Novel dynamics of the fractional KFG equation through the unified and unified solver schemes with stability and multistability analysis

1 Introduction

Nonlinear partial differential models play a vibrant role in detecting the conduct of physical science. During the last two decades, this field has enriched its branches more and more in physical science, namely mathematical physics [1], electromagnetism [2], magnetohydrodynamic [3,4], plasma science [5], solidary physics [6], laser science [7], etc.

The researchers of these fields recognized a vast number of nonlinear models, including the generalized KP model [8], chikungunya infection model [9], the Benjamin–Ono equation [10], the Hirota–Maccari equation [11], the Zoomeron equation [12], the Bogoyavlenskii system [13], the Biswas–Arshed system [14], fractional-calculus [15,16], and Kundu–Eckhaus model [17,18]. In 1995, Doktorov used Klein–Fock–Gordon (KFG) equations to analyze the integrability of this equation [19]. Subsequently, many authors investigated different models with this equation [20,21,22].

The nonlinear equations are solved by various effective methods, such as the WBBM model [23], generalized KP equation [24], new Kudryashov’s methods [25,26,27], the generalized CBS-BK model [28], (G′/G) expansion technique [29], (G′/G ²) expansion technique [30], lie group technique [31], enhanced Kudryashov scheme [32], he first integral method [33], unified technique [34], unified solver technique [35], etc. [36,37].

The nonlinear fractional KFG equation is an elegant expansion of the classical KFG equation, which integrates fractional calculus in field theory. This generalized equation can be applied in various fields, which include quantum mechanics, cosmology, and nonlinear optics [38]. One of the most important factors of this equation is it enhances the understanding level by offering insights into the behavior of fields with fractional dynamics where traditional models have some backlogs. This study explores the soliton outcomes of the fractional KFG model through the unified and unified solver methods with stability and multistability analysis for the conditions of various parameters.

The design of this article is as follows. Section 2 covers the translation of an ordinary differential equation (ODE) from the fractional KFG model. The solution processes are discussed through unified and unified solver methods in Sections 3 and 4. Subsequently, the graphical discussion of the outcomes is shown in Section 5. In Sections 6 and 7, the stability and multistability analyses with a graphical representation of the limitations of the existence of the parameters are explored for the suggested model. Moreover, the novelty of our work is discussed in Section 8. Finally, some conclusions are given in the last section.

The originality of this study is that the outcome from our suggested model through the unified and unified solver methods has not been explored by any other scholars before.

2 Fractional KFG model

The nonlinear fractional KFG time-dependent model [20,39,40] can be written as follows:

Here, c ₃, c ₂, and c ₁ are arbitrary parameters, and w(x, t) indicates a wave amplitude that relies on x, t. Eq. (1) is a nonlinear equation of a relativistic wave in energy-relevant physics. Here, the comfortable fractional derivative [41,42] is represented by D t 2 β and defined by

where f is a function whose domain is non-negative real and whose range is a real number, t > 0, and 0 < β ≤ 1. The consequent formula is found and constructed on the explanation of Eq. (2) as

Some other characteristics are discussed below for r (any constant) and s, m (real constants):

1. D t β t n = n t n − β ∀ n ∈ ℝ .

2. D t β ( r ) = 0 .

3. D t β s f ( t ) = s D t β f ( t ) .

4. D t β ( s f ( t ) + m g ( t ) ) = s D t β f ( t ) + cm g ( t ) .

5. D t β ( g ( t ) f ( t ) ) = g ( t ) D t β f ( t ) + f ( t ) D t β g ( t ) .

6. D t β f ( t ) g ( t ) = g ( t ) D t β f ( t ) − f ( t ) D t β g ( t ) g 2 ( t ) , g ( t ) ≠ 0 .

To convert Eq. (1) into an ODE form, consider the following relation:

Eq. (1) will be converted into the following ODE:

with arbitrary constants b, c ₃, c ₂, and c ₁. Here “′” denotes derivative with respect to χ.

3 Unified technique for the fractional KFG model

Assume a trial solution with an auxiliary equation (Eq. (1)) as follows [43,44]:

and

Eq. (6) gives the following solutions.

Group 1: Hyperbolic function (for s < 0):

Group 2: Trigonometric function (for s > 0):

Group 3: Rational function (for s = 0):

when A ≠ 0, s, E, and B are arbitrary constants.

By balancing R ³ and R″ and taking n = 1, we obtain Eq. (6):

Plugging Eqs. (11) and (7) into Eq. (5),

From Eqs. (4), (5), (8)–(11), and (12), we obtain

where χ = x − b t β β and b = ± 1 4 − 2 s ( 8 s c 1 + c 3 ) s .

From Eqs. (4), (5), (8)–(11) and (13), we obtain

where χ = x − b t β β and b = ± 1 2 − s ( 4 s c 1 − c 3 ) s .

From Eqs. (4), (5), (8)–(11) and (14), we obtain

where χ = x − b t β β and b = ± 1 2 − 2 s ( 2 s c 1 + c 3 ) s .

From Eqs. (4), (5), (8)–(11), and (15), we obtain

where χ = x − b t β β and b = ± 1 2 − 2 s ( 2 s c 1 + c 3 ) s .

4 Unified solver method for the proposed system

From the obtainable solver process in previous studies [45,46], the results for Eq. (5) are shown as follows:

Group 1: Rational function outcomes (for c ₃ = 0):

where χ = x − b t β β , and ξ and b are arbitrary constants.

Group 2: Trigonometric function outcomes (for c 3 b 2 + c 1 < 0 ):