1 Introduction

In the present day of computer networking and communications, the topic of study in the theory of solitons and their utilization in fiber optics is becoming increasingly essential. An optical soliton is a flash of light that travels without distortion owing to dispersion or other causes. Both temporal and spatial solitons will be addressed, combined with the physical components that make them feasible. In this situation, the optical pulse could begin to create a stable nonlinear pulse known as an optical soliton. The dispersion of the fiber material restricts the bit rate of transmission. Fiber loss is the sole element that contributes to the decline of the pulse quality through expansion in the pulse width.

The complex nonlinear (2 + 1)-dimensional Fokas system that demonstrates nonlinear pulse propagation in monomode fiber optics has the following form:

$$\begin{aligned} \begin{aligned}&i u_{t} + \beta _{1} u_{xxx} + \beta _{2} uv =0\\&\beta _{3} v_{y} - \beta _{4} (|u|^2 )_{x} =0, \end{aligned} \end{aligned}$$
(1)

that derived in 1994 by Fokas (1994) employing the inverse spectral method, the non-linear pulse propagation in monomode fiber optic is represented by the complex functions v(x, y, t) and u(x, y, t). \(\beta _{1}\) symbolizes the dispersion coefficient, which characterizes the degree of dispersion in the system, \(\beta _{2}\) denotes the nonlinear pairing parameter, which indicates the intensity of the nonlinear dealing among the two components of the system, u and v, \(\beta _{3}\) symbolizes the transverse diffusion parameter, which specifies the amount of dispersion in the transverse direction, \(\beta _{4}\) illustrates the nonlinear immersion coefficient, which represents the amount of a saturated state of the nonlinear participation. Differing versions of (1) have been examined using various methodologies, including Riccati expansion and Ansatz methods (Khater 2021), the generic Kudryashov’s method, the Sardar sub-equation approach, and Bernoulli sub-equation function method (Ali et al. 2023b), the truncated Painlevé approach (Thilakavathy et al. 2023), generalized Riccati equation mapping and Kudryashov methods (Kumar and Kumar 2023a), using a modified mapping method (Mohammed et al. 2023), the extended rational versions of \(\sinh\)–\(\cosh\) and \(\sin\)–\(\cos\) methods (Wang et al. 2022), the bilinear transformation method (Chen et al. 2019; Rao et al. 2015), the bilinear Kadomtsev-Petviashvili hierarchy reduction method (Rao et al. 2021), the bilinear forms of Hirota’s method (Rao et al. 2019), the exponential function method (Wang 2022), the elliptic function expansion forms of the Jacobian method (Tarla et al. 2022), the singular manifold, and the expansion forms of \(G'/G^2\), Sine-Gordon methods (Alrebdi et al. 2022), the polynomial method that depends on the complete discrimination (Zhang et al. 2023).

Numerous research works examine considerable analytical and semi-analytical techniques for getting the exact solution of NPDEs, including the modified version of the exponential-function method (Muhamad et al. 2023), the extended rational forms of \(\sin\)–\(\cos\) and \(\sinh\)–\(\cosh\) methods (Mahmud et al. 2023a, b), Bernoulli and its improved version (Baskonus et al. 2022a, b; Mahmud et al. 2023c, d), The transformation of Laplace has been used for solving the fractional system in the form of Caputo fractional derivatives (Tanriverdi et al. 2021), it is worth mentioning that the main source of these modifications are (Mahmud 2023; Muhamad 2023f), the extended auxiliary equation mapping and extended direct algebraic methods (Iqbal et al. 2018a, b, 2019; Seadawy et al. 2019, 2020a, b; Seadawy and Iqbal 2021), the extension of the modified rational expansion method (Seadawy et al. 2021), the modification form of extended auxiliary equation mapping method (Lu et al. 2018; Iqbal and Seadawy 2020; Seadawy and Iqbal 2023), the extended modified rational expansion method (Seadawy et al. 2022), the generalized exponential rational function method (Ghanbari and Gómez-Aguilar 2019a, b; Ghanbari and Baleanu 2020; Ghanbari 2019; Ghanbari et al. 2018; Ghanbari and Kuo 2019; Ghanbari and Baleanu 2019), the five methods mentioned therein (Khater and Ghanbari 2021), the reproducing kernel method (Ghanbari and Akgül 2020), the extended rational \(\sinh\)-Gordon method and \(\exp (-\phi (\eta ))\) expansion function method (Shafqat-ur-Rehman and Ahmad 2023; Rehman and Ahmad 2023), the modified generalized exponential rational function method, and the modified rational \(\sinh\)–\(\cosh\) and \(\sin\)–\(\cos\) methods (Rehman et al. 2022, 2023a, b; Ahmad et al. 2023; Ahmad 2023), the modified Sardar sub-equation method (Ali et al. 2023a). Considering this context, we can notice an array of methodologies used by several academics to express their ideas in exploring the mathematical models that describe situations in real life (Gasmi et al. 2023; Jafari et al. 2023; Srinivasa and Mundewadi 2023; Bilal et al. 2023; Kumar and Kumar 2023b; Nasir et al. 2023). Overall, some shortcomings and adverse characteristics in the prior versions of these methods became the motivation for us to come up with these two additional enhancements.

This scholarly investigation has been laid out as follows: Sect. 1 is specialized for listing the literature relevant to the approaches and the examined model in a short overview. The methodologies of the described approaches are detailed in Sect. 2. The formulation of the recommended techniques for constructing specific semi-analytic solutions to Eq. (1) is presented in Sect. 3. In Sect. 4, the concluding remarks of the study have been provided agreeably. Finally, the last Sect. 5, is dedicated to the analysis and discussion of the results that were collected.

2 Formulation of the modification methods

Always, the configuration of the presented approaches commonly depends on the following step:

Step 1 Let the next NPDE be followed.

$$\begin{aligned} {\mathscr {S}} \left( {\mathscr {T}}, {\mathscr {T}}_{x},{\mathscr {T}}_{t},{\mathscr {T}}_{y},{\mathscr {T}}_{xt},{\mathscr {T}}_{xx},{\mathscr {T}}_{yt},{\mathscr {T}}_{xyt}, \ldots \right) = 0, \end{aligned}$$
(2)

wherein \({\mathscr {T}} = {\mathscr {T}}(x,y,t)\). By setting

$$\begin{aligned} {\mathscr {T}}(x,y,t) = {\mathscr {R}}({\mathscr {P}}),\; {\mathscr {P}} = \delta _1 x + \delta _2 y - \delta _3 t, \end{aligned}$$
(3)

where \(\delta _1,\; \delta _2\) and \(\delta _3\) are non-zero arbitrary parameters. If (3) is substituted in (2), then the outcome is presented as follows

$$\begin{aligned} {\mathscr {I}}({\mathscr {R}},{\mathscr {R}}',{\mathscr {R}}'', \ldots ) = 0, \end{aligned}$$
(4)

herein

$$\begin{aligned} {\mathscr {R}} = {\mathscr {R}}({\mathscr {P}}),\; {\mathscr {R}}' = \frac{d {\mathscr {R}}}{d {\mathscr {P}}},\; {\mathscr {R}}'' = \frac{{d^2 {\mathscr {R}}}}{{d {\mathscr {P}} ^2}}, \ldots \end{aligned}$$

Step 2 Initially, we created these two modified solution forms:

  1. 1.

    For the first modification, let the solution to (4) take the following forms:

    $$\begin{aligned} {\mathscr {T}}({\mathscr {P}})= \frac{\gamma _0 + \gamma _1 \sinh ( \mu {\mathscr {P}})}{\gamma _2 \sinh ( \mu {\mathscr {P}}) \pm \gamma _3 \cosh ( \mu {\mathscr {P}})}, \; \gamma _2 \sinh ( \mu {\mathscr {P}}) \pm \gamma _3 \cosh ( \mu {\mathscr {P}}) \ne 0, \end{aligned}$$
    (5)

    or,

    $$\begin{aligned} {\mathscr {T}}({\mathscr {P}})= \frac{\gamma _0 + \gamma _1 \cosh ( \mu {\mathscr {P}})}{\gamma _2 \sinh ( \mu {\mathscr {P}}) \pm \gamma _3 \cosh ( \mu {\mathscr {P}})}, \; \gamma _2 \sinh ( \mu {\mathscr {P}}) \pm \gamma _3 \cosh ( \mu {\mathscr {P}}) \ne 0, \end{aligned}$$
    (6)
  2. 2.

    For the second modification, suppose that the solutions to (4) take the following forms:

    $$\begin{aligned} {\mathscr {T}}({\mathscr {P}})= \frac{\gamma _0 + \gamma _1 \sin ( \mu {\mathscr {P}})}{\gamma _2 \sin ( \mu {\mathscr {P}}) \pm \gamma _3 \cos ( \mu {\mathscr {P}})}, \; \gamma _2 \sin ( \mu {\mathscr {P}}) \pm \gamma _3 \cos ( \mu {\mathscr {P}}) \ne 0, \end{aligned}$$
    (7)

    or,

    $$\begin{aligned} {\mathscr {T}}({\mathscr {P}})= \frac{\gamma _0 + \gamma _1 \cos ( \mu {\mathscr {P}})}{\gamma _2 \sin ( \mu {\mathscr {P}}) \pm \gamma _3 \cos ( \mu {\mathscr {P}})}, \; \gamma _2 \sin ( \mu {\mathscr {P}}) \pm \gamma _3 \cos ( \mu {\mathscr {P}}) \ne 0, \end{aligned}$$
    (8)

where in (5–8), the \(\mu , \; \gamma _{i},\; \text {for}\; i=0,1,2,3\) are intended coefficients that will be identified later such that

$$\begin{aligned} \gamma _{0}^2 + \gamma _{1}^2 \ne 0, \; \gamma _{2}^2 + \gamma _{3}^2 \ne 0, \end{aligned}$$

and a wave number \(\mu \ne 0\).

Step 3 Anonymous, also known as parameters, might be found by substituting one of (5–8) into (4), putting together all the terms that have the same powers as and equating to zero all the coefficients for the same power terms, this process produces a set of algebraic equations. Identifying the solutions to the obtained algebraic system using different symbolic computing tools is possible.

Step 4 By re-installing the obtained results of \(\gamma _0, \gamma _1, \gamma _2, \gamma _3\) and \(\mu\) into one of (5–8), the solution to (4) will be derived, and thereafter, the solution to (2) is obtained.

3 Implementations of the recommended methods

Implementing waveform transformation

$$\begin{aligned} u(x,y,t) = U(\xi )e^{i\kappa \xi }, v(x,y,t) = V(\xi ),\; \xi = \delta _{1} x + \delta _{2} y - \delta _{3} t, \end{aligned}$$
(9)

to (1), then one gets the following:

$$\begin{aligned} \begin{aligned}&\kappa \delta _3 U - i (\delta _3 - 2 \beta _1 \kappa \delta _1 ^2 )U' -\beta _1 \kappa ^2 \delta _1 ^2 U + \beta _1 \delta _1 ^2 U'' + \beta _2 UV =0\\&\quad \beta _3 \delta _2 V' -2 \beta _4 \delta _1 U U'=0. \end{aligned} \end{aligned}$$
(10)

directly from the second part of (10), one obtains:

$$\begin{aligned} V= \frac{\beta _4 \delta _1}{\beta _3 \delta _2} U^2 \cdot \end{aligned}$$
(11)

By substituting (11) into the first part of (10), the following is the outcome:

$$\begin{aligned} \kappa \delta _3 U - i (\delta _3 - 2 \beta _1 \kappa \delta _1 ^2 )U' -\beta _1 \kappa ^2 \delta _1 ^2 U + \beta _1 \delta _1 ^2 U'' + \frac{\beta _2 \beta _4 \delta _1}{\beta _3 \delta _2} U^3 =0. \end{aligned}$$
(12)

By splitting the real and imagined components of (12), the operators end up with:

$$\begin{aligned} \begin{aligned}&\kappa \delta _3 U -\beta _1 \kappa ^2 \delta _1 ^2 U + \beta _1 \delta _1 ^2 U'' + \frac{\beta _2 \beta _4 \delta _1}{\beta _3 \delta _2} U^3 =0\\&\quad - i (\delta _3 - 2 \beta _1 \kappa \delta _1 ^2 )U'=0. \end{aligned} \end{aligned}$$
(13)

From the imaginary part of (13), one immediately obtains:

$$\begin{aligned} \kappa = \frac{\delta _3}{2 \beta _1 \delta _1 ^2} \cdot \end{aligned}$$
(14)

By substituting (14) into the real part of (13) after simplifications, the following is the result:

$$\begin{aligned} \frac{\delta _3 ^2}{4 \beta _1 \delta _1 ^2} U + \beta _1 \delta _1 ^2 U'' + \frac{\beta _2 \beta _4 \delta _1}{\beta _3 \delta _2} U^3 =0. \end{aligned}$$
(15)

A recommended equation to suppose the trial solution is the ordinary differential equation (15).

3.1 Implementation of MER \(\sinh\)–\(\cosh\) M to the examined model

To solve (1) by employing the MER \(\sinh\)-\(\cosh\) M, suppose that (15) has a solution with the following form:

$$\begin{aligned} \frac{\gamma _1 \sinh (\mu \xi )+\gamma _0}{\gamma _2 \sinh (\mu \xi )+\gamma _3 \cosh (\mu \xi )}. \end{aligned}$$
(16)

In (16), \(\mu , \gamma _{0}, \gamma _{1}, \gamma _{2}, \; \text {and}\; \gamma _{3}\) are unknown purposeful parameters that must be demonstrated later by taking into account that

$$\begin{aligned} \mu \ne 0, \; \gamma _{0}^2 + \gamma _{1}^2 \ne 0, \; \gamma _{2}^2 + \gamma _{3}^2 \ne 0, \end{aligned}$$

and \(\mu\) is a wave number. Moreover, the derivatives of (16) with respect to \(\xi\) are taking the following forms:

$$\begin{aligned} U' = \frac{\gamma _1 \gamma _3 \mu -\gamma _0 \mu \left( \gamma _3 \sinh (\mu \xi )+\gamma _2 \cosh (\mu \xi )\right) }{\left( \gamma _2 \sinh (\mu \xi ) +\gamma _3 \cosh (\mu \xi )\right) {}^2}, \end{aligned}$$
(17)

and

$$\begin{aligned} \begin{aligned} U''&= -\frac{2 \left( \gamma _3 \mu \sinh (\mu \xi )+\gamma _2 \mu \cosh (\mu \xi )\right) \left( \gamma _1 \gamma _3 \mu -\gamma _0 \mu \left( \gamma _3 \sinh (\mu \xi )+ \gamma _2 \cosh (\mu \xi )\right) \right) }{\left( \gamma _2 \sinh (\mu \xi )+\gamma _3 \cosh (\mu \xi )\right) ^3}\\&\quad -\frac{\gamma _0 \mu \left( \gamma _2 \mu \sinh (\mu \xi )+\gamma _3 \mu \cosh (\mu \xi )\right) }{\left( \gamma _2 \sinh (\mu \xi )+\gamma _3 \cosh (\mu \xi )\right) ^2}. \end{aligned} \end{aligned}$$
(18)

Subbing (16)–(18) into (15), one gets the following:

$$\begin{aligned} \left. \begin{aligned}&-4 \beta _1^2 \beta _3 \gamma _0 \gamma _2^2 \delta _2 \delta _1^4 \mu ^2 \sinh ^2(\mu \xi )-4 \beta _1^2 \beta _3 \gamma _0 \gamma _3^2 \delta _2 \delta _1^4 \mu ^2 \sinh ^2(\mu \xi )-4 \beta _1 \beta _2 \beta _4 \gamma _0^3 \delta _1^3\\&\quad +8 \beta _1^2 \beta _3 \gamma _1 \gamma _3^2 \delta _2 \delta _1^4 \mu ^2 \sinh (\mu \xi )+8 \beta _1^2 \beta _3 \gamma _1 \gamma _2 \gamma _3 \delta _2 \delta _1^4 \mu ^2 \cosh (\mu \xi )-\beta _3 \gamma _0 \gamma _3^2 \delta _2 \delta _3^2\\&\quad +4 \beta _1^2 \beta _3 \gamma _0 \gamma _3^2 \delta _2 \delta _1^4 \mu ^2-4 \beta _1 \beta _2 \beta _4 \gamma _1^3 \delta _1^3 \sinh ^3(\mu \xi )-\beta _3 \gamma _1 \gamma _2^2 \delta _2 \delta _3^2 \sinh ^3(\mu \xi )\\&\quad -\beta _3 \gamma _1 \gamma _3^2 \delta _2 \delta _3^2 \sinh ^3(\mu \xi )-12 \beta _1 \beta _2 \beta _4 \gamma _0 \gamma _1^2 \delta _1^3 \sinh ^2(\mu \xi ) -\beta _3 \gamma _0 \gamma _2^2 \delta _2 \delta _3^2 \sinh ^2(\mu \xi )\\&\quad -\beta _3 \gamma _0 \gamma _3^2 \delta _2 \delta _3^2 \sinh ^2(\mu \xi )-12 \beta _1 \beta _2 \beta _4 \gamma _0^2 \gamma _1 \delta _1^3 \sinh (\mu \xi ) -\beta _3 \gamma _1 \gamma _3^2 \delta _2 \delta _3^2 \sinh (\mu \xi )\\&\quad -2 \beta _3 \gamma _1 \gamma _2 \gamma _3 \delta _2 \delta _3^2 \sinh ^2(\mu \xi ) \cosh (\mu \xi )-2 \beta _3 \gamma _0 \gamma _2 \gamma _3 \delta _2 \delta _3^2 \sinh (\mu \xi ) \cosh (\mu \xi )\\&\quad -8 \beta _1^2 \beta _3 \gamma _0 \gamma _2 \gamma _3 \delta _2 \delta _1^4 \mu ^2 \sinh (\mu \xi ) \cosh (\mu \xi )-8 \beta _1^2 \beta _3 \gamma _0 \gamma _2^2 \delta _2 \delta _1^4 \mu ^2 =0. \end{aligned} \right\} \end{aligned}$$
(19)

In (19) collecting all the coefficients with the same powers of \(\cosh ^{\tau _1}(\mu {\mathscr {P}}) \sinh ^{\tau _2}(\mu {\mathscr {P}})\) where \(\tau _1, \tau _2 = 0, 1, 2, 3\) and equating them to zero. From the coefficients of \(\cosh ^{\tau _1}(\mu {\mathscr {P}}) \sinh ^{\tau _2}(\mu {\mathscr {P}})\), one creates a system as given below:

$$\begin{aligned} \left. \begin{aligned}&-8 \beta _1^2 \beta _3 \gamma _0 \gamma _2^2 \delta _2 \delta _1^4 \mu ^2+4 \beta _1^2 \beta _3 \gamma _0 \gamma _3^2 \delta _2 \delta _1^4 \mu ^2 -4 \beta _1 \beta _2 \beta _4 \gamma _0^3 \delta _1^3-\beta _3 \gamma _0 \gamma _3^2 \delta _2 \delta _3^2=0,\\&\quad 8 \beta _1^2 \beta _3 \gamma _1 \gamma _2 \gamma _3 \delta _1^4 \delta _2 \mu ^2=0, \\&\quad 8 \beta _1^2 \beta _3 \gamma _1 \gamma _3^2 \delta _2 \delta _1^4 \mu ^2-12 \beta _1 \beta _2 \beta _4 \gamma _0^2 \gamma _1 \delta _1^3 -\beta _3 \gamma _1 \gamma _3^2 \delta _2 \delta _3^2 =0,\\&\quad -8 \beta _1^2 \beta _3 \gamma _0 \gamma _2 \gamma _3 \delta _2 \delta _1^4 \mu ^2-2 \beta _3 \gamma _0 \gamma _2 \gamma _3 \delta _2 \delta _3^2 =0,\\&\quad -4 \beta _1^2 \beta _3 \gamma _0 \gamma _2^2 \delta _2 \delta _1^4 \mu ^2-4 \beta _1^2 \beta _3 \gamma _0 \gamma _3^2 \delta _2 \delta _1^4 \mu ^2 -12 \beta _1 \beta _2 \beta _4 \gamma _0 \gamma _1^2 \delta _1^3-\beta _3 \gamma _0 \gamma _2^2 \delta _2 \delta _3^2\\&\quad -\beta _3 \gamma _0 \gamma _3^2 \delta _2 \delta _3^2 =0,\\&\quad -2 \beta _3 \gamma _1 \gamma _2 \gamma _3 \delta _2 \delta _3^2=0,\\&\quad -4 \beta _1 \beta _2 \beta _4 \gamma _1^3 \delta _1^3-\beta _3 \gamma _1 \gamma _2^2 \delta _2 \delta _3^2-\beta _3 \gamma _1 \gamma _3^2 \delta _2 \delta _3^2=0. \end{aligned} \right\} \end{aligned}$$
(20)

One creates the following cases by solving (20).

Case 1 The following are the parameters that were obtained from solving (20):

$$\begin{aligned} \gamma _2= & {} -\frac{\sqrt{2 \beta _1 \beta _3 \gamma _3^2 \delta _1 \delta _2 \mu ^2-\beta _2 \beta _4 \gamma _0^2}}{\sqrt{2} \sqrt{\beta _1} \sqrt{\beta _3} \sqrt{\delta _1} \sqrt{\delta _2} \mu }; \nonumber \\ \gamma _1= & {} 0; \delta _3=-2 i \beta _1 \delta _1^2 \mu . \end{aligned}$$
(21)

The following set of solutions to (1) has been identified by replacing (21) gathering with (16) into (15).

$$\begin{aligned} \begin{aligned} u_1 = \frac{\gamma _0 e^{\delta _2 \mu y+\delta _1 \mu \left( x+2 i \beta _1 \delta _1 \mu t\right) }}{\gamma _3 \cosh \left( \delta _2 \mu y+\delta _1 \mu \left( x+2 i \beta _1 \delta _1 \mu t\right) \right) -\frac{\sqrt{2 \beta _1 \beta _3 \gamma _3^2 \delta _1 \delta _2 \mu ^2-\beta _2 \beta _4 \gamma _0^2} \sinh \left( \delta _2 \mu y+\delta _1 \mu \left( x+2 i \beta _1 \delta _1 \mu t\right) \right) }{\sqrt{2} \sqrt{\beta _1} \sqrt{\beta _3} \sqrt{\delta _1} \sqrt{\delta _2} \mu }}, \end{aligned} \end{aligned}$$
(22)

and

$$\begin{aligned} \begin{aligned} v_1 = \frac{\beta _4 \gamma _0^2 \delta _1}{\beta _3 \delta _2 \left( \gamma _3 \cosh \left( \mu \left( 2 i \beta _1 \delta _1^2 \mu t+\delta _1 x+\delta _2 y\right) \right) - \frac{\sqrt{2 \beta _1 \beta _3 \gamma _3^2 \delta _1 \delta _2 \mu ^2-\beta _2 \beta _4 \gamma _0^2} \sinh \left( \mu \left( 2 i \beta _1 \delta _1^2 \mu t+\delta _1 x +\delta _2 y\right) \right) }{\sqrt{2} \sqrt{\beta _1} \sqrt{\beta _3} \sqrt{\delta _1} \sqrt{\delta _2} \mu }\right) ^2}. \end{aligned} \end{aligned}$$
(23)

Graphs of (22) and (23) where \(\; \beta _1=-\frac{8}{3}; \beta _2=\frac{1}{2}; \beta _3=\frac{9}{4}; \beta _4=\frac{2}{3}; \mu =-\frac{2}{3}; \delta _1=\frac{2}{5}; \delta _2=-\frac{1}{2}; y=-\frac{3}{2}; \gamma _0=\frac{5}{2}; \gamma _3=\frac{1}{2},\) and \(\; - 20 \le x \le 20,\; -20 \le t \le 20 \;\) are given in the following:

Fig. 1
figure 1

3D figures of (22)

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Fig. 2
figure 2

Contour surfaces of (22)

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For the values of t that are mentioned below, one reaches:

Fig. 3
figure 3

2D graphs of (22)

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Fig. 4
figure 4

3D figures of (23)

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Fig. 5
figure 5

Contour surfaces of (23)

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The values of t are mentioned in the legend below.

Fig. 6
figure 6

2D graphs of (23)

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Case 2 The following are the parameters that were obtained from solving (20):

$$\begin{aligned} \begin{aligned} \gamma _0&=-\frac{\sqrt{\beta _1} \sqrt{\beta _3} \gamma _3 \sqrt{\delta _1} \sqrt{\delta _2} \mu }{\sqrt{2} \sqrt{\beta _2} \sqrt{\beta _4}};\\ \gamma _1&=-\frac{i \sqrt{\beta _1} \sqrt{\beta _3} \gamma _3 \sqrt{\delta _1} \sqrt{\delta _2} \mu }{\sqrt{2} \sqrt{\beta _2} \sqrt{\beta _4}};\gamma _2=0;\delta _3=-\sqrt{2} \beta _1 \delta _1^2 \mu . \end{aligned} \end{aligned}$$
(24)

The following set of solutions to (1) has been determined by re-installing (24) with (16) into (15).

$$\begin{aligned} \begin{aligned} u_2&= -\frac{\sqrt{\beta _1} \sqrt{\beta _3} \sqrt{\delta _1} \sqrt{\delta _2} \mu \exp \left( -\frac{i \mu \left( \sqrt{2} \beta _1 \delta _1^2 \mu t+\delta _1 x+\delta _2 y \right) }{\sqrt{2}}\right) \text {sech}\left( \mu \left( \sqrt{2} \beta _1 \delta _1^2 \mu t+\delta _1 x+\delta _2 y\right) \right) }{\sqrt{2} \sqrt{\beta _2} \sqrt{\beta _4}}\\&\quad -\frac{\sqrt{\beta _1} \sqrt{\beta _3} \sqrt{\delta _1} \sqrt{\delta _2} \mu \exp \left( -\frac{i \mu \left( \sqrt{2} \beta _1 \delta _1^2 \mu t+\delta _1 x+\delta _2 y \right) }{\sqrt{2}}\right) \left( i \tanh \left( \mu \left( \sqrt{2} \beta _1 \delta _1^2 \mu t+\delta _1 x+\delta _2 y\right) \right) \right) }{\sqrt{2} \sqrt{\beta _2} \sqrt{\beta _4}}, \end{aligned} \end{aligned}$$
(25)

and

$$\begin{aligned} v_2 = \frac{\beta _1 \delta _1^2 \mu ^2 \left( \text {sech}\left( \mu \left( \sqrt{2} \beta _1 \delta _1^2 \mu t+\delta _1 x+\delta _2 y\right) \right) +i \tanh \left( \mu \left( \sqrt{2} \beta _1 \delta _1^2 \mu t+\delta _1 x+\delta _2 y\right) \right) \right) {}^2}{2 \beta _2}. \end{aligned}$$
(26)

Profile of the solutions in (25) and (26) where \(\; \beta _1=\frac{8}{3}; \beta _2=\frac{1}{2}; \beta _3=\frac{5}{4}; \beta _4=\frac{2}{5}; \mu =-\frac{3}{4}; \delta _1=\frac{5}{2}; \delta _2=\frac{3}{2}; y=-\frac{3}{2}; \gamma _0=\frac{5}{2}; \gamma _3=\frac{1}{2};\) and \(\; - 20 \le x \le 20,\) are given bellow for the different values of t that mentioned in the legend

Fig. 7
figure 7

2D graphs to (25)

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Fig. 8
figure 8

2D graphs to (26)

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Case 3 The following are the parameters that were reached from solving (20):

$$\begin{aligned} \begin{aligned} \gamma _0=-\frac{i \sqrt{\beta _3} \gamma _3 \sqrt{\delta _2} \delta _3}{\sqrt{2} \sqrt{\beta _1} \sqrt{\beta _2} \sqrt{\beta _4} \delta _1^{3/2}}; \gamma _1=0;\gamma _2=0;\mu =\frac{i \delta _3}{2 \beta _1 \delta _1^2}. \end{aligned} \end{aligned}$$
(27)

By inserting (27) and (16) into (15), the following set of solutions to (1) have been gained:

$$\begin{aligned} u_3 = -\frac{i \sqrt{\beta _3} \sqrt{\delta _2} \delta _3 \exp \left( \frac{i \delta _3 \left( -\delta _3 t+\delta _1 x+\delta _2 y\right) }{2 \beta _1 \delta _1^2}\right) \sec \left( \frac{\delta _3 \left( -\delta _3 t+\delta _1 x+\delta _2 y\right) }{2 \beta _1 \delta _1^2}\right) }{\sqrt{2} \sqrt{\beta _1} \sqrt{\beta _2} \sqrt{\beta _4} \delta _1^{3/2}}, \end{aligned}$$
(28)

and

$$\begin{aligned} v_3 = -\frac{\delta _3^2 \sec ^2\left( \frac{\delta _3 \left( -\delta _3 t+\delta _1 x+\delta _2 y\right) }{2 \beta _1 \delta _1^2}\right) }{2 \beta _1 \beta _2 \delta _1^2}. \end{aligned}$$
(29)

Remark 1

Similarly, by assuming that (6) is the trial solution to (15), some other set solutions to (1) may be obtained using the same prior process.

3.2 Implementation of MER \(\sin\)–\(\cos\) M to the examined model

To solve (1) by employing the MER \(\sin\)–\(\cos\) M, suppose that (15) has a solution with the following structure:

$$\begin{aligned} \frac{\lambda _1 \cos (\mu \xi )+\lambda _0}{\lambda _3 \sin (\mu \xi )+\lambda _2 \cos (\mu \xi )}. \end{aligned}$$
(30)

In (16), \(\mu , \lambda _{0}, \lambda _{1}, \lambda _{2}, \; \text {and}\; \lambda _{3}\) are unknown purposeful parameters that must be demonstrated later by taking into account that

$$\begin{aligned} \mu \ne 0, \; \lambda _{0}^2 + \lambda _{1}^2 \ne 0, \; \lambda _{2}^2 + \lambda _{3}^2 \ne 0, \end{aligned}$$

and \(\mu\) is a wave number. Moreover, the successive derivatives of (16) according to \(\xi\) are taking the forms below.

$$\begin{aligned} U' = \frac{\lambda _0 \lambda _2 \mu \sin (\mu \xi )-\lambda _3 \mu \left( \lambda _0 \cos (\mu \xi )+\lambda _1\right) }{\left( \lambda _3 \sin (\mu \xi ) +\lambda _2 \cos (\mu \xi )\right) {}^2}, \end{aligned}$$
(31)

and

$$\begin{aligned} \begin{aligned} U''&= -\frac{2 \left( \lambda _3 \mu \cos (\mu \xi )-\lambda _2 \mu \sin (\mu \xi )\right) \left( \lambda _0 \lambda _2 \mu \sin (\mu \xi )-\lambda _3 \mu \left( \lambda _0 \cos (\mu \xi )+\lambda _1\right) \right) }{\left( \lambda _3 \sin (\mu \xi )+\lambda _2 \cos (\mu \xi )\right) {}^3}\\&\quad +\frac{\lambda _0 \lambda _3 \mu ^2 \sin (\mu \xi )+\lambda _0 \lambda _2 \mu ^2 \cos (\mu \xi )}{\left( \lambda _3 \sin (\mu \xi )+\lambda _2 \cos (\mu \xi )\right) ^2}. \end{aligned} \end{aligned}$$
(32)

Subbing (30)–(32) into (15), one gets the following:

$$\begin{aligned} \left. \begin{aligned}&4 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _2^2 \mu ^2 \sin ^2(\mu \xi )-4 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _3^2 \mu ^2 \sin ^2(\mu \xi )+4 \beta _1 \beta _2 \beta _4 \delta _1^3 \lambda _0^3\\&\quad -8 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _1 \lambda _2 \lambda _3 \mu ^2 \sin (\mu \xi )+8 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _1 \lambda _3^2 \mu ^2 \cos (\mu \xi )\\&\quad -8 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _2 \lambda _3 \mu ^2 \sin (\mu \xi ) \cos (\mu \xi )+4 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _2^2 \mu ^2\\&\quad +8 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _3^2 \mu ^2+\beta _3 \delta _2 \delta _3^2 \lambda _0 \lambda _3^2 \sin ^2(\mu \xi )+4 \beta _1 \beta _2 \beta _4 \delta _1^3 \lambda _1^3 \cos ^3(\mu \xi )\\&\quad +\beta _3 \delta _2 \delta _3^2 \lambda _1 \lambda _2^2 \cos ^3(\mu \xi )+12 \beta _1 \beta _2 \beta _4 \delta _1^3 \lambda _0 \lambda _1^2 \cos ^2(\mu \xi ) +\beta _3 \delta _2 \delta _3^2 \lambda _0 \lambda _2^2 \cos ^2(\mu \xi )\\&\quad +12 \beta _1 \beta _2 \beta _4 \delta _1^3 \lambda _0^2 \lambda _1 \cos (\mu \xi )+2 \beta _3 \delta _2 \delta _3^2 \lambda _1 \lambda _2 \lambda _3 \sin (\mu \xi ) \cos ^2(\mu \xi )\\&\quad +\beta _3 \delta _2 \delta _3^2 \lambda _1 \lambda _3^2 \sin ^2(\mu \xi ) \cos (\mu \xi )+2 \beta _3 \delta _2 \delta _3^2 \lambda _0 \lambda _2 \lambda _3 \sin (\mu \xi ) \cos (\mu \xi ) =0. \end{aligned} \right\} \end{aligned}$$
(33)

In (33), by collecting all the coefficients with the same powers of \(\cos ^{\tau _1}(\mu {\mathscr {P}}) \sin ^{\tau _2}(\mu {\mathscr {P}})\) where \(\tau _1, \tau _2 = 0, 1, 2, 3\) and equating them to zero. From the coefficients of \(\cos ^{\tau _1}(\mu {\mathscr {P}}) \sin ^{\tau _2}(\mu {\mathscr {P}})\), one creates a system as given below:

$$\begin{aligned} \left. \begin{aligned}&4 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _2^2 \mu ^2+8 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _3^2 \mu ^2 +4 \beta _1 \beta _2 \beta _4 \delta _1^3 \lambda _0^3=0,\\&\quad -8 \beta _1^2 \beta _3 \delta _1^4 \delta _2 \lambda _1 \lambda _2 \lambda _3 \mu ^2 =0, \\&\quad 4 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _2^2 \mu ^2-4 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _0 \lambda _3^2 \mu ^2 +\beta _3 \delta _2 \delta _3^2 \lambda _0 \lambda _3^2 =0,\\&\quad 8 \beta _1^2 \beta _3 \delta _2 \delta _1^4 \lambda _1 \lambda _3^2 \mu ^2+12 \beta _1 \beta _2 \beta _4 \delta _1^3 \lambda _0^2 \lambda _1 =0,\\&\quad 2 \beta _3 \delta _2 \delta _3^2 \lambda _0 \lambda _2 \lambda _3-8 \beta _1^2 \beta _3 \delta _1^4 \delta _2 \lambda _0 \lambda _2 \lambda _3 \mu ^2=0,\\&\quad \beta _3 \delta _2 \delta _3^2 \lambda _1 \lambda _3^2 =0,\\&\quad 12 \beta _1 \beta _2 \beta _4 \delta _1^3 \lambda _0 \lambda _1^2+\beta _3 \delta _2 \delta _3^2 \lambda _0 \lambda _2^2 =0,\\&\quad 2 \beta _3 \delta _2 \delta _3^2 \lambda _1 \lambda _2 \lambda _3 =0,\\&\quad 4 \beta _1 \beta _2 \beta _4 \delta _1^3 \lambda _1^3+\beta _3 \delta _2 \delta _3^2 \lambda _2^2 \lambda _1=0. \end{aligned} \right\} \end{aligned}$$
(34)

By solving (34), the following cases are created:

Case 1 The following are the parameters that were obtained from solving (34):

$$\begin{aligned} \beta _1=\frac{\delta _3}{2 \delta _1^2 \mu }; \lambda _1=0; \lambda _2=0; \lambda _0=\frac{i \sqrt{\beta _3} \sqrt{\delta _2} \sqrt{\delta _3} \lambda _3 \sqrt{\mu }}{\sqrt{\beta _2} \sqrt{\beta _4} \sqrt{\delta _1}}. \end{aligned}$$
(35)

The following set of solutions to (1) has been identified by replacing (35) gathering with (30) into (15).

$$\begin{aligned} \begin{aligned} u_4 = \frac{i \sqrt{\beta _3} \sqrt{\delta _2} \sqrt{\delta _3} \sqrt{\mu } \left( \cot \left( \mu \left( -\delta _3 t+\delta _1 x+\delta _2 y\right) \right) +i\right) }{\sqrt{\beta _2} \sqrt{\beta _4} \sqrt{\delta _1}}, \end{aligned} \end{aligned}$$
(36)

and

$$\begin{aligned} \begin{aligned} v_4 = -\frac{\delta _3 \mu }{\beta _2} \csc ^2\left( \mu \left( -\delta _3 t+\delta _1 x+\delta _2 y\right) \right) . \end{aligned} \end{aligned}$$
(37)

Graphs of (36) and (37) where \(\; \beta _2=\frac{1}{4}; \beta _3=\frac{7}{2}; \beta _4=\frac{5}{3}; \mu =\frac{1}{2}; \delta _1=\frac{2}{5}; \delta _2=\frac{3}{2}; \delta _3=\frac{3}{4}; y=\frac{3}{2},\) and \(\; - 10 \le x \le 10,\; -10 \le t \le 10 \;\) are given in the following:

Fig. 9
figure 9

3D figures represent the imaginary part of (36) and the real part of (37)

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Fig. 10
figure 10

Contour surfaces represent the imaginary part of (36) and the real part of (37)

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Where the values of t are mentioned in the legend, one gets:

Fig. 11
figure 11

2D graphs represent the imaginary part of (36) and the real part of (37)

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Case 2 The following are the parameters that were obtained from solving (34):

$$\begin{aligned} \begin{aligned} \delta _1=\frac{i \sqrt{\delta _3}}{\sqrt{2} \sqrt{\beta _1} \sqrt{\mu }}; \lambda _1=0; \lambda _2=0; \lambda _0=\frac{i \root 4 \of {-2} \root 4 \of {\beta _1} \sqrt{\beta _3} \sqrt{\delta _2} \root 4 \of {\delta _3} \lambda _3 \mu ^{3/4}}{\sqrt{\beta _2} \sqrt{\beta _4}}. \end{aligned} \end{aligned}$$
(38)

The following set of solutions to (1) has been determined by re-installing (38) with (30) into (15).

$$\begin{aligned} \begin{aligned} u_5&= \frac{2 \root 4 \of {-2} \root 4 \of {\beta _1} \sqrt{\beta _3} \sqrt{\delta _2} \root 4 \of {\delta _3} \mu ^{3/4} e^{\frac{\sqrt{2} \sqrt{\delta _3} \sqrt{\mu } x}{\sqrt{\beta _1}}+2 i \delta _3 \mu t}}{\sqrt{\beta _2} \sqrt{\beta _4} \left( e^{\frac{\sqrt{2} \sqrt{\delta _3} \sqrt{\mu } x}{\sqrt{\beta _1}}+2 i \delta _3 \mu t} -e^{2 i \delta _2 \mu y}\right) }, \end{aligned} \end{aligned}$$
(39)

and

$$\begin{aligned} v_5 = \frac{\delta _3 \mu }{\beta _2} \csc ^2\left( \mu \left( -\delta _3 t+\frac{i \sqrt{\delta _3} x}{\sqrt{2} \sqrt{\beta _1} \sqrt{\mu }}+\delta _2 y\right) \right) . \end{aligned}$$
(40)

Profile of the solutions in (39) and (40) where \(\beta _1=\frac{8}{3};\beta _2=\frac{1}{8};\beta _3=-\frac{1}{4};\beta _4=\frac{2}{5};\mu =\frac{1}{2}; \delta _1=\frac{5}{2};\delta _2=-\frac{1}{2};\delta _3=\frac{5}{2};y=-\frac{3}{2}\) and \(\; - 20 \le x \le 20,\; - 20 \le t \le 20 \;\) are given below:

Fig. 12
figure 12

3D figures of (39)

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Fig. 13
figure 13

Contour surfaces of (39)

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For the values of t that are mentioned in the legend, one reaches:

Fig. 14
figure 14

2D graphs of (39)

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Fig. 15
figure 15

3D figures of (40)

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Fig. 16
figure 16

Contour surfaces of (40)

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For the values of t that are mentioned in the legend, one obtains:

Fig. 17
figure 17

2D graphs of (40)

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Remark 2

Similarly, by assuming that (7) is the trial solution to (15), some other set solutions to (1) may be obtained using the same prior process.

4 Conclusion

The present study describes the first implementation of two modified trigonometric analytic methods on a complex nonlinear (2 + 1)-dimensional Fokas system. The studied model is constructed to explain the nonlinear pulsed transmission in monomode fibers with optical features. Our novel modification approaches are the modified extended rational \(\sinh\)–\(\cosh\) method and the modified extended rational \(\sin\)–\(\cos\) method. The outcomes have been illustrated by numerous innovative and unique solutions that have been stated by traveling waves, oscillating, soliton types, and exponential rational functions blended with trigonometric and hyperbolic trigonometric functions. The updated approaches are trustworthy, influential, and straightforward in discovering semi-analytic solutions to mathematical models in numerous domains, such as mathematics, physics, biology, and engineering. The detected results have been detailed in three dimensions, contour surfaces, and two-dimensional graphs that represent the impact of temporal progression. The two- and three-dimensional displays help us better appreciate the qualities of the acquired outcomes. The obtained outcomes have all been properly validated by putting the created findings back into their linked equations. The functioning and behavior of the graphs mostly rely on the specified numerical values that are supplied for the optional coefficients. For the future scope of the work, we recommend that the authors use these two modifications, which we believe are useful, practical, and effective. It will play a significant role in forthcoming research related to applied science.

5 Results and discussion

The following statements have been added to clarify the distinguishing characteristics of our updating methods: We have acquired a collection of solutions that are difficult to get through the utilization of prior iterations of these techniques. The adjustments we have made are dependable, efficient, and swiftly adaptable to many mathematical models. Some shortcomings and unfavorable variables in the past versions of these procedures supplied the impetus for us to arrive at these two further enhancements. Although no analytical technique is devoid of drawbacks, positively, there are major benefits to our modifications for portraying the formulated solution in (22) and in (23) that are unreachable to acquire by employing the prior old versions. The singular breather solitons in both x and t are shown in Figs. 1, 2, 3, 4 and 5. Figure 6 represents a solitary wave on the left and a bright soliton on the right-hand side. Figures 7, 8, 9, 10 and 11 represent periodic and traveling wave solutions. The two interacting breather solitons are illustrated in Figs. 12, 13 and 14. The dark soliton on the right-hand side and the solitary waveform on the left-hand side can be observed in Figs. 15, 16 and 17.