Construction of new exact solutions of the resonant fractional NLS equation with the extended Fan sub-equation method

1 Introduction

In recent decades, it has been noticed that nonlinear phenomena have remarkable properties in physics and mathematical engineering. The phenomenon of nonlinear evaluation equations (NLEE) has received a lot of attention and has become one of the most interesting areas of research. Such type of models are widely used to elucidate many complex physical phenomena that occur in fluid and plasma wave mechanics, fiber optic communications, biophysics, Soliton’s theory, and many more (Kalim et al., 2018; Ghanbari et al., 2019a; Ghanbari et al., 2019b; Munusamy et al., 2020; Rezazadeh, 2018).

The area of nonlinear partial differential equations is becoming one of the most essential disciplines describing dynamics of key phenomena of nature and has attracted renowned researchers and scientists in the theory of optical fibre arising in telecommunications, spectroscopy, plasma physics and many more. Optical solitons are basically localized electromagnetic waves which can transmit large amount of information through optical fibers across the trans-oceanic distance in femto-seconds. In the modern era of science and technology, the theory of solitons has created revolutionary developments in the telecommunication engineering and is one of the most blistering field of research over the past few decades and reckon as the technology of future generation for high-speed communication systems (Ghanbari et al., 2020; Hosseini et al., 2020a; Hosseini et al., 2020b; Korpinar et al., 2019; Hashemi et al., 2019).

Many complex phenomena in diverse areas of nonlinear science are usually described by nonlinear models. One of the most famous of these nonlinear equations is the Schrödinger’s equation. This equation has been used in several domains among which: fiber optics, hydrodynamics, Plasma physics, nonlinear electrical transmission lines, and so on (Liu et al., 2019). However, there are a multitude of nonlinear equations such as the Ginzburg–Landau equation (Wazwaz, 2006), the Korteweg de Vries equation (Seadawy et al., 2020), the Zoomeron equation (Hosseini et al., 2020c), the Boussinesq equation (Mehdinejadiani and Parviz, 2020), to mention a few, that can also help to describe many systems. It’s worth mentioning that integer order derivatives are previously utilised to analyze these problems. However, in order to gain a better understanding of the dynamics of these nonlinear equations, a common type of derivative called fractional derivatives should be presented (Manafian and Lakestani, 2017; Miller and Ross, 1993).

This concept is a generalization of derivatives. Thus, we observe that the fractional NLEE is a natural extension of a NLEE of integeral order. These fractional differential equations are generally not easy to solve and methods that have been proposed in the literature to transform them are used. Among these methods, we can list the Caputo fractional derivatives, the Riemann–Liouville fractional derivatives, modified Riemann–Liouville derivative, conformable fractional derivative. The enactment of such methods is just a beginning for a revolution of recent era towards fractional calculus. To date, there are several methods in the literature that can be implemented to deal with the nonlinear models arising in diverse disciplines (Goswami et al., 2020; Singh et al., 2021a; Singh et al., 2021b; Singh et al., 2021c).

The outline of this study is given as follows: in Section 2, a brief introduction to the R-FNLSE with quadratic-cubic nonlinearity is illustrated whereas in Section 3 some noval anlytical solutions to the nonlinear complex model (1) are established. The geometrical behavior of the solutions is demonstrated in Section 4. At the end, the conclusions have been extracted.

2 The R-FNLSE with quadratic-cubic nonlinearity

Over the last few decades, the study solitonic and optical behaviour of nonlinear models is becoming one of the most excited topics in the diverse disciplines of contemporary science. Our prospective is applicable until the given model imply more even odd order partial derivative terms (Seadawy and Lu, 2017; Younis et al., 2020).

Over the last couple of years, many computational analysis have been established effectively to describe the soliton solutions of various type of nonlinear Schrödinger equation (NLSE) among these are, the extended sinh-Gordon equation expansion method (Baskonus et al., 2018), extended Jocobi’s elliptic approach (Hong and Lu, 2009), the modified auxiliary equation mapping method (Seadawy et al., 2020), the inverse scattering transformation method (Zhang and Chen, 2019), extended rational sine–cosine method (Mahak and Akram, 2019), and many more.

The purpose of the study is to investigate the NLSE of fractional order (Bhrawy et al., 2014) with quadratic-cubic nonlinearity, as well as perturbation terms and higher order dispersions (third and fourth order dispersions). The R-FNLSE with quadratic-cubic nonlinearity is studied by using extended Fan sub-equation (EFSE) approach.

3 Applications to the EFSE method

The transformation

The solution for Eq. (4) is of the fashion (Kalim et al., 2021; El-Wakil and Abdou, 2008)

Inserting Eq. (6) along Eq. (7) in Eq. (4) and picking the coefficients of ${\varphi }^j{\varphi }^{(k)}$ , $$\begin{array}{c}{\mathfrak{a}}_0{\gamma }_1+{\mathfrak{a}}_0^3\left(-\left({\varsigma }_3-\kappa \varrho \right)\right)-{\mathfrak{a}}_0^2{\varsigma }_2-2{\mathfrak{a}}_2{\gamma }_2{\ell }_0-\frac{1}{2}{\mathfrak{a}}_1{\gamma }_2{\ell }_1\hfill \\ +\frac{3}{2}{\mathfrak{a}}_2{\ell }_1^2\sigma +8{\mathfrak{a}}_2{\ell }_0{\ell }_2\sigma +\frac{1}{2}{\mathfrak{a}}_1{\ell }_1{\ell }_2\sigma +3{\mathfrak{a}}_1{\ell }_0{\ell }_3\sigma =0,\hfill \\ {\mathfrak{a}}_1{\gamma }_1-3{\mathfrak{a}}_1{\mathfrak{a}}_0^2\left({\varsigma }_3-\kappa \varrho \right)-2{\mathfrak{a}}_1{\mathfrak{a}}_0{\varsigma }_2-3{\mathfrak{a}}_2{\gamma }_2{\ell }_1-{\mathfrak{a}}_1{\gamma }_2{\ell }_2\hfill \\ +{\mathfrak{a}}_1{\ell }_2^2\sigma +15{\mathfrak{a}}_2{\ell }_1{\ell }_2\sigma +30{\mathfrak{a}}_2{\ell }_0{\ell }_3\sigma +\frac{9}{2}{\mathfrak{a}}_1{\ell }_1{\ell }_3\sigma +12{\mathfrak{a}}_1{\ell }_0{\ell }_4\sigma =0,\hfill \\ {\mathfrak{a}}_2{\gamma }_1-3{\mathfrak{a}}_2{\mathfrak{a}}_0^2\left({\varsigma }_3-\kappa \varrho \right)-3{\mathfrak{a}}_1^2{\mathfrak{a}}_0\left({\varsigma }_3-\kappa \varrho \right)-2{\mathfrak{a}}_2{\mathfrak{a}}_0{\varsigma }_2\hfill \\ -{\mathfrak{a}}_1^2{\varsigma }_2-4{\mathfrak{a}}_2{\gamma }_2{\ell }_2-\frac{3}{2}{\mathfrak{a}}_1{\gamma }_2{\ell }_3+16{\mathfrak{a}}_2{\ell }_2^2\sigma +42{\mathfrak{a}}_2{\ell }_1{\ell }_3\sigma \hfill \\ +\frac{15}{2}{\mathfrak{a}}_1{\ell }_2{\ell }_3\sigma +72{\mathfrak{a}}_2{\ell }_0{\ell }_4\sigma +15{\mathfrak{a}}_1{\ell }_1{\ell }_4\sigma =0,\hfill \\ \hfill a_1^3\left(-\left({\varsigma }_3-\kappa \varrho \right)\right)-6{\mathfrak{a}}_0{\mathfrak{a}}_2{\mathfrak{a}}_1\left({\varsigma }_3-\kappa \varrho \right)-2{\mathfrak{a}}_2{\mathfrak{a}}_1{\varsigma }_2-2{\mathfrak{a}}_1{\gamma }_2{\ell }_4& \hfill \\ \hfill -5a_2{\gamma }_2{\ell }_3+\frac{15}{2}{\mathfrak{a}}_1{\ell }_3^2\sigma +20{\mathfrak{a}}_1{\ell }_2{\ell }_4\sigma +65{\mathfrak{a}}_2{\ell }_2{\ell }_3\sigma +90{\mathfrak{a}}_2{\ell }_1{\ell }_4\sigma & =0,\hfill \\ -3{\mathfrak{a}}_2{\mathfrak{a}}_1^2\left({\varsigma }_3-\kappa \varrho \right)-3{\mathfrak{a}}_0{\mathfrak{a}}_2^2\left({\varsigma }_3-\kappa \varrho \right)-{\mathfrak{a}}_2^2{\ell }_2-6{\mathfrak{a}}_2{\gamma }_2{\ell }_4\hfill \\ +30{\mathfrak{a}}_1{\ell }_3{\ell }_4\sigma +\frac{105}{2}{\mathfrak{a}}_2{\ell }_3^2\sigma +120{\mathfrak{a}}_2{\ell }_2{\ell }_4\sigma =0,\hfill \\ -3{\mathfrak{a}}_1{\mathfrak{a}}_2^2\left({\varsigma }_3-\kappa \varrho \right)+168{\mathfrak{a}}_2{\ell }_3{\ell }_4\sigma +24{\mathfrak{a}}_1{\ell }_4^2\sigma =0,\hfill \\ 120{\mathfrak{a}}_2{\ell }_4^2\sigma -{\mathfrak{a}}_2^3\left({\varsigma }_3-\kappa \varrho \right)=0\text{.}\hfill \end{array}$$

We select variables suitably which gives

We have soliton solutions

Case I

If ${\ell }_0={\vartheta }_3^2,{\ell }_1=2{\vartheta }_1{\vartheta }_3,{\ell }_2=2{\vartheta }_2{\vartheta }_3+{\vartheta }_1^2,{\ell }_3=2{\vartheta }_1{\vartheta }_2,{\ell }_4={\vartheta }_2^2$ , there may exist parameters ${\vartheta }_1,{\vartheta }_2$ , satisfy ${\vartheta }_3$ , the solutions of (1) are ${\varphi }_{\eta }^I,(\eta =1,2,\dots ,24)$ .

Type I.

${\vartheta }_1^2-4{\vartheta }_2{\vartheta }_3>0$ ,   ${\vartheta }_1{\vartheta }_2\ne 0$ ,   ${\vartheta }_2{\vartheta }_3\ne 0$ . We obtain the dark optical solitons in the form as follows

We obtain the combined bright-dark optical soliton in the form as follows

We obtain the combined dark-singular optical solitons in the form as follows

Type II.

${\vartheta }_1^2-4{\vartheta }_2{\vartheta }_3<0$ ,   ${\vartheta }_1{\vartheta }_2\ne 0$ ,   ${\vartheta }_2{\vartheta }_3\ne 0$ . The collection of periodic solitons are obtained

If ${\ell }_0={\vartheta }_3^2,{\ell }_1=2{\vartheta }_1{\vartheta }_3,{\ell }_2=0,{\ell }_3=2{\vartheta }_1{\vartheta }_2,{\ell }_4={\vartheta }_2^2,\varphi$ is one of the ${\varphi }_{\eta }^{\mathit{II}},(\eta =1,2,\dots ,12)$ . A collection of dark optical soliton is observed

${\ell }_0={\ell }_1=0$ , we have the following solution of (1) in the form ${\varphi }_{\eta }^{\mathit{III}},(\eta =1,2,\dots ,10)$ .

Type I.

${\ell }_2=1,{\ell }_3=\frac{-2{\varrho }_3}{{\varrho }_1},{\ell }_4=\frac{{\varrho }_3^2-{\varrho }_2^2}{{\varrho }_1^2}$ , where ${\varrho }_1,{\varrho }_2,{\varrho }_3$ are arbitrary constants.

${\ell }_2=1,{\ell }_3=\frac{-2{\varrho }_3}{{\varrho }_1},{\ell }_4=\frac{{\varrho }_3^2+{\varrho }_2^2}{{\varrho }_1^2}$ , where ${\varrho }_1,{\varrho }_2,{\varrho }_3$ are arbitrary constants.

Eqs. (21), (22) give a set of bright and singular optical solitons for ${\varrho }_2=0$

${\ell }_2=4,{\ell }_3=-\frac{4\left(2{\varrho }_2+{\varrho }_4\right)}{{\varrho }_1},{\ell }_4=\frac{4{\varrho }_2^2+4{\varrho }_4{\varrho }_2+{\varrho }_3^2}{{\varrho }_1^2}$ , where ${\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4$ are arbitrary constants.

${\ell }_2=4,{\ell }_3=\frac{4\left({\varrho }_4-2{\varrho }_2\right)}{{\varrho }_1},{\ell }_4=\frac{4{\varrho }_2^2-4{\varrho }_4{\varrho }_2+{\varrho }_3^2}{{\varrho }_1^2}$ , where ${\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4$ are arbitrary constants.

Another class of dark and singular optical solitons is obtained for ${\varrho }_2={\varrho }_4$

${\ell }_2=-1,{\ell }_3=\frac{2{\varrho }_3}{{\varrho }_1},{\ell }_4=\frac{{\varrho }_3^2-{\varrho }_2^2}{{\varrho }_1^2}$ , where ${\varrho }_1,{\varrho }_2,{\varrho }_3$ are arbitrary constants.

${\ell }_2=4,{\ell }_3=\frac{-2{\varrho }_3}{{\varrho }_1},{\ell }_4=\frac{{\varrho }_3^2-{\varrho }_2^2}{{\varrho }_1^2}$ , where ${\varrho }_1,{\varrho }_2,{\varrho }_3$ are arbitrary constants.

${\ell }_2=-4,{\ell }_3=\frac{4\left(2{\varrho }_2+{\varrho }_4\right)}{{\varrho }_1},{\ell }_4=-\frac{4{\varrho }_2^2+4{\varrho }_4{\varrho }_2-{\varrho }_3^2}{{\varrho }_1^2}$ , where ${\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4$ are arbitrary constants.

${\ell }_1={\ell }_3=0$ , Eq. (1) have solutions of the form ${\varphi }_{\eta }^{\mathit{IV}},(\eta =1,2,\dots ,16)$ .

Type I

${\ell }_0=\frac{1}{4},{\ell }_2=\frac{1-2m^2}{2},{\ell }_4=\frac{1}{4}$ ,

${\ell }_0=\frac{1}{4},{\ell }_2=\frac{1-2m^2}{2},{\ell }_4=\frac{1}{4}$ ,

4 Discussion and results

To display a set of novel travelling wave solutions to the Eq. (1), Mathematica 11.0 is executed to demonstrate the behaviour of bit flow and to understand the complex structure of solitons for a set of limitations. The parameter $\alpha \in (0,1]$ is the core constraint which simulate the flow rate propagation, plays a key rote in telecommunications and the theory of optical fibres.

Fig. 1 illustrates the solution $|{\varphi }_1^I(x,t)|$ constructed in Case I (Type I) for $\kappa$  = −2, $\omega$  = 0.1, $\nu$  = 0.1, $\alpha$  = 0.5, $\sigma$  = 2, $\varrho$  = 2, $\theta$  = 2, $\gamma$  = 1, ${\varsigma }_1$  = 1, ${\varsigma }_2$  = 2, ${\varsigma }_3$  = 3, ${\vartheta }_1$  = 5, ${\vartheta }_2$  = 1, ${\vartheta }_3$  = 2, while Fig. 2 illustrates the solution $|{\varphi }_3^I(x,t)|$ constructed in Case I (Type I) for $\kappa$  = −2, $\omega$  = 0.1, $\nu$  = −0.5, $\alpha$  = 1, $\sigma$  = 1, $\varrho$  = 2, $\theta$  = 2, $\gamma$  = 1, ${\varsigma }_1$  = 1, ${\varsigma }_2$  = 2, ${\varsigma }_3$  = 3, ${\vartheta }_1$  = 3, ${\vartheta }_2$  = 1, ${\vartheta }_3$  = 1, whereas Fig. 3 visualizes the solution $|{\varphi }_{10}^I(x,t)|$ produced in Case I (Type I) for $\kappa$  = −2, $\omega$  = 0.1, $\nu$  = 0.1, $\alpha$  = 0.5, $\sigma$  = 2, $\varrho$  = 2, $\theta$  = 2, $\gamma$  = 1, ${\varsigma }_1$  = 1, ${\varsigma }_2$  = 2, ${\varsigma }_3$  = 3, ${\vartheta }_1$  = 5, ${\vartheta }_2$  = 3, ${\vartheta }_3$  = 2. Similarly, Fig. 4 illustrates the solution $|{\varphi }_{11}^I(x,t)|$ constructed in Case I (Type I) for $\kappa$  = −2, $\omega$  = 0.1, $\nu$  = 0.1, $\alpha$  = 0.5, $\sigma$  = 2, $\varrho$  = 2, $\theta$  = 2, $\gamma$  = 2, ${\varsigma }_1$  = 1, ${\varsigma }_2$  = 2, ${\varsigma }_3$  = 3, ${\vartheta }_1$  = 3, ${\vartheta }_2$  = 1, ${\vartheta }_3$  = −1, while Fig. 5 visualizes the solution $|{\varphi }_{17}^I(x,t)|$ constructed in Case I (Type II) for $\kappa$  = 0.1, $\omega$  = 1, $\nu$  = 0.1, $\alpha$  = 1, $\sigma$  = 2, $\varrho$  = 2, $\theta$  = 2, $\gamma$  = 2, ${\varsigma }_1$  = 1, ${\varsigma }_2$  = 2, ${\varsigma }_3$  = 1, ${\vartheta }_1$  = 1, ${\vartheta }_2$  = 1, ${\vartheta }_3$  = 1.