In the context of the dispersive concatenation model with multiplicative white noise, the polarization properties of the solitons can be systematically characterized to enhance the understanding of their dynamics:

The solitons in this study exhibit polarization states dependent on the interplay of the dispersive effects and the multiplicative white noise. For bright and dark solitons, the polarization tends to align linearly, as the interaction between the modes preserves phase coherence in a deterministic manner. For singular solitons and complexitons, the polarization is more likely elliptical due to the intricate coupling between the amplitude and phase variations in both polarization components.

To provide a quantitative characterization of the polarization, the Stokes parameters \((S_0, S_1, S_2, S_3)\) are estimated: \(S_0\) represents the total intensity of the soliton, calculated as \(S_0 = |E_x|^2 + |E_y|^2\), where \(E_x\) and \(E_y\) are the electric field components in the orthogonal polarization modes. \(S_1\) captures the difference in intensity between the two polarization components: \(S_1 = |E_x|^2 - |E_y|^2\). \(S_2\) measures the linear polarization at \(45^\circ\): \(S_2 = 2 \Re (E_x E_y^*)\). \(S_3\) quantifies the degree of circular polarization: \(S_3 = 2 \Im (E_x E_y^*)\). These parameters can be further normalized to the total intensity \(S_0\) to provide a dimensionless depiction of the polarization state. In bright and dark solitons, \(S_3 \approx 0\) confirms negligible circular polarization, indicating dominantly linear polarization. Conversely, nonzero \(S_3\) in singular and complexitons suggests ellipticity in the polarization state.

The study finds that multiplicative white noise predominantly influences the phase of the polarization components, inducing phase modulation without significantly altering the Stokes parameters. This robustness implies that the fundamental polarization states, whether linear or elliptical, remain intact under noise perturbations.

Bright and dark solitons retain linear polarization due to symmetric dispersive effects across the two modes. Their polarization angle is determined by the relative amplitudes of \(E_x\) and \(E_y\), while the noise-induced phase shifts introduce minor perturbations without disrupting the linearity.

Singular solitons and complexitons exhibit ellipticity in their polarization states, characterized by a nonzero ellipticity angle. This is reflected in the non-zero contributions of \(S_2\) and \(S_3\), representing the coupling effects induced by dispersive concatenation.

By estimating the Stokes parameters, the study not only confirms the robustness of the solitons but also provides a practical framework for analyzing polarization in experimental setups. This characterization facilitates the design of optical systems that leverage these soliton solutions for robust signal transmission.

As a result, the detailed polarization analysis, supplemented by Stokes parameters, reinforces the stability and versatility of soliton solutions within the dispersive concatenation model. This work highlights the resilience of soliton structures against white noise while uncovering the intricate dynamics of their polarization properties.

To address the coupled systems (2) and (3), we make the assumption of the following solution structure:

$$\begin u(x,t)= & U_1(\zeta ) e^,\nonumber \\ v(x,t)= & U_2(\zeta ) e^, \end$$

(10)

The wave variable \(\zeta\) is defined as

$$\begin \zeta =k(x-\nu t). \end$$

(11)

In this case, the variables \(U_j(\zeta )\) (where \(j=1,2\)) represent the amplitude component of the soliton solution, whereas \(\nu\) represents the speed of the soliton. Furthermore, the phase component \(\phi (x,t)\) is defined as.

$$\begin \phi (x,t)=-\kappa x+\omega t+\sigma W(t)-\sigma ^2 t+ \theta _0. \end$$

(12)

In this context, \(\kappa\) represents the frequency of the solitons, \(\omega\) relates to the wave number, and \(\theta _0\) specifies the phase constant. The influence of white noise is restricted to the phase components of the solitons across both polarization vectors, leaving their amplitude and structural integrity largely unaffected. This insight reinforces the robustness of solitons in stochastic environments and suggests their potential viability in practical applications where noise is inevitable. By inserting Eq. (10) into Eqs. (2) and (3) and then decomposing them into their real and imaginary components, the resultant real parts provide

$$\begin & -\kappa ^3 \sigma _^ U_2^3 \delta _3^-2 k^2 \kappa \sigma _^ U_2 U_2'^2 \delta _3^-2 k^2 \kappa \sigma _^ U_1 U_1' U_2' \delta _3^\nonumber \\ & +2 k^2 \kappa \left( \sigma _^+\sigma _^\right) U_2 U_1' U_2' \delta _3^\nonumber \\ & +k^2 \kappa \sigma _^ U_2^2 U_2'' \delta _3^+\left( \kappa \delta _3^ \sigma _^-\delta _2^ \sigma _^\right) U_1^5\nonumber \\ & +\left( \kappa \delta _3^ \sigma _^-\delta _2^ \sigma _^\right) U_1 U_2^4\nonumber \\ & +\left( \kappa ^2 \left[ \delta _2^ \left( \sigma _^-\sigma _^+\sigma _^+\sigma _^\right) \right. \right. \nonumber \\ & \left. \left. -\kappa \delta _3^ \left( \sigma _^+\sigma _^-\sigma _^-\sigma _^\right) \right] -b_1^\right) U_1^3\nonumber \\ & +\left[ \kappa ^2 \left\ \left( \sigma _^-\sigma _^+\sigma _^\right) \right. \right. \nonumber \\ & \left. \left. -\kappa \delta _3^ \left( \sigma _^+\sigma _^-\sigma _^-\sigma _^\right) \right\} -b_2^\right] U_1 U_2^2\nonumber \\ & -k^2 \left[ \delta _2^ \left( \sigma _^+\sigma _^\right) -\kappa \delta _3^ \left( 2 \sigma _^+2 \sigma _^+\sigma _^\right) \right] U_1 U_1'^2-k^4 \left( \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^\right) U_1^\nonumber \\ & -k^2 \left( \delta _2^ \left( \sigma _^+\sigma _^\right) -\kappa \delta _3^ \sigma _^\right) U_1 U_2'^2\nonumber \\ & -k^2 \left( \delta _2^ \sigma _^-\kappa \delta _3^ \left[ 3 \sigma _^+\sigma _^-\sigma _^\right] \right) U_2^2 U_1''\nonumber \\ & +\left( \kappa ^2 a^-\kappa ^3 \delta _1^ \sigma _1^-\kappa ^4 \delta _2^ \sigma _3^+\kappa ^5 \delta _3^ \sigma _9^+\omega -\sigma ^2 \right) U_1\nonumber \\ & +\kappa ^2 \left( \delta _2^ \sigma _^-\kappa \delta _3^ \sigma _^\right) U_1^2 U_2\nonumber \\ & -k^2 \left( \delta _2^ \left( \sigma _^+\sigma _^\right) -\kappa \delta _3^ \left[ 3 \sigma _^+\sigma _^-\sigma _^\right] \right) U_1^2 U_1''\nonumber \\ & +k^ 2 \left( \kappa \delta _3^ \sigma _^-\delta _2^ \sigma _^\right) U_1^2 U_2''\nonumber \\ & -k^2 \left( a^-3 \kappa \delta _1^ \sigma _1^-6 \kappa ^2 \delta _2^ \sigma _3^+10 \kappa ^3 \delta _3^ \sigma _9^\right) U_1''\nonumber \\ & +\left( \kappa \delta _3^ \sigma _^-\delta _2^ \sigma _^\right) U_1^3 U_2^2=0, \end$$

(13)

and

$$\begin & \kappa ^3 \sigma _^ U_1^3 \delta _3^+2 k^2 \kappa \sigma _^ U_1 U_1'^2 \delta _3^\nonumber \\ & -2 k^2 \kappa \left( \sigma _^+\sigma _^\right) U_1 U_1' U_2' \delta _3^+k^4 \left( \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^\right) U_2^\nonumber \\ & +2 k^2 \kappa \sigma _^ U_2 U_1' U_2' \delta _3^-k^2 \kappa \sigma _^ U_1^2 U_1'' \delta _3^\nonumber \\ & +\left( \delta _2^ \sigma _^-\kappa \delta _3^ \sigma _^\right) U_2^5+\left( \delta _2^ \sigma _^-\kappa \delta _3^ \sigma _^\right) U_1^2 U_2^3\nonumber \\ & +\left( b_1^+\kappa ^2 \left\ \left( \sigma _^+\sigma _^-\sigma _^-\sigma _^\right) \right. \right. \nonumber \\ & \left. \left. -\delta _2^ \left[ \sigma _^-\sigma _^+\sigma _^+\sigma _^\right] \right\} \right) U_2^3\nonumber \\ & +\kappa ^2 \left( \kappa \delta _3^ \sigma _^-\delta _2^ \sigma _^\right) U_1 U_2^2\nonumber \\ & +k^2 \left( \delta _2^ \left( \sigma _^+\sigma _^2\right) -\kappa \delta _3^ \sigma _^\right) U_2 U_1'^2\nonumber \\ & +k^2 \left( \delta _2^ \left( \sigma _^+\sigma _^\right) -\kappa \delta _3^ \left[ 2 \sigma _^+2 \sigma _^+\sigma _^\right] \right) U_2 U_2'^2\nonumber \\ & +\left( \delta _2^ \sigma _^-\kappa \delta _3^ \sigma _^\right) U_1^4 U_2\nonumber \\ & +\left( b_2^+\kappa ^2 \left\ \left( \sigma _^+\sigma _^-\sigma _^-\sigma _^\right) \right. \right. \nonumber \\ & \left. \left. -\delta _2^ \left[ \sigma _^-\sigma _^+\sigma _^2\right] \right\} \right) U_1^2 U_2\nonumber \\ & +\left( -\kappa ^2 a^+\kappa ^3 \delta _1^ \sigma _1^+\kappa ^4 \delta _2^ \sigma _3^-\kappa ^5 \delta _3^ \sigma _9^-\omega +\sigma ^2 \right) U_2\nonumber \\ & +k^2 \left( \delta _2^ \sigma _^-\kappa \delta _3^ \sigma _^\right) U_2^2 U_1''\nonumber \\ & +k^2 \left( \delta _2^ \sigma _^-\kappa \delta _3^ \left[ 3 \sigma _^+\sigma _^-\sigma _^\right] \right) U_1^2 U_2''\nonumber \\ & +k^2 \left( a^-3 \kappa \delta _1^ \sigma _1^-6 \kappa ^2 \delta _2^ \sigma _3^+10 \kappa ^3 \delta _3^ \sigma _9^\right) U_2''\nonumber \\ & +k^2 \left( \delta _2^ \left( \sigma _^+\sigma _^\right) -\kappa \delta _3^ \left[ 3 \sigma _^+\sigma _^-\sigma _^\right] \right) U_2^2 U_2''=0, \end$$

(14)

and the imaginary parts give

$$\begin & k U_1' \left( 2 \kappa a^+5 \kappa ^4 \delta _3^ \sigma _9^-4 \kappa ^3 \delta _2^ \sigma _3^\right. \nonumber \\ & \left. -3 \kappa ^2 \delta _1^ \sigma _1^+\nu \right) +k^5 \delta _3^ \sigma _9^ U_1^+\delta _1^ \sigma _^ U_1^3\nonumber \\ & +k^3 U_1^ \left( 4 \kappa \delta _2^ \sigma _3^+\delta _1^ \sigma _1^+\delta _3^ \left[ \sigma _^ U_2^2-10 \kappa ^2 \sigma _9^\right] \right) \nonumber \\ & +k^3 \delta _3^ \left( \sigma _^+\sigma _^\right) U_1 U_1' U_1''\nonumber \\ & +k^3 \delta _3^ \left( \sigma _^+\sigma _^\right) U_2 U_2' U_1''+k^3 \delta _3^ \sigma _^ U_1 U_1' U_2''\nonumber \\ & +k^3 \delta _3^ \sigma _^ U_2 U_2' U_2''+k^3 \delta _3^ \sigma _^ U_1'^3\nonumber \\ & +k^3 \delta _3^ \sigma _^ U_1^2 U_1^-\kappa ^2 k \delta _3^ \sigma _^ U_1 U_2 U_1'\nonumber \\ & +\kappa ^2 k \delta _3^ \sigma _^ U_2^2 U_2'+k \delta _3^ \sigma _^ U_1^4 U_1'+k \delta _3^ \sigma _^ U_2^4 U_1'+\nonumber \\ & +k U_1^2 U_1' \left( 2 \kappa \delta _2^ \left( \sigma _^+\sigma _^-\sigma _^\right) \right. \nonumber \\ & \left. +\delta _3^ \left[ -3 \kappa ^2 \sigma _^-3 \kappa ^2 \sigma _^+\kappa ^2 \sigma _^+\kappa ^2 \sigma _^+\sigma _^ U_2^2\right] \right) \nonumber \\ & +\kappa k U_2^2 U_1' \left( 2 \delta _2^ \sigma _^-\kappa \delta _3^ \left[ 3 \sigma _^+2 \sigma _^-2 \sigma _^-\sigma _^\right] \right) \nonumber \\ & + \delta _1^ \sigma _^ U_1 U_2^2+k^3 \delta _3^ \sigma _^ U_1' U_2'^2\nonumber \\ & +2 \kappa k U_1^2 U_2' \left( \kappa \delta _3^ \sigma _^-\delta _2^ \sigma _^\right) \nonumber \\ & -\kappa k U_1 U_2 U_2' \left( \kappa \delta _3^ \left( \sigma _^+\sigma _^\right) -2 \delta _2^ \sigma _^\right) =0, \end$$

(15)

and

$$\begin & -k U_2' \left( 2 \kappa a^+5 \kappa ^4 \delta _3^ \sigma _9^-4 \kappa ^3 \delta _2^ \sigma _3^\right. \nonumber \\ & \left. -3 \kappa ^2 \delta _1^ \sigma _1^+\nu \right) -k \delta _3^ \sigma _^ U_1^4 U_2'-k \delta _3^ \sigma _^ U_2^4 U_2'\nonumber \\ & -k^5 \delta _3^ \sigma _9^ U_2^-k^3 U_2^ \left( 4 \kappa \delta _2^ \sigma _3^+\delta _1^ \sigma _1^\right. \nonumber \\ & \left. +\delta _3^ \left[ \sigma _^ U_1^2-10 \kappa ^2 \sigma _9^\right] \right) -\delta _1^ \sigma _^ U_1^2 U_2\nonumber \\ & -k^3 \delta _3^ \sigma _^ U_1 U_1' U_1''-k^3 \delta _3^ \sigma _^ U_2 U_2' U_1''-k^3 \delta _3^ \left( \sigma _^\right. \nonumber \\ & \left. +\sigma _^\right) U_1 U_1' U_2''-k^3 \delta _3^ \left( \sigma _^+\sigma _^\right) U_2 U_2' U_2''\nonumber \\ & -k^3 \delta _3^ \sigma _^ U_2'^3-k^3 \delta _3^ \sigma _^ U_1'^2 U_2'+U_2^2 \left( -k^3 \delta _3^ \sigma _^ U_2^\right. \nonumber \\ & \left. -k \delta _3^ \sigma _^ U_1^2 U_2'\right) -\kappa ^2 k \delta _3^ \sigma _^ U_1^2 U_1'\nonumber \\ & +\kappa ^2 k \delta _3^ \sigma _^ U_1 U_2 U_2'+2 \kappa k U_2^2 U_1' \left( \delta _2^ \sigma _^-\kappa \delta _3^ \sigma _^\right) \nonumber \\ & +\kappa k U_1 U_2 U_1' \left( \kappa \delta _3^ \left( \sigma _^+\sigma _^\right) -2 \sigma _^2 \delta _2^\right) \nonumber \\ & +\kappa k U_2^2 U_2' \left( \kappa \delta _3^ \left( 3 \sigma _^+3 \sigma _^-\sigma _^-\sigma _^\right) \right. \nonumber \\ & \left. -2 \delta _2^ \left[ \sigma _^+\sigma _^-\sigma _^\right] \right) -\delta _1^ \sigma _^ U_2^3\nonumber \\ & +\kappa k U_1^2 U_2' \left( \kappa \delta _3^ \left( 3 \sigma _^+2 \sigma _^-2 \sigma _^-\sigma _^\right) \right. \nonumber \\ & \left. -2 \delta _2^ \sigma _^\right) =0. \end$$

(16)

This coupled system of equations can be simply uncoupled under the assumption \(U_2=\lambda ~ U_1\). Consequently, Eqs. (13) and (14) can be expressed as:

$$\begin & -k^2 U_1'' \left( a^+10 \kappa ^3 \delta _3^ \sigma _9^-6 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^\right) \nonumber \\ & +U_1 \bigg (\kappa ^2 a^+\kappa ^5 \delta _3^ \sigma _9^-\kappa ^4 \delta _2^ \sigma _3^\nonumber \\ & -\kappa ^3 \delta _1^ \sigma _1^+\omega -\sigma ^2 \bigg )+U_1^5 \left( \kappa \delta _3^ \left( \lambda ^4 \sigma _^\right. \right. \nonumber \\ & \left. \left. +\lambda ^2 \sigma _^+\sigma _^\right) -\delta _2^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) \right) \nonumber \\ & +U_1^3 \bigg (-\lambda ^2 b_2^-b_1^+\kappa ^2 \bigg [\delta _2^ \left( \lambda ^2 \sigma _^\right. \nonumber \\ & \left. -\lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^-\sigma _^+\sigma _^+\sigma _^\right) \nonumber \\ & -\kappa \delta _3^ \left\^+\lambda ^2 \sigma _^+\lambda ^2 \sigma _^\right. \nonumber \\ & \left. -\lambda ^2 \sigma _^-\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^-\sigma _^-\sigma _^\right\} \bigg ]\bigg )\nonumber \\ & - k^4 U_1^ \left( \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^\right) \nonumber \\ & + k^2 U_1 U_1'^2 \bigg (\kappa \delta _3^ \bigg (-2 \lambda ^3 \sigma _^+2 \lambda ^2 \sigma _^+2 \lambda ^2 \sigma _^+\lambda ^2 \sigma _^\nonumber \\ & -2 \lambda \sigma _^+2 \sigma _^+2 \sigma _^+\sigma _^\bigg )-\delta _2^ \left[ \lambda ^2 \sigma _^\right. \nonumber \\ & \left. +\lambda ^2 \sigma _^+\sigma _^+\sigma _^\right] \bigg )\nonumber \\ & +k^2 U_1^2 U_1'' \bigg (\kappa \delta _3^ \left( \lambda ^3 \sigma _^+3 \lambda ^2 \sigma _^+\lambda ^2 \sigma _^\right. \nonumber \\ & \left. -\lambda ^2 \sigma _^+\lambda \sigma _^+3 \sigma _^+\sigma _^-\sigma _^\right) \nonumber \\ & -\delta _2^ \left[ \lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^\right] \bigg )=0, \end$$

(17)

and

$$\begin & k^2 \lambda U_1'' \left( a^+10 \kappa ^3 \delta _3^ \sigma _9^-6 \kappa ^2 \delta _2^ \sigma _3^\right. \nonumber \\ & \left. -3 \kappa \delta _1^ \sigma _1^\right) +\lambda U_1 \bigg (-\kappa ^2 a^-\kappa ^5 \delta _3^ \sigma _9^+\kappa ^4 \delta _2^ \sigma _3^\nonumber \\ & +\kappa ^3 \delta _1^ \sigma _1^-\omega +\sigma ^2 \bigg )+U_1^3 \bigg (\lambda ^3 b_1^+\lambda b_2^\nonumber \\ & +\kappa ^2 \bigg [\kappa \delta _3^ \bigg (\lambda ^3 \sigma _^+\lambda ^3 \sigma _^-\lambda ^3 \sigma _^-\lambda ^3 \sigma _^+\lambda ^2 \sigma _^\nonumber \\ & +\lambda \sigma _^+\lambda \sigma _^-\lambda \sigma _^-\lambda \sigma _^+\sigma _^\bigg )\nonumber \\ & -\lambda \delta _2^ \bigg \^-\lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\lambda \sigma _^\nonumber \\ & +\sigma _^-\sigma _^+\sigma _^2\bigg \}\bigg ]\bigg )+k^4 \lambda U_1^ \left( \delta _2^ \sigma _3^\right. \nonumber \\ & \left. -5 \kappa \delta _3^ \sigma _9^\right) \nonumber \\ & +k^2 U_1 U_1'^2 \bigg (\kappa \delta _3^ \bigg (2 \lambda ^2 \sigma _^-\lambda \bigg (2 \lambda ^2 \sigma _^\nonumber \\ & +2 \lambda ^2 \sigma _^+\lambda ^2 \sigma _^+2 \sigma _^+2 \sigma _^+\sigma _^\bigg )+2 \sigma _^\bigg )\nonumber \\ & +\lambda \delta _2^ \bigg [\lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\sigma _^+\sigma _^2\bigg ]\bigg )\nonumber \\ & +k^2 U_1^2 U_1'' \bigg (\lambda \delta _2^ \left( \lambda ^2 \sigma _^+\lambda \left( \lambda \sigma _^+\sigma _^\right) +\sigma _^\right) \nonumber \\ & -\kappa \delta _3^ \left[ 3 \lambda ^3 \sigma _^+\lambda ^3 \sigma _^-\lambda ^3 \sigma _^\right. \nonumber \\ & \left. +\lambda ^2 \sigma _^+3 \lambda \sigma _^+\lambda \sigma _^-\lambda \sigma _^+\sigma _^\right] \bigg )\nonumber \\ & +\lambda U_1^5 \left( \delta _2^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) \right. \nonumber \\ & \left. -\kappa \delta _3^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) \right) =0, \end$$

(18)

and Eqs. (15) and (16) become

$$\begin & k U_1' \left( 2 \kappa a^+5 \kappa ^4 \delta _3^ \sigma _9^-4 \kappa ^3 \delta _2^ \sigma _3^\right. \nonumber \\ & \left. -3 \kappa ^2 \delta _1^ \sigma _1^+\nu \right) +U_1^2 \left( k^3 \lambda ^2 \delta _3^ \sigma _^ U_1^+k \lambda ^2 \delta _3^ \sigma _^ U_1^2 U_1'\right) \nonumber \\ & +k^5 \delta _3^ \sigma _9^ U_1^+k^3 U_1^ \left( 2 \kappa \left( 2 \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^\right) \right. \nonumber \\ & \left. +\delta _1^ \sigma _1^\right) +k^3 \delta _3^ \left( \lambda ^2 \sigma _^+\sigma _^\right) U_1'^3\nonumber \\ & +k^3 \delta _3^ U_1 \left( \lambda ^3 \sigma _^+\lambda ^2 \sigma _^+\lambda ^2 \sigma _^\right. \nonumber \\ & \left. +\lambda \sigma _^+\sigma _^+\sigma _^\right) U_1' U_1''+k^3 \delta _3^ \sigma _^ U_1^2 U_1^\nonumber \\ & +\kappa k U_1^2 U_1' \bigg (2 \delta _2^ \left\^+\lambda ^2 \sigma _^-\lambda \sigma _^\right. \nonumber \\ & \left. +\sigma _^+\sigma _^-\sigma _^\right\} -\kappa \delta _3^ \bigg [-\lambda ^3 \sigma _^+3 \lambda ^2 \sigma _^\nonumber \\ & +3 \lambda ^2 \sigma _^-\lambda ^2 \sigma _^-\lambda ^2 \sigma _^-\lambda \sigma _^+3 \sigma _^\nonumber \\ & +3 \sigma _^-\sigma _^-\sigma _^\bigg ]\bigg )\nonumber \\ & +k \delta _3^ U_1^4 \left( \lambda ^4 \sigma _^+\sigma _^\right) U_1'\nonumber \\ & +\delta _1^ U_1^3 \left( \lambda ^2 \sigma _^+\sigma _^\right) =0, \end$$

(19)

and

$$\begin & -k \lambda U_1' \left( 2 \kappa a^+5 \kappa ^4 \delta _3^ \sigma _9^-4 \kappa ^3 \delta _2^ \sigma _3^\right. \nonumber \\ & \left. -3 \kappa ^2 \delta _1^ \sigma _1^+\nu \right) -k^5 \lambda \delta _3^ \sigma _9^ U_1^\nonumber \\ & -k^3 \lambda U_1^ \left( 2 \kappa \left( 2 \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^\right) \right. \nonumber \\ & \left. +\delta _1^ \sigma _1^\right) -k \lambda \delta _3^ U_1^4 \left( \lambda ^4 \sigma _^+\sigma _^\right) U_1'\nonumber \\ & -k^3 \delta _3^ U_1 \left( \lambda ^2 \sigma _^+\lambda \left( \lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\sigma _^+\sigma _^\right) \right. \nonumber \\ & \left. +\sigma _^\right) U_1' U_1''-k^3 \lambda \delta _3^ \left( \lambda ^2 \sigma _^+\sigma _^\right) U_1'^3\nonumber \\ & +U_1^2 \left( -k^3 \lambda \delta _3^ U_1^ \left( \lambda ^2 \sigma _^+\sigma _^\right) \right. \nonumber \\ & \left. -k \lambda ^3 \delta _3^ \sigma _^ U_1^2 U_1'\right) -\lambda \delta _1^ U_1^3 \left( \lambda ^2 \sigma _^+\sigma _^\right) \nonumber \\ & +\kappa k U_1^2 U_1' \bigg (-\kappa \delta _3^ \bigg \^-3 \lambda ^3 \sigma _^\nonumber \\ & +\lambda ^3 \sigma _^+\lambda ^3 \sigma _^+\lambda ^2 \sigma _^-3 \lambda \sigma _^-3 \lambda \sigma _^\nonumber \\ & +\lambda \sigma _^+\lambda \sigma _^+\sigma _^\bigg \}-2 \lambda \delta _2^ \bigg [\lambda ^2 \sigma _^\nonumber \\ & +\lambda ^2 \sigma _^-\lambda ^2 \sigma _^-\lambda \sigma _^+\sigma _^+\sigma _^2\bigg ]\bigg )=0. \end$$

(20)

By comparing the coefficients of the linearly independent functions in Eqs. (19) and (20), we may set them equal to zero. The speed of the two components and the parametric constraints are obtained as follows:

Upon setting the coefficients of \(U_1'\) to zero results in the velocity along the two components to be

$$\begin & \nu =-\kappa \left( 2 a^+5 \kappa ^3 \delta _3^ \sigma _9^-4 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^\right) , \end$$

(21)

and

$$\begin & \nu =-\kappa \left( 2 a^+5 \kappa ^3 \delta _3^ \sigma _9^-4 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^\right) . \end$$

(22)

Equating the speed of the solitons along the two components in (21) and (22) gives the parameter constraint.

$$\begin & 2 a^+5 \kappa ^3 \delta _3^ \sigma _9^-4 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^ \nonumber \\ & =2 a^+5 \kappa ^3 \delta _3^ \sigma _9^-4 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^. \end$$

(23)

For integrability the coefficients of a few functions are to be set to zero. These result in additional parameter constraints whose details are enumerated as follows:

Setting the coefficients of \(^3\) to zero results in

$$\begin & \lambda ^2 \sigma _^+\sigma _^=\lambda ^2 \sigma _^+\sigma _^=0. \end$$

(24)

Setting the coefficients of \(U_1'^2\) to zero results in

$$\begin & -\kappa \delta _3^ \left( -\lambda ^3 \sigma _^+3 \lambda ^2 \sigma _^+3 \lambda ^2 \sigma _^\right. \nonumber \\ & \left. -\lambda ^2 \sigma _^-\lambda ^2 \sigma _^-\lambda \sigma _^+3 \sigma _^+3 \sigma _^-\sigma _^-\sigma _^\right) \nonumber \\ & + 2 \delta _2^ \left( \lambda ^2 \sigma _^+\lambda ^2 \sigma _^-\lambda \sigma _^+\sigma _^+\sigma _^-\sigma _^\right) =0, \end$$

(25)

and

$$\begin & -\kappa \delta _3^ \left( -3 \lambda ^3 \sigma _^-3 \lambda ^3 \sigma _^+\lambda ^3 \sigma _^+\lambda ^3 \sigma _^+\lambda ^2 \sigma _^\right. \nonumber \\ & \left. -3 \lambda \sigma _^-3 \lambda \sigma _^+\lambda \sigma _^+\lambda \sigma _^+\sigma _^\right) \nonumber \\ & -2 \lambda \delta _2^ \left( \lambda ^2 \sigma _^+\lambda ^2 \sigma _^-\lambda ^2 \sigma _^-\lambda \sigma _^+\sigma _^+\sigma _^2\right) =0. \end$$

(26)

Setting the coefficients of \(^4 U_1'\) to zero results in

$$\begin & \lambda ^4 \sigma _^+\sigma _^+\lambda ^2 \sigma _^=\lambda ^4 \sigma _^+\sigma _^+\lambda ^2 \sigma _^=0. \end$$

(27)

Setting the coefficients of \(^3\) to zero results in

$$\begin & \lambda ^2 \sigma _^+\sigma _^=\lambda ^2 \sigma _^+\sigma _^=0. \end$$

(28)

Setting the coefficients of \(U_1 U_1' U_1''\) to zero results in

$$\begin & \lambda ^3 \sigma _^+\lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^=0, \end$$

(29)

and

$$\begin & \lambda ^2 \sigma _^+\lambda \left( \lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\sigma _^+\sigma _^\right) +\sigma _^=0. \end$$

(30)

Setting the coefficients of \(U_1^\) to zero results in

$$\begin & 2 \kappa \left( 2 \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^\right) +\delta _1^ \sigma _1^=0, \end$$

(31)

and

$$\begin & 2 \kappa \left( 2 \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^\right) +\delta _1^ \sigma _1^=0. \end$$

(32)

Setting the coefficients of \(U_1^2 U_1^\) to zero results in

$$\begin & \lambda ^2 \sigma _^+\sigma _^=\lambda ^2 \sigma _^+\sigma _^=0. \end$$

(33)

Setting the coefficients of \(U_1^\) to zero results in

$$\begin & \sigma _^=\sigma _^=0. \end$$

(34)

By comparing Eqs. (17) with (18) and reducing them to a single equation, we are able to derive the following parametric restrictions:

By comparing the coefficients of \(U_1^\), we obtain

$$\begin & \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^=\lambda \left( \delta _2^ \sigma _3^-5 \kappa \delta _3^ \sigma _9^\right) . \end$$

(35)

By comparing the coefficients of \(U_1^U_1''\), we obtain

$$\begin & \kappa \delta _3^ \left( \lambda ^3 \sigma _^+3 \lambda ^2 \sigma _^+\lambda ^2 \sigma _^\right. \nonumber \\ & \left. -\lambda ^2 \sigma _^+\lambda \sigma _^+3 \sigma _^+\sigma _^-\sigma _^\right) \nonumber \\ & -\delta _2^ \left( \lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^\right) \nonumber \\ & =\lambda \delta _2^ \left( \lambda ^2 \sigma _^+\lambda \left( \lambda \sigma _^+\sigma _^\right) +\sigma _^\right) \nonumber \\ & -\kappa \delta _3^ \left( 3 \lambda ^3 \sigma _^+\lambda ^3 \sigma _^-\lambda ^3 \sigma _^\right. \nonumber \\ & \left. +\lambda ^2 \sigma _^+3 \lambda \sigma _^+\lambda \sigma _^-\lambda \sigma _^+\sigma _^\right) . \end$$

(36)

By comparing the coefficients of \(U_1''\), we obtain

$$\begin & a^+10 \kappa ^3 \delta _3^ \sigma _9^-6 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^=\nonumber \\ & -\lambda \left( a^+10 \kappa ^3 \delta _3^ \sigma _9^-6 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^\right) . \end$$

(37)

By comparing the coefficients of \(U_1 U_1'^2\), we obtain

$$\begin & \kappa \delta _3^ \left( -2 \lambda ^3 \sigma _^+2 \lambda ^2 \sigma _^+2 \lambda ^2 \sigma _^\right. \nonumber \\ & \left. +\lambda ^2 \sigma _^-2 \lambda \sigma _^+2 \sigma _^+2 \sigma _^+\sigma _^\right) \nonumber \\ & -\delta _2^ \left( \lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\sigma _^+\sigma _^\right) =\nonumber \\ & \kappa \delta _3^ \left( 2 \lambda ^2 \sigma _^-\lambda \left( 2 \lambda ^2 \sigma _^+2 \lambda ^2 \sigma _^\right. \right. \nonumber \\ & \left. \left. +\lambda ^2 \sigma _^+2 \sigma _^+2 \sigma _^+\sigma _^\right) +2 \sigma _^\right) \nonumber \\ & +\lambda \delta _2^ \left( \lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\sigma _^+\sigma _^2\right) . \end$$

(38)

By comparing the coefficients of \(U_1^5\), we obtain

$$\begin & \kappa \delta _3^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) -\delta _2^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) =\nonumber \\ & \lambda \left( \delta _2^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) -\kappa \delta _3^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) \right) . \end$$

(39)

By comparing the coefficients of \(U_1\), we obtain

$$\begin & \kappa ^2 a^+\kappa ^5 \delta _3^ \sigma _9^-\kappa ^4 \delta _2^ \sigma _3^-\kappa ^3 \delta _1^ \sigma _1^+\omega -\sigma ^2 =\nonumber \\ & \lambda \left( -\kappa ^2 a^-\kappa ^5 \delta _3^ \sigma _9^+\kappa ^4 \delta _2^ \sigma _3^+\kappa ^3 \delta _1^ \sigma _1^-\omega +\sigma ^2 \right) . \end$$

(40)

By comparing the coefficients of \(U_1^3\), we obtain

$$\begin & -\lambda ^2 b_2^-b_1^+\kappa ^2 \bigg (\delta _2^ \left( \lambda ^2 \sigma _^-\lambda ^2 \sigma _^\right. \nonumber \\ & \left. +\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^-\sigma _^+\sigma _^+\sigma _^\right) \nonumber \\ & -\kappa \delta _3^ \bigg [\lambda ^3 \sigma _^+\lambda ^2 \sigma _^+\lambda ^2 \sigma _^\nonumber \\ & -\lambda ^2 \sigma _^-\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^-\sigma _^-\sigma _^\bigg ]\bigg )=\nonumber \\ & \lambda ^3 b_1^+\lambda b_2^+\kappa ^2 \bigg (\kappa \delta _3^ \bigg \^\nonumber \\ & +\lambda ^3 \sigma _^-\lambda ^3 \sigma _^-\lambda ^3 \sigma _^+\lambda ^2 \sigma _^+\lambda \sigma _^+\lambda \sigma _^\nonumber \\ & -\lambda \sigma _^-\lambda \sigma _^+\sigma _^\bigg \}-\lambda \delta _2^ \bigg [\lambda ^2 \sigma _^-\lambda ^2 \sigma _^\nonumber \\ & +\lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^-\sigma _^+\sigma _^2\bigg ]\bigg ). \end$$

(41)

The transformed nonlinear ordinary differential equation that we will analyze by the F–expansion technique is

$$\begin & -k^2 U_1'' \left( a^-6 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^\right) \nonumber \\ & +U_1 \bigg (\kappa ^2 a^-\kappa ^4 \delta _2^ \sigma _3^-\kappa ^3 \delta _1^ \sigma _1^+\omega -\sigma ^2 \bigg )\nonumber \\ & +U_1^5 \left( \kappa \delta _3^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) \right. \nonumber \\ & \left. -\delta _2^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) \right) \nonumber \\ & +U_1^3 \bigg (-\lambda ^2 b_2^-b_1^+\kappa ^2 \bigg [\delta _2^ \left( \lambda ^2 \sigma _^\right. \nonumber \\ & \left. -\lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^-\sigma _^+\sigma _^+\sigma _^\right) \nonumber \\ & -\kappa \delta _3^ \left\^+\lambda ^2 \sigma _^+\lambda ^2 \sigma _^\right. \nonumber \\ & \left. -\lambda ^2 \sigma _^-\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^-\sigma _^-\sigma _^\right\} \bigg ]\bigg )\nonumber \\ & - k^4 U_1^ \left( \delta _2^ \sigma _3^\right) + k^2 U_1 U_1'^2 \nonumber \\ & \bigg (\kappa \delta _3^ \bigg (-2 \lambda ^3 \sigma _^+2 \lambda ^2 \sigma _^+2 \lambda ^2 \sigma _^+\lambda ^2 \sigma _^\nonumber \\ & -2 \lambda \sigma _^+2 \sigma _^+2 \sigma _^+\sigma _^\bigg )\nonumber \\ & -\delta _2^ \left[ \lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\sigma _^+\sigma _^\right] \bigg )\nonumber \\ & +k^2 U_1^2 U_1'' \bigg (\kappa \delta _3^ \left( \lambda ^3 \sigma _^+3 \lambda ^2 \sigma _^\right. \nonumber \\ & \left. +\lambda ^2 \sigma _^-\lambda ^2 \sigma _^+\lambda \sigma _^+3 \sigma _^+\sigma _^-\sigma _^\right) \nonumber \\ & -\delta _2^ \left[ \lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^\right] \bigg )=0, \end$$

(42)

which can be rewritten in a compact structure as

$$\begin & B_4 U_1 U_1'^2+B_6 U_1^2 U_1''+B_5 U_1''+B_3 U_1^5\nonumber \\ & \quad +B_2 U_1^3+B_1 U_1+k^2 U_1^=0, \end$$

(43)

with

$$\begin B_1=-\frac-\kappa ^4 \delta _2^ \sigma _3^-\kappa ^3 \delta _1^ \sigma _1^+\omega -\sigma ^2 } \sigma _3^},\\ B_2=-\frac & \lambda ^2 b_2^+b_1^+\kappa ^2 \bigg [\kappa \delta _3^ \big (\lambda ^3 \sigma _^+\lambda ^2 \sigma _^+\lambda ^2 \sigma _^-\lambda ^2 \sigma _^-\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^-\sigma _^-\sigma _^\big )\\ & -\delta _2^ \left( \lambda ^2 \sigma _^-\lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^-\sigma _^+\sigma _^+\sigma _^\right) \bigg ] \end\right) } \sigma _3^},\\ B_3=-\frac \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) -\delta _2^ \left( \lambda ^4 \sigma _^+\lambda ^2 \sigma _^+\sigma _^\right) } \sigma _3^},\\ B_4=-\frac \left( -2 \lambda ^3 \sigma _^+2 \lambda ^2 \sigma _^+2 \lambda ^2 \sigma _^+\lambda ^2 \sigma _^-2 \lambda \sigma _^+2 \sigma _^+2 \sigma _^+\sigma _^\right) -\delta _2^ \left( \lambda ^2 \sigma _^+\lambda ^2 \sigma _^+\sigma _^+\sigma _^\right) } \sigma _3^},\\ B_5=\frac-6 \kappa ^2 \delta _2^ \sigma _3^-3 \kappa \delta _1^ \sigma _1^} \sigma _3^},\\ B_6=-\frac \left( \lambda ^3 \sigma _^+3 \lambda ^2 \sigma _^+\lambda ^2 \sigma _^-\lambda ^2 \sigma _^+\lambda \sigma _^+3 \sigma _^+\sigma _^-\sigma _^\right) -\delta _2^ \left( \lambda ^2 \sigma _^+\lambda \sigma _^+\sigma _^+\sigma _^\right) } \sigma _3^}, \end\right. }} \end$$

(44)

provided \(\delta _2^ \sigma _3^\ne 0\).

Balancing \(_}^\) with \(U_^\) in Eq. (43) yields \(N=1\), thus formulating the solution as:

$$\begin }\left( \xi \right) =A_+A_F\left( \xi \right) . \end$$

(45)

Upon substituting (45) along with (8) into (43), we derive the equations as:

$$\begin & }}^B_}+RA_}}}^B_}+}}^B_}+A_ }B_}=0, \\ & 12\,PR^A_}+Q}}^A_}B_}+^^A_}+QA_}B_}+R}}^B_}\\ & +5\,}}^A_}B_}+3\,}} ^A_}B_}+A_}B_}=0, \\ & 10\,}}^}}^B_}+QA_}}}^B_}+2\,QA_ }}}^B_}+3\,A_}}}^B_}=0, \\ & 10\,}}^}}^B_}+20\,PQ^A_}+2\,P}} ^A_}B_}+Q}}^B_}\\ & +Q}}^B_}+}} ^B_}+2\,PA_}B_}=0, \\ & 5\,A_}}}^B_}+PA_}}}^B_}+4\,PA_}}}^B_}=0, \end$$

$$\begin }}^B_}+24\,^^A_}+P}}^B_}+2\,P}}^B_}=0. \end$$

(46)

Through the solution of these Eq. (46), the outcomes are revealed:

$$\begin k=\pm \sqrt}RB_}-^B_}B_}-^B_ }B_}-QB_}B_}-QB_}B_}-QB_}B_}-B_}B_}}}-12\,PQRB_}-^B_}-^B_}-12\,PRB_}- ^B_}}}}, \\ A_}=0,\ \ A_}=\pm \sqrt^RB_}+18\,P^B_}+20\,PQB_}}}-12\,PQRB_}-^B_}-^B_ }-12\,PRB_}-^B_}}}}, \end$$

$$\begin B_}= 12\,PB_}RB_}-24\,PRB_}B_}-3\,^B_}B_}+6\, ^B_}B_} \\ -2\,QB_}B_}+8\,QB_}B_}-12\,QB_}B_}-12\,B_}B_} \end \right) \left( \begin 8\,PQRB_}-12\,PQRB_}-^B_} \\ -^B_}-12\,PRB_}-^B_} \end \right) }^^}}^-216\,P^R}} ^+81\,^}}^-240\,PQRB_}B_}+180\,^B_}B_}+100\,^}}^\right) }.} \end$$

(47)

Analyzing the behaviors of a dark soliton, with an emphasis on its magnitude, imaginary and real components given \(\sigma =0\)

Analyzing the behaviors of a dark soliton, with an emphasis on its imaginary and real components given \(\sigma =2\)

Analyzing the behaviors of a dark soliton, with an emphasis on its imaginary and real components given \(\sigma =3\)

Analyzing the behaviors of a dark soliton, with an emphasis on its imaginary and real components given \(\sigma =4\)

Analyzing the behaviors of a dark soliton, with an emphasis on its imaginary and real components given \(\sigma =5\)

Result–1:

Incorporating the effects of Eq. (9) into Eq. (47) leads to a transformation that results in:

$$\begin k=\pm \sqrt}B_}-2\,B_}B_}-2\,B_}B_ }-2\,B_}B_}+2\,B_}B_}+4\,B_}B_}}}+8\,B_}-32\,B_}}}},A_}=0, \end$$

$$\begin A_}= & \pm \sqrt}-6\,B_}}}+B_}-4\,B_}}}},\nonumber \\ B_}= & }B_}-B_}B_}+4\,B_}B_}-6\,B_}B_}\right) \left( 2\,B_}+B_}-4\,B_}\right) }}}^-60\,B_}B_}+36\,}}^}.} \end$$

(48)

Consequently, the dark and singular soliton solutions are formulated as:

$$\begin & u(x,t)=\pm \sqrt}-6\,B_}}}+B_}-4\,B_ }}}}\text \\ & \left[ \begin \sqrt}B_}-2\,B_}B_}-2\,B_}B_}-2\,B_ }B_}+2\,B_}B_}+4\,B_}B_}}}+8\,B_}-32\,B_}}}} \\ \\ \times \left( x+\kappa \left( 2a^+5\kappa ^\delta _^\sigma _^-4\kappa ^\delta _^\sigma _^-3\kappa \delta _^\sigma _^\right) t\right) \end \right] \end$$

$$\begin \times e^t+\theta _\right) }, \end$$

(49)

$$\begin & v(x,t)=\pm \lambda \sqrt}-6\,B_}}}+B_ }-4\,B_}}}}\text \\ & \left[ \begin \sqrt}B_}-2\,B_}B_}-2\,B_}B_}-2\,B_ }B_}+2\,B_}B_}+4\,B_}B_}}}+8\,B_}-32\,B_}}}} \\ \\ \times \left( x+\kappa \left( 2a^+5\kappa ^\delta _^\sigma _^-4\kappa ^\delta _^\sigma _^-3\kappa \delta _^\sigma _^\right) t\right) \end \right] \end$$

$$\begin \times e^t+\theta _\right) }, \end$$

(50)

and

$$\begin & u(x,t)=\pm \sqrt}-6\,B_}}}+B_}-4\,B_ }}}}\text \\ & \left[ \begin \sqrt}B_}-2\,B_}B_}-2\,B_}B_}-2\,B_ }B_}+2\,B_}B_}+4\,B_}B_}}}+8\,B_}-32\,B_}}}} \\ \\ \times \left( x+\kappa \left( 2a^+5\kappa ^\delta _^\sigma _^-4\kappa ^\delta _^\sigma _^-3\kappa \delta _^\sigma _^\right) t\right) \end \right] \end$$

$$\begin \times e^t+\theta _\right) }, \end$$

(51)

$$\begin & v(x,t)=\pm \lambda \sqrt}-6\,B_}}}+B_ }-4\,B_}}}}\text \\ & \left[ \begin \sqrt}B_}-2\,B_}B_}-2\,B_}B_}-2\,B_ }B_}+2\,B_}B_}+4\,B_}B_}}}+8\,B_}-32\,B_}}}} \\ \\ \times \left( x+\kappa \left( 2a^+5\kappa ^\delta _^\sigma _^-4\kappa ^\delta _^\sigma _^-3\kappa \delta _^\sigma _^\right) t\right) \end \right] \end$$

$$\begin \times e^t+\theta _\right) }. \end$$

(52)

The waveforms presented in Eqs. (49)–(52) are defined by the parameters:

$$\begin \left( 5\,B_}-6\,B_}\right) \left( 2\,B_}+B_}-4\,B_ }\right) <0, \end$$

$$\begin & \left( B_}B_}-2\,B_}B_}-2\,B_}B_}-2\,B_}B_ }+2\,B_}B_}+4\,B_}B_}\right) \nonumber \\ & \left( 16\,B_}+8\,B_ }-32\,B_}\right) <0. \end$$

(53)

Analyzing the behaviors of a singular soliton, with an emphasis on its magnitude, imaginary and real components given \(\sigma =0\)

Analyzing the behaviors of a singular soliton, with an emphasis on its imaginary and real components given \(\sigma =2\)

Analyzing the behaviors of a singular soliton, with an emphasis on its imaginary and real components given \(\sigma =3\)

Analyzing the behaviors of a singular soliton, with an emphasis on its imaginary and real components given \(\sigma =4\)

Analyzing the behaviors of a singular soliton, with an emphasis on its imaginary and real components given \(\sigma =5\)

Result–2:

By applying the relationship established in Eqs. (9), (47) is modified and takes the form of:

$$\begin k=\pm \sqrt}+B_}\right) },A_}=0,A_}=\pm \sqrt}+18\,B_}}}+B_}+B_}}}}, \end$$

$$\begin B_}= 12\,B_}}}^+14\,B_}B_}B_}+4\,B_}B_}B_ }+2\,B_}}}^-6\,B_}B_}B_}-8\,B_}}}^ \\ +12\,}}^B_}+15\,B_}B_}B_}+6\,B_}B_}B_ }+3\,}}^B_}-3\,B_}B_}B_}-6\,B_}}}^ \end \right) }}}^+180\,B_}B_}+81\,}} ^\right) }.} \end$$

(54)

Therefore, the solutions exhibit bright solitons:

$$\begin & u(x,t)=\pm \sqrt}+18\,B_}}}+B_}+B_}}}} \text \left[ \sqrt}+B_}\right) }\right. \\ & \left. \left( x+\kappa \left( 2a^+5\kappa ^\delta _^\sigma _^-4\kappa ^\delta _^\sigma _^-3\kappa \delta _^\sigma _^\right) t\right) \right] \end$$

$$\begin \times e^t+\theta _\right) }, \end$$

(55)

and

$$\begin & v(x,t)=\pm \lambda \sqrt}+18\,B_}}}+B_ }+B_}}}}\text \left[ \sqrt}+B_}\right) }\right. \\ & \left. \left( x+\kappa \left( 2a^+5\kappa ^\delta _^\sigma _^-4\kappa ^\delta _^\sigma _^-3\kappa \delta _^\sigma _^\right) t\right) \right] \end$$

$$\begin \times e^t+\theta _\right) }. \end$$

(56)

The wave structures depicted in (55) and (56) are determined by the constraints given below:

$$\begin \left( 20\,B_}+18\,B_}\right) \left( B_}+B_}+B_}\right)<0,\ \ \ B_}+B_}<0. \end$$

(57)

Analyzing the behaviors of a bright soliton, with an emphasis on its magnitude, imagin