Analytical technique and diverse chaos-identifying tools for the time fractional symmetric regularized long wave equation

Introduction

Recently, nonlinear evolution equations have become a significant tool for modeling diverse real-life phenomena in the areas of contemporary science, including fluid dynamics¹, coastal engineering², nonlinear optics³, plasma physics⁴, and others. There have been multiple mathematical models introduced to explain these intricate systems, including the Yajima–Oikawa model⁵, the Oskolkov Eq⁶., the Kudryashov-Sinelshchikov Eq⁷., the Jaulent–Miodek hierarchy Eq⁸., and many others. To discover the soliton dynamics and novel characteristics of these nonlinear problems, researchers have developed multiple techniques including the modified simple equation technique⁹, the Wang’s direct mapping method¹⁰, ansatz approach¹¹, the extended auxiliary equation technique¹², $$\text{tan}\left(F\left(\frac{\xi }{2}\right)\right)$$-expansion method¹³, modified generalized exponential rational function method¹⁴, the extension exponential rational function process¹⁵, unified solver method¹⁶, and many others.

Fractional calculus is at the forefront of ground-breaking research and is becoming increasingly widespread in various physical issues in nonlinear fields. Numerous researchers have also come up with fractional approaches, such as the Caputo-Fabrizio derivative¹⁷, the modified Riemann-Liouville derivative¹⁸, the local fractional derivative¹⁹, the truncated M-fractional derivative²⁰, the conformable fractional derivative²¹, and more.

The conformable fractional derivative is a forthright definition of the fractional derivative. A conformable fractional derivative is less complex than using a fractional derivative. The conformable fractional derivative obeys certain standard principles that other fractional derivatives do not satisfy, including the chain rule. The sole limitation is that the conformable fractional derivative of every differentiable function at zero is equal to zero. To overcome this problem, a suitable fractional derivative is suggested in^22,23.

In this work, we utilize $$\left(\frac{G^{\prime }}{G^2}\right)$$-expansion technique²⁴ to derive the exact solutions of the conformable time-fractional nonlinear SRLW problem. This research also uses strange attractors, recurrence plots, return maps, Lyapunov exponents, power spectrum plots, 3D and 2D phase diagrams, and fractal dimensions to identify and inspect the chaos of this nonlinear problem. To the best of the author’s knowledge, there has never been an investigation into the relationship between these types of studies and the analysis of this nonlinear model.

where $$p=p(x,t)$$ is the amplitude portion of solitons, $$u$$, $$v$$ and $$w$$ real parameters, $$D_{tt}^{2\beta }$$ is the differential operator with fractions of order $$2\beta$$ in conformable perception. With its modest nonlinearity, this equation offers an intriguing model for electrostatics and charge density fluctuations.

This article is structured into 12 sections, where Sect. 1 contains the introduction with the graphical abstract (see Fig. 1) of this work. Section 2 describes the conformable fractional derivatives. Section 3 contains the $$\left(\frac{G^{\prime }}{G^2}\right)$$expansion algorithm, and Sect. 4 comprises its applications. Section 5 provides the graphical representations of the solutions. Section 6 corresponds to the stability analysis of the obtained solution. Section 7 delivers the chaotic analysis by Lyapunov exponents and other tools. Section 8 includes multi-stability analysis. Section 9 explains the uniqueness of our approach and the obtained results. Section 10 covers the future-oriented studies. Finally, Sect. 11 is for the conclusion.

The conformable fractional derivative’s preface

where the function $$p=p\left(t\right):[0,\mathrm{\infty })\to \mathbb{R}$$ exist.

The $$\left(\frac{{\text{varvec}G}^{\prime }}{{\text{varvec}G}^2}\right)$$-expansion algorithm

where $$L$$ is a polynomial in $$p$$, incorporating numerous partial derivatives, employing nonlinear elements and the higher-order derivatives, and $$p(x,t)$$ is an unidentified function of the independent variables $$x$$, $$t$$. The fundamental processes for obtaining exact solutions to NPDEs are as follows.

Application of $$\left(\frac{\text{varvec}G\prime }{{\text{varvec}G}^2}\right)$$ algorithm

where $$q$$ and $$a$$ are the wave velocity and the wave height, respectively.

We find the following six solution sets when we use Maple to solve the obtained algebraic system (15).

Graphical representation

Figure 2 illustrates that $$Re\left(p_{11}\right)$$ displays local breathers with a kink wave, while $$Im\left(p_{11}\right)$$ displays breather wave, and $$\left|p_{11}\right|$$ denotes local breathers with dark soliton, using the values $$\alpha =1$$, $$q=1$$, $$u=1$$,$$a=2$$, $$w=-1$$,$$v=2.5$$,$$\beta =0.5$$, $$f=1$$, and $$g=1$$. Additionally, the patterns are the same for $$p_{21}$$, $$p_{31}$$, $$p_{41}$$, $$p_{43}$$, $$p_{51}$$, $$p_{53}$$, $$p_{61}$$, and $$p_{63}$$.

Figure 3 displays local breathers with lump waves. Using the values $$\alpha =1$$, $$q=1$$, $$u=1$$, $$a=1$$, $$w=-1$$, $$v=1$$, $$\beta =0.5$$, $$f=1$$, and $$g=1$$. Additionally, the patterns are the same for $$p_{12}$$.

Figure 4 illustrates that $$Re\left(p_{33}\right)$$ displays kinky breathers, while $$Im\left(p_{33}\right)$$ and $$\left|p_{33}\right|$$ display double kink and local breather wave, respectively, using the values $$\alpha =0$$, $$q=1$$, $$u=-1$$, $$a=1$$, $$w=-1$$, $$v=1$$, $$\beta =0.5$$, $$f=1$$, and $$g=1$$. Additionally, the patterns are the same for $$p_{32}$$, $$p_{42}$$, $$p_{52}$$, and $$p_{62}$$.

Figure 5 illustrates that $$Re\left(p_{52}\right)$$ displays a local breather with a dark soliton, while $$Im\left(p_{52}\right)$$ and $$\left|p_{52}\right|$$ displays local breather with dark & bright soliton and breather wave, respectively, using the values $$\alpha =1$$, $$q=1$$, $$u=1$$, $$a=1$$, $$w=4$$, $$v=1$$, $$\beta =0.5$$, $$f=1$$, and $$g=1$$. Additionally, the patterns are the same for $$p_{32}$$, $$p_{42}$$, $$p_{52}$$, and $$p_{62}$$.

Stability analysis of the solution