{0}^{2}\alpha _{1}+8n\alpha _{0}^{3}\alpha _{1}=0,\)

\(H^{0}:2n\alpha _{0}^{4}+2\,m\alpha _{0}^{3}+2\,l\alpha _{0}^{2}+2\alpha _{2}C_{0}\alpha _{0}-\frac{1}{2}C_{0}\alpha _{1}^{2}=0.\)

According to the extended auxiliary equation approach, the algebraic equations system \(H^{0}-H^{8}\) has several families of results as follows:

Family–1: If we substitute \(C_{0}=\frac{C_{4}^{3}\left( r^{2}-1\right) }{32C_{6}^{2}r^{2}},~C_{2}=\frac{C_{4}^{2}\left( 5\,r^{2}-1\right) }{16C_{6}r^{2}}\), \(0<r<1,\) \(C_{6}>0\) in Eqs. \((H^{0}-H^{8})\) and solve them using the Maple, we have the result:

$$\begin{aligned} \alpha _{0}=-\frac{3m}{4n},~\alpha _{1}=0,~\alpha _{2}=\pm \sqrt{-\frac{ 3C_{6}}{n}},~l=\frac{3m^{2}\left( 5r^{2}-1\right) }{64nr^{2}},~C_{4}=\pm \frac{m\,}{2}\sqrt{-\frac{3C_{6}}{n}}, \end{aligned}$$
(56)

provided \(n<0.\)

Now, substituting (56) along with (55) into (53), one arrives the JEF solutions of Eqs. (2) and (3) in the forms:

$$\begin{aligned} u_{8}\left( x,y,t\right)= & {} \pm \left\{ \frac{-3m}{8n}\left( 1\pm \text{ sn } \left( \frac{\sqrt{3}m}{4r\sqrt{-n}}\xi ,r\right) \right) \right\} ^{\frac{1 }{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(57)
$$\begin{aligned} v_{8}\left( x,y,t\right)= & {} \pm \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \text{ sn }\left( \frac{\sqrt{3}m}{4r\sqrt{-n}}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(58)

In particular, if \(r\rightarrow 1^{\mathbf {-}}\) in (57) and (58), then one gets the same dark dromion solutions (39) and (40), respectively.

The JEF solutions are

$$\begin{aligned} u_{9}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\pm \frac{1}{r} \text{ ns }\left( \frac{\sqrt{3}m}{4r\sqrt{-n}}\xi ,r\right) \right) \right\} ^{ \frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(59)
$$\begin{aligned} v_{9}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \frac{1 }{r}\text{ ns }\left( \frac{\sqrt{3}m}{4r\sqrt{-n}}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(60)

In particular, if \(r\rightarrow 1^{\mathbf {-}}\) in (59) and (60), then one gets the same singular dromion solutions (48) and (49), respectively.

Family–2: If we substitute \(C_{0}=\frac{C_{4}^{3}\left( 1-r^{2}\right) }{32C_{6}^{2}},~C_{2}=\frac{C_{4}^{2}\left( 5\,-r^{2}\right) }{16C_{6}}\), \(0<r<1,\) \(C_{6}>0\) in Eqs. \((H^{0}-H^{8})\) and solve them using the Maple, we have the result:

$$\begin{aligned} \alpha _{0}=-\frac{3m}{4n},~\alpha _{1}=0,~\alpha _{2}=\pm \sqrt{-\,\frac{ 3C_{6}}{n}},~l=-\frac{3\,m^{2}\left( r^{2}-5\right) }{64n},~C_{4}=\pm \frac{ m\,}{2}\sqrt{-\,\frac{3C_{6}}{n}}, \end{aligned}$$
(61)

provided \(n<0.~\)Substituting (61) along with (55) into (53), one obtains JEF solutions of Eqs. (2) and (3) in the forms:

$$\begin{aligned} u_{10}\left( x,y,t\right) =\left\{ \frac{-3m}{8n}\left( 1\mp r\text{ sn } \left( \frac{\sqrt{3}m}{4\sqrt{-n}}\xi ,r\right) \right) \right\} ^{\frac{1}{ 2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(62)
$$\begin{aligned} v_{10}\left( x,y,t\right) =\varkappa \left\{ \frac{-3m}{8n}\left( 1\mp r \text{ sn }\left( \frac{\sqrt{3}m}{4\sqrt{-n}}\xi ,r\right) \right) \right\} ^{ \frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(63)

In particular, if \(r\rightarrow 1^{\mathbf {-}}\) in (62) and (63), then one gets the same dark dromion solutions (39) and (40), respectively.

The JEF solutions are

$$\begin{aligned} u_{11}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\mp \text{ ns }\left( \frac{\sqrt{3}m}{4\sqrt{-n}}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i \left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(64)
$$\begin{aligned} v_{11}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\mp \text{ ns }\left( \frac{\sqrt{3}m}{4\sqrt{-n}}\xi ,r\right) \right) \right\} ^{\frac{ 1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(65)

In particular, if \(r\rightarrow 1^{\mathbf {-}}\) in (64) and (65), then one gets the same singular dromion solutions (48) and (49), respectively.

Family–3: If we substitute \(C_{0}=\frac{C_{4}^{3}}{32r^{2}C_{6}^{2}},~C_{2}=\frac{C_{4}^{2}\left( 4r^{2}+1\right) }{16r^{2}C_{6}},~0<r<1,~C_{6}<0\) in Eqs. \((H^{0}-H^{8})\) and solve them using the Maple, we have the result:

$$\begin{aligned} \alpha _{0}=-\frac{3m}{4n},~\alpha _{1}=0,~\alpha _{2}=\pm \sqrt{-\,\frac{ 3C_{6}}{n}},~l=\frac{3m^{2}\left( 4r^{2}+1\right) }{64nr^{2}},~C_{4}=\pm \frac{m}{2}\sqrt{-\,\frac{3C_{6}}{n}}, \end{aligned}$$
(66)

provided \(n>0.~\)Substituting (66) along with (55) into (53), one gets JEF solutions of Eqs. (2) and (3) in the forms:

$$\begin{aligned} u_{12}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\pm \text{ cn }\left( \frac{\sqrt{3}m}{4r\sqrt{n}}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i \left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(67)
$$\begin{aligned} v_{12}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \text{ cn }\left( \frac{\sqrt{3}m}{4r\sqrt{n}}\xi ,r\right) \right) \right\} ^{\frac{ 1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(68)

In particular, if \(r\rightarrow 1^{\mathbf {-}}\) in (67) and (68), then one gets the bright dromion solutions:

$$\begin{aligned} u_{13}\left( x,y,t\right)= & {} \pm \left\{ \frac{-3m}{8n}\left( 1\pm \text{ sech } \left( \frac{\sqrt{3}m}{4\sqrt{n}}\xi \right) \right) \right\} ^{\frac{1}{2} }e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(69)
$$\begin{aligned} v_{13}\left( x,y,t\right)= & {} \pm \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \text{ sech }\left( \frac{\sqrt{3}m}{4\sqrt{n}}\xi \right) \right) \right\} ^{ \frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(70)

And the JEF solutions are

$$\begin{aligned} u_{14}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\pm \sqrt{1-r^{2}} \text{ sd }\left( \frac{\sqrt{3}m}{4r\sqrt{n}}\xi ,r\right) \right) \right\} ^{ \frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(71)
$$\begin{aligned} v_{14}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \sqrt{ 1-r^{2}}\text{ sd }\left( \frac{\sqrt{3}m}{4r\sqrt{n}}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(72)

Family–4: If we substitute \(C_{0}=\frac{C_{4}^{3}r^{2}}{ 32C_{6}^{2}\left( r^{2}-1\right) },~C_{2}=\frac{C_{4}^{2}\left( 5r^{2}-4\right) }{16C_{6}\left( r^{2}-1\right) },~0<r<1,~C_{6}<0\) in Eqs. \((H^{0}-H^{8})\) and solve them using the Maple, we have the result:

$$\begin{aligned} \alpha _{0}=-\frac{3m}{4n},~\alpha _{1}=0,~\alpha _{2}=\pm \sqrt{-\,\frac{ 3C_{6}}{n}},~l=\frac{3m^{2}\left( 5r^{2}-4\right) }{64n\left( r^{2}-1\right) },~C_{4}=\pm \frac{m}{2}\sqrt{-\,\frac{3C_{6}}{n}}, \end{aligned}$$
(73)

provided \(n>0.~\)Substituting (73) along with (55) into (53), one gets JEF solutions of Eqs. (2) and (3) in the forms:

$$\begin{aligned} u_{15}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\pm \frac{1}{\sqrt{ 1-r^{2}}}\text{ dn }\left( \frac{\sqrt{3}m}{4\sqrt{n\left( 1-r^{2}\right) }} \xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(74)
$$\begin{aligned} v_{15}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \frac{ 1}{\sqrt{1-r^{2}}}\text{ dn }\left( \frac{\sqrt{3}m}{4\sqrt{n\left( 1-r^{2}\right) }}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(75)

and

$$\begin{aligned} u_{16}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\pm \text{ nd }\left( \frac{\sqrt{3}m}{4\sqrt{n\left( 1-r^{2}\right) }}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(76)
$$\begin{aligned} v_{16}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \text{ nd }\left( \frac{\sqrt{3}m}{4\sqrt{n\left( 1-r^{2}\right) }}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(77)

Family–5: If we substitute \(C_{0}=\frac{C_{4}^{3}}{ 32C_{6}^{2}\left( 1-r^{2}\right) },~C_{2}=\frac{C_{4}^{2}\left( 4r^{2}-5\right) }{16C_{6}\left( r^{2}-1\right) },~0<r<1,C_{6}>0\) in Eqs. \((H^{0}-H^{8})\) and solve them using the Maple, we have the result:

$$\begin{aligned} \alpha _{0}=-\frac{3m}{4n},~\alpha _{1}=0,~\alpha _{2}=\pm \sqrt{-\,\frac{ 3C_{6}}{n}},~l=\frac{3m^{2}\left( 4r^{2}-5\right) }{64n\left( r^{2}-1\right) },~C_{4}=\pm \frac{m}{2}\sqrt{-\,\frac{3C_{6}}{n}}, \end{aligned}$$
(78)

provided \(n<0.~\)Substituting (78) along with (55) into (53), one gets JEF solutions of Eqs. (2) and (3) in the form:

$$\begin{aligned} u_{17}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\pm \text{ nc }\left( \frac{\sqrt{3}m}{4\sqrt{n\left( r^{2}-1\right) }}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(79)
$$\begin{aligned} v_{17}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \text{ nc }\left( \frac{\sqrt{3}m}{4\sqrt{n\left( r^{2}-1\right) }}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(80)

and

$$\begin{aligned} u_{18}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\pm \frac{1}{\sqrt{ 1-r^{2}}}\text{ ds }\left( \frac{\sqrt{3}m}{4\sqrt{n\left( r^{2}-1\right) }} \xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(81)
$$\begin{aligned} v_{18}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \frac{ 1}{\sqrt{1-r^{2}}}\text{ ds }\left( \frac{\sqrt{3}m}{4\sqrt{n\left( r^{2}-1\right) }}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(82)

Family–6: If we substitute \(C_{0}=\frac{r^{2}C_{4}^{3}}{32C_{6}^{2}},~C_{2}=\frac{C_{4}^{2}\left( r^{2}+4\right) }{16C_{6}},~0<r<1,~C_{6}<0\) in Eqs. \((H^{0}-H^{8})\) and solve them using the Maple, we have the result:

$$\begin{aligned} \alpha _{0}=-\frac{3m}{4n},~\alpha _{1}=0,~\alpha _{2}=\pm \sqrt{-\,\frac{ 3C_{6}}{n}},~l=\frac{3m^{2}\left( r^{2}+4\right) }{64n},~C_{4}=\pm \frac{m}{2 }\sqrt{-\,\frac{3C_{6}}{n}}, \end{aligned}$$
(83)

provided \(n>0.~\)Substituting (83) along with (55) into (53), one arrives the JEF solutions

$$\begin{aligned} u_{19}\left( x,y,t\right)= & {} \pm \left\{ \frac{-3m}{8n}\left( 1\pm \text{ dn } \left( \frac{\sqrt{3}m}{4\sqrt{n}}\xi ,r\right) \right) \right\} ^{\frac{1}{2 }}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(84)
$$\begin{aligned} v_{19}\left( x,y,t\right)= & {} \pm \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \text{ dn }\left( \frac{\sqrt{3}m}{4\sqrt{n}}\xi ,r\right) \right) \right\} ^{ \frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }, \end{aligned}$$
(85)

and

$$\begin{aligned} u_{20}\left( x,y,t\right)= & {} \left\{ \frac{-3m}{8n}\left( 1\pm \sqrt{1-r^{2}} \text{ nd }\left( \frac{\sqrt{3}m}{4\sqrt{n}}\xi ,r\right) \right) \right\} ^{ \frac{1}{2}}e^{i\left[ -\kappa _{11}x-\kappa _{12}y+\omega _{1}t+\sigma _{1}W_{1}(t)-\sigma _{1}^{2}t\right] }, \end{aligned}$$
(86)
$$\begin{aligned} v_{20}\left( x,y,t\right)= & {} \varkappa \left\{ \frac{-3m}{8n}\left( 1\pm \sqrt{ 1-r^{2}}\text{ nd }\left( \frac{\sqrt{3}m}{4\sqrt{n}}\xi ,r\right) \right) \right\} ^{\frac{1}{2}}e^{i\left[ -\kappa _{21}x-\kappa _{22}y+\omega _{2}t+\sigma _{2}W_{2}(t)-\sigma _{2}^{2}t\right] }. \end{aligned}$$
(87)

Conclusions

The current paper revealed optical dromions with differential group delay with the usage of two integration algorithms,. They are the generalized projective Riccati equation approach and the extended auxiliary equation algorithm. These schemes give a plethora of dromions that are enlisted in the paper. The parameter constraints also do guarantee the existence of such dromions. The results are indeed promising and can be generalized further along. Next up, the model will be addressed in dispersion–flattened fibers and the results of those research activities will be recovered and reported. The numerical analysis of the model will also be carried out using a number of algorithms. These are Laplace–Adomian decomposition scheme as well as variational iteration method. Once these results are aligned with the pre–existing works [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65], they will be disseminated across various journals and made visible.