Dynamical wave structures for some diffusion--reaction equations with quadratic and quartic nonlinearities

Nauman Ahmed; Jorge E. Macías-Díaz; Makhdoom Ali; Muhammad Jawaz; Muhammad Z. Baber; María G. Medina-Guevara

doi:10.1515/phys-2024-0115

Article Open Access

Dynamical wave structures for some diffusion--reaction equations with quadratic and quartic nonlinearities

Nauman Ahmed , Jorge E. Macías-Díaz , Makhdoom Ali , Muhammad Jawaz , Muhammad Z. Baber and María G. Medina-Guevara

Published/Copyright: February 14, 2025

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From the journal Open Physics Volume 23 Issue 1

Abstract

This work investigates the quadratic and quartic nonlinear diffusion–reaction equations with nonlinear convective flux terms, which are investigated analytically. Diffusion–reaction equations have a wide range of applications in several scientific areas, such as chemistry, biology, and population dynamics of the species. The new extended direct algebraic method is applied to obtain abundant families of solitary wave solutions. Different types of solitary wave solutions are obtained by applying this analytical method. This approach provides the solutions in the form of single and combined wave structures, which are observed in shock, complex solitary-shock, shock-singular, and periodic-singular forms. Some of the solutions are depicted graphically to illustrate the fact that they are, indeed, wave solutions of the mathematical model.

Keywords: extended direct algebraic method; quadratic and quartic nonlinearities; soliton solutions; traveling-wave solutions

1 Introduction

In the physical sciences, differential equations (DEs) are essential to describe some physical phenomena. Many important evolutionary occurrences are often modeled by nonlinear partial differential equations (NPDEs). Most physical phenomena in various fields, such as field theory, optical fiber [1,2], mathematical physics, plasma physics [3], biophysics, fluid dynamics, mechanics, aerospace industry [4], chemistry reactions [5–7], metrology, and many other fields [8–10] are described by using the NPDEs. For instance, the nonlinear Schrödinger equation (NLSE) has numerous applications in scientific fields such as nonlinear optics, mathematical finance, plasma physics, nuclear and solid-state physics, biochemistry, superconductivity, and matter.

As most of the physical occurrences are modeled by NPDEs. It is essential to find the solutions to NPDEs. Finding the solutions to the NPDEs is crucial to having a thorough understanding of the phenomenon. To obtain the exact solution, a range of analytical techniques have been developed and reported in the literature. The Kudryashov method [11], the homogenous balance method [12,13], the generalized Kudryashov method [14,15], the tanh–coth method [16,17], the direct algebraic method, extended modified auxiliary equation mapping method, the unified method, the modified and extended rational expansion method, the extended ( G ′ G ) -expansion technique, the variational principle method [18,19], the amplitude ansatz method [20], the bilinear neural network method [21], the auxiliary equation method [22], the improved P-expansion approach [23], the improved modified extended tanh-function method [24], the generalized algebraic method, the Jacobian elliptic functions technique [25], the improved exp ( − F ( η ) ) -expansion method [26], the Riccati equation method [27], the modified ( G ′ G ) -expansion method [28] among others, have been developed to this end.

It is worth pointing out that Yan and Lou worked on the soliton molecules of Sharma–Tasso–Olver–Burgers equation [29], Wu et al. investigated the nonlinear von Karman equations for the three-layer microplates [30]. Meanwhile, Wang et al. considered the diffusively delay-coupled memristive Chialvo neuron map for the network patterns [31], while Dai et al. worked on the macrodispersivity models with an analytical solution [32]. Kai and Yin worked on the Gaussian traveling wave solution to a special kind of Schrödinger equation [33], Zhu et al. used the logarithmic transformation for the modified Schrödinger’s equation [34] and nonlinear Zakharov system [35] to obtain analytical solutions. Berkal and Almatrafi worked on the bifurcation and stability of a two-dimensional activator–inhibitor model [36], and some researchers have worked on the approximate solutions of reaction–diffusion models [37–40]. in the present study, we will derive exact solutions for a nonlinear reaction–diffusion system.

Our current research focuses on nonlinear diffusion–reaction (DR) equations, which have wide applications in biology, chemical, physical, and logical systems. Various reduced versions of the DR equations have been investigated in the literature. In the study by Triki et al. [41], the auxiliary equation method was used to study three nonlinear DR equations in inhomogeneous mediums, including derivative-type and algebraic-type nonlinearities. Furthermore, nonlinear reaction diffusion equations with cubic and quantic nonlinearities, nonlinear DR equations including quadratic-cubic nonlinearities, and nonlinear diffusion–reaction equations with a nonlinear convective flux term have been investigated by Bhardwaj et al. [42], Malik et al. [43,44]. We examine the DR equation for a case where the diffusion coefficient D is unaffected by density.

In the present work, we investigate the dynamics through the solutions of particular nonlinear DR equations, which include a nonlinear convective flux term along with quadratic and biquadratic nonlinearities and are represented by the following equations:

(1) φ t + k φ φ x = D φ x x + α φ − β φ 2 ,

(2) φ t + k φ 2 φ x = D φ x x + α φ − β φ 4 ,

where α , β , and k are physical constants that need to be determined, D is the diffusion coefficient, and φ = φ ( x , t ) has different interpretations according to the phenomenon under investigation. These equations describe transport phenomena where diffusion and convection processes are equally important and nonlinear diffusion is assumed to be similar to nonlinear convection effects. The ( G ′ G ) -expansion method and the Kudryashov method have been recently applied to solve this system of equations.

The purpose of this study is to improve the accuracy of possible soliton solutions of the DR equations through a new methodology. Being a novel analytical technique, the new extended direct algebraic method (NEDAM) has not been applied to solve nonlinear DR equations with a nonlinear convective flux term. In domains including engineering, physics, and applied mathematics, NEDAM has demonstrated its reliability and effectiveness in resolving NPDEs. For example, Vahidi [45], Mirhosseini-Alizamini et al. [46], and Munawar et al. [47] have used NEDAM to solve different types of NPDEs. Furthermore, Kurt et al. used NEDAM to find traveling and solitary wave solutions of the potential Kadomtsev–Petviashvili equation. The author found that the NEDAM is accurate, effective, and applicable for solving problems. There are many other methods that provided us with hyperbolic, trigonometric, and rational function solutions. As we will see, single and combined wave structures are observed in shock, complex solitary-shock, shock-singular, and periodic-singular forms. Rational solutions also emerged during the derivation.

The present work is organized as follows. Section 2 is devoted to provide a brief summary of the NEDAM. To that end, we employ [48] as a guiding reference. The methodology is explained in its most general setting. We provide therein a set of steps that lead to the derivation of exact traveling-wave solutions of general systems of partial DEs in two variables, namely, space and time. Moreover, various cases are fully described to reach exact solutions for those models. In turn, Section 3 is devoted to derive the exact traveling-wave solutions for the mathematical models (1) and (2). From our discussion, it can be readily checked the existence of abundant solutions of this form for our mathematical models. For the sake of convenience, we illustrate the behavior of just some of those solutions by means of three-dimensional plots, contour plots, and two-dimensional plots. As we can see from the figures, all of the solutions obtained through the NEDAM are traveling waves. Finally, we close this work with a section of concluding remarks.

2 Method

In this section, we provide an outline of the NEDAM [48]. This method consists of the next steps.

Step 1. We suppose that a NLPDE can be expressed as follows:

(3) H ( M , M t , M x , M t t , M x t , M x x , … ) = 0 ,

where M is an unknown function and H is a polynomial of M ( x , t ) .

Step 2. Consider the transformation M ( x , t ) = ψ ( ζ ) , where ζ = x − w t . Here, w indicates the wave velocity. By substituting this transformation into (3), we obtain

(4) ψ ( φ , φ ′ , φ ″ , φ ‴ , … ) = 0 ,

where φ = φ ( ζ ) , φ ′ = d φ d ζ , φ ″ = d 2 φ d ζ 2 , etc.

Step 3. We assume that (4) has a solution of the form

(5) φ ( ζ ) = ∑ i = 0 N b i Ω i ( ζ ) ,

where b i ( 0 ≤ i ≤ N ) are unknown constants, b N ≠ 0 , and Ω ( ζ ) satisfies the auxiliary ordinary differential equation (ODE):

(6) Ω ′ ( ζ ) = ln F ( m + n Ω ( ζ ) + ν Ω ( ζ ) 2 ) , F ≠ 0 , 1 ,

where m , n , and ν are constants. The solution to Eq. (6) can be expressed as follows:

If n 2 − 4 m ν < 0 and ν ≠ 0 , then we have
(7) Ω 1 ( ζ ) = − n 2 ν + Φ 1 tan F − ( n 2 − 4 m ν ) 2 ζ ,

(8) Ω 2 ( ζ ) = − n 2 ν + Φ 1 cot F − ( n 2 − 4 m ν ) 2 ζ ,

(9) Ω 3 ( ζ ) = − n 2 ν + Φ 1 tan F ( − ( n 2 − 4 m ν ) ) ± p q sec F ( − n 2 − 4 m ν ζ ) ,

(10) Ω 4 ( ζ ) = − n 2 ν + Φ 1 cot F ( − ( n 2 − 4 m ν ) ) ± p q csc F ( − n 2 − 4 m ν ζ ) ,

(11) Ω 5 ( ζ ) = − n 2 ν + Φ 1 2 tan F − ( n 2 − 4 m ν ) 4 − cot F − n 2 − 4 m ν 4 ζ ,
where
(12) Φ 1 = − ( n 2 − 4 m ν ) 2 ν .

If n 2 − 4 m ν > 0 and ν ≠ 0 , then
(13) Ω 6 ( ζ ) = − n 2 ν − Φ 2 tanh F n 2 − 4 m ν 2 ζ ,

(14) Ω 7 ( ζ ) = − n 2 ν − Φ 2 coth F n 2 − 4 m ν 2 ζ ,

(15) Ω 8 ( ζ ) = − n 2 ν − Φ 2 tanh F ( n 2 − 4 m ν ) ± ι p q sech F ( n 2 − 4 m ν ζ ) ,

(16) Ω 9 ( ζ ) = − n 2 ν − Φ 1 coth F ( − ( n 2 − 4 m ν ) ) ± p q csch F ( n 2 − 4 m ν ζ ) ,

(17) Ω 10 ( ζ ) = − n 2 ν + Φ 1 2 tanh F − ( n 2 − 4 m ν ) 4 − coth F − n 2 − 4 m ν 4 ζ ,
where
(18) Φ 2 = n 2 − 4 m ν 2 ν .

If m ν > 0 and n = 0 , then
(19) Ω 11 ( ζ ) = m ν tan F ( m ν ζ ) ,

(20) Ω 12 ( ζ ) = − m ν cot F ( m ν ζ ) ,

(21) Ω 13 ( ζ ) = m ν ( tan F ( 2 m ν ζ ) ± p q sec F ( 2 m ν ζ ) ) ,

(22) Ω 14 ( ζ ) = m ν ( − cot F ( 2 m ν ζ ) ± p q csc F ( 2 m ν ζ ) ) ,

(23) Ω 15 ( ζ ) = 1 2 m ν tan F m ν 2 ζ − cot F m ν 2 ζ .

If m ν < 0 and n = 0 , then
(24) Ω 16 ( ζ ) = − − m ν tanh F ( − m ν ζ ) ,

(25) Ω 17 ( ζ ) = − − m ν coth F ( − m ν ζ ) ,

(26) Ω 18 ( ζ ) = − m ν ( − tanh F ( 2 − m ν ζ ) ± ι m ν sech F ( 2 − m ν ) ) ,

(27) Ω 20 ( ζ ) = − 1 2 − m ν tan F − m ν 2 ζ + cot F − m ν 2 ζ .

If n = 0 and ν = m , then
(28) Ω 21 ( ζ ) = tan F ( m ζ ) ,

(29) Ω 22 ( ζ ) = − cot F ( m ζ ) ,

(30) Ω 23 ( ζ ) = tan F ( 2 m ζ ) ± p q sec F ( 2 m ξ ) ,

(31) Ω 24 ( ζ ) = − cot F ( 2 m ζ ) ± p q csc F ( 2 m ζ ) ,

(32) Ω 25 ( ζ ) = 1 2 tan F m 2 ζ − cot F m 2 ζ .

If n = 0 and ν = − m , then
(33) Ω 26 ( ζ ) = − tan F ( m ζ ) ,

(34) Ω 27 ( ζ ) = − cot F ( m ζ ) ,

(35) Ω 28 ( ζ ) = − tanh F ( 2 m ζ ) ± ι p q sec F ( 2 m ζ ) ,

(36) Ω 29 ( ζ ) = − coth F ( 2 m ζ ) ± p q csch F ( 2 m ζ ) ,

(37) Ω 30 ( ζ ) = − 1 2 tanh F m 2 ζ + coth F m 2 ζ .

If n 2 = 4 m ν , then
(38) Ω 31 ( ζ ) = − 2 m ( n ζ ln F + 2 ) n 2 ζ ln F .
If n = χ , m = r χ with ( r ≠ 0 ) , and ν = 0 , then Ω 32 ( ζ ) = F χ ζ − r .
If n = ν = 0 , then Ω 33 ( ξ ) = μ ξ ln F .
If n = m = 0 , then
(39) Ω 34 ( ζ ) = − 1 ν ζ ln F .
If = m = 0 and ν ≠ 0 , then
(40) Ω 35 ( ζ ) = − p n ν ( cosh F ( n ζ ) − sinh F ( n ζ ) + p ) ,

(41) Ω 36 ( ζ ) = − n ( sinh F ( n ζ ) + c o s h F ( n ζ ) ) ν ( sinh F ( n ζ ) + cosh F ( n ζ ) + q ) .
If n = χ , ν = r χ ( r ≠ 0 ) and m = 0 , then
(42) Ω 37 ( ζ ) = p F χ ζ p − r q I χ ζ .

It is worth recalling that the generalized hyperbolic and triangular functions are defined as follows:

(43) sinh F ( ζ ) = p F ζ − q F − ζ 2 , cosh F ( ζ ) = p F ζ + q F − ζ 2 ,

(44) tanh F ( ζ ) = p F ζ − q F − ζ p F ζ + q F − ζ , coth F ( ζ ) = p F ζ + q F − ζ p F ζ − q F − ζ ,

(45) sech F ( ζ ) = 2 p F ζ + q F − ζ , csch F ( ζ ) = 2 p F ζ − q F − ζ ,

(46) sin F ( ζ ) = p F ι ζ − q F − ι ζ 2 , cos F ( ζ ) = p F ι ζ + q F − ι ζ 2 ,

(47) tan F ( ζ ) = − ι p F ζ − q F − ι ζ p F ζ + q F − ι ζ , cot F ( ζ ) = − ι p F ζ + q F − ι ζ p F ζ − q F − ι ζ ,

(48) sec F ( ζ ) = 2 p F ι ζ + q F − ι ζ , csc F ( ζ ) = 2 ι p F ι ζ − q F − ι ζ ,

where ζ is an independent variable and p , q > 0 .

Step 4. Calculate N using the balancing method on Eq. (4).

Step 5. Combine Eq. (5) together with Eq. (4), set coefficients of Ω ( ζ ) equal to zero, and determine the values of the unknowns.

3 Results

The present section is divided into two stages. In Section 3.1, we derive exact solutions for the quadratic model and, Section 3.2, we obtain solutions for the quartic model.

3.1 Quadratic model

To obtain soliton solutions of the DR equations under study in the present work, we employ the traveling-wave transformation ζ = x − w t , so that φ ( x , t ) = ψ ( ζ ) . This change of variable transforms Eqs. (1) and (2) into the following nonlinear ODE:

(49) D ψ ″ + w ψ ′ − k ψ 2 ψ ′ + α ψ − β ψ 2 = 0 ,

where the prime symbol denotes the order of derivative of the function ψ with respect to ζ . By applying the homogeneous balancing criterion to Eq. (49), one can readily find that N = 1 . Therefore, we can write the solutions of Eq. (49) as follows:

(50) ψ ( ζ ) = b 0 + b 1 Ω ( ζ ) .

By using Eq. (50) and its derivative in Eq. (49) and by equating the coefficient of the powers of Ω ( ζ ) zero, one can obtain a system of algebraic equations. It is easy to check that the solution of this system of equations is given by

(51) b 0 = D k n log F − D 2 k 2 n 2 log 2 F − 4 D 2 k 2 m ν log 2 F k 2 ,

(52) b 1 = 2 D ν log F k ,

(53) w = 2 D β − D 2 k 2 log 2 F ( n 2 − 4 m ν ) k ,

(54) α = − 2 β D 2 k 2 log 2 F ( n 2 − 4 m ν ) k 2 .

By manipulating Eq. (50) along with the different forms of Ω ( ζ ) described in Section 2, we can obtain the following forms of traveling wave solutions for Eq. (1):

When n 2 − 4 m ν < 0 and ν ≠ 0 , then
(55) φ 1 ( x , t ) = 1 k 2 ( D k n ln F − D σ k ln ( F ) ) + 1 k 2 D ν ln ( F ) − α tan F ζ − σ 2 2 ν − n 2 ν ,

(56) φ 2 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) − − σ cot F ζ − σ 2 2 ν − n 2 ν ,

(57) φ 3 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) − σ ( tan F ( ζ − σ ) − sec F ( ζ − σ ) ) 2 ν − n 2 ν ,

(58) φ 4 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) − σ ( cot F ( ζ − σ ) − csc F ( ζ − σ ) ) 2 ν − n 2 ν ,

(59) φ 5 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) − σ tan F ζ − σ 4 − cot F ζ − σ 4 4 ν − n 2 ν ,
where σ = n 2 − 4 m ν . For illustration purposes, Figure 1 behavior of the solution φ 1 ( x , t ) for the parameter values n = 1.5 , m = 1.5 , ν = 1.5 , β = 1.9 , k = 1.4 , F = 1.5 , and D = 1.4 .

When n 2 − 4 m ν > 0 and ν ≠ 0 , then
(60) φ 6 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) − n 2 ν − σ tanh F ζ σ 2 2 ν ,

(61) φ 7 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) − σ coth F ζ σ 2 2 ν − n 2 ν ,

(62) φ 8 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) − n 2 ν − σ ( tanh F ( ζ σ ) − i sech F ( ζ σ ) ) 2 ν ,

(63) φ 9 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D σ k ln ( F ) ) + 1 k 2 D ν ln ( F ) − σ ( coth F ( ζ σ ) − csch F ( ζ σ ) ) 2 ν − n 2 ν ,

(64) φ 10 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D σ k ln ( F ) ) + 1 k 2 D ν ln ( F ) − σ coth F ζ σ 4 + tanh F ζ σ 4 4 ν − n 2 ν .
Figure 2 shows the graphical behavior of the solution φ 6 ( x , t ) for the parameter values n = 0.1 , m = − 1 , ν = 0.2 , β = 1.2 , k = 0.1 , F = 1.5 , and D = 1.1 .

If m ν > 0 and n = 0 , then
(65) φ 11 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) m ν tan F ( ζ m ν ) ,

(66) φ 12 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) − 1 k 2 D ν ln ( F ) m ν cot F ( ζ m ν ) ,

(67) φ 13 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) m ν tan F ( 2 ζ m ν ) − p q sec F ( 2 ζ m ν ) ,

(68) φ 14 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k 2 D ν ln ( F ) p q csc F ( 2 ζ m ν ) − m ν cot F ( 2 ζ m ν ) ,

(69) φ 15 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k D ν ln ( F ) m ν tan F 1 2 ζ m ν − cot F 1 2 ζ m ν .
Figure 3 shows the graphical behavior of the solution φ 11 ( x , t ) for the parameter values n = 0 , m = 0.1 , ν = 1.5 , β = 0.4 , k = 0.2 , F = 1.2 , and D = 2 .

If m ν < 0 and n = 0 , then
(70) φ 16 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D σ k ln ( F ) ) − 1 k 2 D ν ln ( F ) − m ν tanh F ( ζ − m ν ) ,

(71) φ 17 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) − 1 k 2 D ν ln ( F ) − m ν coth F ( ζ − m ν ) ,

(72) φ 18 ( x , t ) = 1 k 2 ( D k n log F − D k σ ln ( F ) ) − 1 k 2 D ν ln ( F ) − m ν ( tanh F ( 2 ζ − m ν ) + i p q sech F ( 2 ζ − m ν ) ) ,

(73) φ 19 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D σ k L n ( F ) ) − 1 k 2 D ν ln ( F ) − m ν ( p q coth F ( 2 − m ν ) + csch F ( 2 − m ν ) ) ,

(74) φ 20 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D σ k ln ( F ) ) − 1 k D ν ln ( F ) − m ν tanh F 1 2 ζ − m ν − coth F 1 2 ζ − m ν .
For illustration purposes, Figure 4 shows the graphical behavior of the solution φ 16 ( x , t ) for the parameter values n = 0 , m = − 1 , ν = 1.5 , β = 0.5 , k = 0.2 , F = 1.5 , and D = 1 .

When n = 0 and m = ν , then
(75) φ 21 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D σ k ln ( F ) ) + 1 k ( 2 D ν ln ( F ) tan F ( ζ m ) ) ,

(76) φ 22 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) − 1 k ( 2 D ν ln ( F ) cot F ( ζ m ) ) ,

(77) φ 23 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k { 2 D ν ln ( F ) ( sec F ( 2 ζ m ) + tan F ( 2 ζ m ) ) } ,

(78) φ 24 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k { 2 D ν ln ( F ) ( csc F ( 2 ζ m ) − cot F ( 2 ζ m ) ) } ,

(79) φ 25 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k D ν ln ( F ) tan F ζ m 2 − cot F ζ m 2 .

When n = 0 and m = − ν , then
(80) φ 26 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D σ k ln ( F ) ) − 1 k ( 2 D ν ln ( F ) tanh F ( ζ m ) ) ,

(81) φ 27 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) − 1 k ( 2 D ν ln ( F ) coth F ( ζ m ) ) ,

(82) φ 28 ( x , t ) = 1 k 2 ( D k n ln ( F ) − σ D k ln ( F ) ) + 1 k { 2 D ν ln ( F ) ( − tanh F ( 2 ζ m ) + i p q sech F ( 2 ζ m ) ) } ,

(83) φ 29 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k { 2 D ν ln ( F ) ( − coth F ( 2 ζ m ) + i p q csch F ( 2 ζ m ) ) } ,

(84) φ 30 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k D ν ln ( F ) − coth F ζ m 2 − tanh F ζ m 2 .

When n 2 = 4 m ν , then
(85) φ 31 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) − 1 ζ k n 2 { 4 D m ν ( ζ n ln ( F ) + 2 ) } .
If n = χ , m = r , χ ( r ≠ 0 ) , and ν = 0 , then
(86) φ 32 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k { 2 D ν ln ( F ) ( F ζ χ − r ) } .

When n = ν = 0 , then
(87) φ 33 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) + 1 k ( 2 D ζ m ν ln 2 ( F ) ) .
When n = m = 0 , then
(88) φ 34 ( x , t ) = 1 k 2 ( D k n ln ( F ) − D k σ ln ( F ) ) − 2 D ζ k .

When m = 0 and n ≠ 0 , then
(89) φ 35 ( x , t ) = D k n ln ( F ) − D k σ ln ( F ) k 2 − 2 D n p ln ( F ) k ( cosh F ( ζ n ) − sinh F ( ζ n ) + p ) ,

(90) φ 36 ( x , t ) = D k n ln ( F ) − D k σ ln ( F ) k 2 − 2 D n ln ( F ) ( cosh F ( ζ n ) + sinh F ( ζ n ) ) k ( cosh F ( ζ n ) + sinh F ( ζ n ) + q ) .
When n = χ , ν = r χ ( r ≠ 0 ) , and m = 0 , then
(91) φ 37 ( x , t ) = D k n ln ( F ) − D k σ ln ( F ) k 2 − 2 D ν ln ( F ) p F ζ χ k ( p − rq F ζ χ ) .

3.2 Quartic model

By applying the traveling wave transformation, we readily obtain the nonlinear ODE

(92) D ψ ″ + w ψ ′ − k ψ ψ ′ + α ψ − β ψ 4 = 0 .

Observe that we obtain that N = 1 by using the balance between the dispersive and nonlinear terms in Eq. (92). Therefore, the solutions of this equation are expected to be expressed in similar fashion as for Eq. (50). By substituting Eq. (50) together with its derivatives into Eq. (92) and by comparing the coefficients of the indices of Ω ( ζ ) , we can obtain an algebraic system of equations. The solutions lead to two possible families of parameters

Family 1.

(93) b 0 = − β 2 k 2 ln 2 F ( n 2 − 4 m ν ) + β k n ln ( F ) 2 β 2 ,

(94) b 1 = − k ν ln ( F ) β ,

(95) w = − 3 k 3 ln 2 F ( n 2 − 4 m ν ) 2 β 2 ,

(96) D = − k 2 β 2 k 2 ln 2 F ( n 2 − 4 m ν ) 2 β 3 ,

(97) α = − ( β 2 k 2 ln 2 F ( n 2 − 4 m ν ) ) 3 ∕ 2 β 5 .

Family 2.

(98) b 0 = − 3 β 2 ( − k 2 ) ln 2 F ( n 2 − 4 m ν ) + 3 β k n ln ( F ) 6 β 2 ,

(99) b 1 = − k ν ln ( F ) β ,

(100) w = k 3 ln 2 F ( n 2 − 4 m ν ) 6 β 2 ,

(101) D = − k 2 β 2 ( − k 2 ) ln 2 F ( n 2 − 4 m ν ) 2 3 β 3 ,

(102) α = ( β 2 ( − k 2 ) ln 2 F ( n 2 − 4 m ν ) ) 3 ∕ 2 3 3 β 5 .

Proceeding as in the case of the model with quadratic reaction law, we obtain the following traveling wave solutions for Eq. (2).

When n 2 − 4 m ν < 0 and ν ≠ 0 , then
(103) φ 1 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln F ) − 1 β k ν ln F − σ tan F ζ − σ 2 2 ν − n 2 ν ,

(104) φ 2 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln F ) − 1 β k ν ln F − − σ cot F ζ − σ 2 2 ν − n 2 ν ,

(105) φ 3 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β k ν ln ( F ) − σ ( tan F ( ζ − σ ) − sec F ( ζ − σ ) ) 2 ν − n 2 ν ,

(106) φ 4 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β k ν ln ( F ) − − σ ( cot F ( ζ − σ ) − p q csc F ( ζ − σ ) ) 2 ν − n 2 ν ,

(107) φ 5 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β k ν ln ( F ) − σ tan F ζ − σ 4 − cot F ζ − σ 4 4 ν − n 2 ν .
For the sake of convenience, Figure 5 shows graphs of the solution φ 1 ( x , t ) for the parameter values n = 1 , m = 1 , ν = 1.5 , β = 1 , k = 1.2 , and F = 1.5 .

When n 2 − 4 m ν > 0 and ν ≠ 0 , then
(108) φ 6 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln F ) − 1 β k ν ln F − n 2 ν − σ tanh F ζ σ 2 2 ν ,

(109) φ 7 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln F ) − 1 β k ν ln F − σ coth F ζ σ 2 2 ν − n 2 ν ,

(110) φ 8 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n L n ( F ) ) − 1 β k ν ln ( F ) − n 2 ν − σ ( tanh F ( ζ σ ) ± i p q sech F ( ζ σ ) ) 2 ν ,

(111) φ 9 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β k ν ln ( F ) − σ ( coth F ( ζ σ ) ± p q csch F ( ζ σ ) ) 2 ν − n 2 ν ,

(112) φ 10 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β k ν ln ( F ) − σ coth F ζ σ 4 + tanh F ζ σ 4 4 ν − n 2 ν .
For illustration purposes, Figure 6 shows the graphical behavior of the solution φ 6 ( x , t ) for the parameter values n = 1 , m = 2 , ν = − 1.5 , β = 1 , k = 1.7 , and F = 1.5 .

If m ν > 0 and n = 0 , then
(113) φ 11 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β k ν ln ( F ) m ν tan F ( ζ m ν ) ,

(114) φ 12 ( x , t ) = 1 β k ν ln ( F ) m ν cot F ( ζ m ν ) − 1 2 β 2 ( β 2 k 2 σ log 2 F + β k n ln ( F ) ) ,

(115) φ 13 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β k ν ln ( F ) m ν ( tan ( 2 ζ m ν ) ± p q sec F ( 2 ζ m ν ) ) ,

(116) φ 14 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + 1 β k ν ln ( F ) m ν ( cot F ( 2 ζ m ν ) ± p q csc F ( 2 ζ m ν ) ) ,

(117) φ 15 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 2 β k ν ln ( F ) m ν tan F 1 2 ζ m ν − cot F 1 2 ζ m ν .

When m ν < 0 and n = 0 , then
(118) φ 16 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + 1 β k ν ln ( F ) − m ν tanh F ( ζ − m ν ) ,

(119) φ 17 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + 1 β k ν ln ( F ) − m ν coth F ( ζ − m ν ) ,

(120) φ 18 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + 1 β k ν ln ( F ) − m ν ( tanh F ( 2 ζ − m ν ) + i p q sech F ( 2 ζ − m ν ) ) ,

(121) φ 19 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + 1 β k ν ln ( F ) − m ν ( coth F ( 2 − m ν ) ± p q csch F ( 2 − m ν ) ) ,

(122) φ 20 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + 1 2 β k ν ln ( F ) − m ν tanh F 1 2 ζ − m ν − coth F 1 2 ζ − m ν .
Figure 7 shows the solution φ 16 ( x , t ) for the parameter values n = 0 , m = 1 , ν = − 1.5 , β = 1.5 , k = 1.5 , and F = 1.5 .

If n = 0 and m = ν , then
(123) φ 21 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β ( k ν ln ( F ) tan F ( ζ m ) ) ,

(124) φ 22 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ Ln F 2 + β k n ln ( F ) ) + 1 β ( k ν ln ( F ) cot F ( ζ m ) ) ,

(125) φ 23 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β { k ν ln ( F ) ( tan F ( 2 ζ m ) ± p q sec F ( 2 ζ m ) ) } ,

(126) φ 24 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β { k ν ln ( F ) ( − cot F ( 2 ζ m ) ± p q csc F ( 2 ζ m ) ) } ,

(127) φ 25 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 2 β k ν ln ( F ) tan F ζ m 2 − cot F ζ m 2 .
Figure 8 shows the graphical behavior of the solution φ 21 ( x , t ) for the parameter values n = 0 , m = 1.5 , ν = 1.5 , β = 1.5 , k = 1.1 , and F = 1.5 .

When n = 0 and ν = − m , then
(128) φ 26 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ Ln F 2 + β k n ln ( F ) ) + 1 β ( k ν ln ( F ) tanh F ( ζ m ) ) ,

(129) φ 27 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + 1 β ( k ν ln ( F ) coth F ( ζ m ) ) ,

(130) φ 28 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β { k ν ln ( F ) ( − tanh F ( 2 ζ m ) + i p q sech F ( 2 ζ m ) ) } ,

(131) φ 29 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β { k ν ln ( F ) ( p q csch F ( 2 ζ m ) − coth F ( 2 ζ m ) ) } ,

(132) φ 30 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 2 β k ν ln ( F ) − coth F ζ m 2 − tanh F ζ m 2 .

If n 2 = 4 m ν , then
(133) φ 31 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + 1 β ζ n 2 { 2 k m ν ( ζ n ln ( F ) + 2 ) } .
When n = χ , m = r χ with r ≠ 0 and ν = 0 , then
(134) φ 32 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β { k ν ln ( F ) ( F ζ χ − r ) } .
When n = ν = 0 then
(135) φ 33 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) − 1 β ( ζ k m ν ln 2 F ) .
If n = m = 0 , then
(136) φ 34 ( x , t ) = − 1 2 β 2 ( β 2 k 2 σ ln 2 F + β k n ln ( F ) ) + k β ζ .

When = m = 0 and ν ≠ 0 , then
(137) φ 35 ( x , t ) = − β 2 k 2 σ ln 2 F + β k n ln ( F ) 2 β 2 + k n p ln ( F ) β ( cosh F ( ζ n ) − sinh F ( ζ n ) + p ) ,

(138) φ 36 ( x , t ) = − β k σ ln ( F ) + β k n ln ( F ) 2 β 2 + k n ln ( F ) ( cosh F ( ζ n ) + sinh F ( ζ n ) ) β ( cosh F ( ζ n ) + sinh F ( ζ n ) + q ) .
If n = χ , ν = r χ with r ≠ 0 , and m = 0 , then
(139) φ 37 ( x , t ) = k ν p ln ( F ) F ζ χ β ( p − q r F ζ χ ) − β k n ln ( F ) + σ 2 β 2 .

4 Discussion

In this section, we will discuss the graphical representation of the results obtained in the previous section by using the new NEDAM. Different types of soliton solutions are seen, including solitary wave, kink, anti-kink, and rational function solutions. Using the Mathematica 11.1 software, graphs have been created to show the solutions physical behavior in the form of 3D, 2D and contour plots. When modeling a variety of physical phenomena (such as chemical reactions, the creation of biological patterns, and heat conduction), reaction–diffusion equations with quadratic and quartic nonlinearities may essential.

It is well known that solitary waves are examples of coherent structures that propagate without distortion because of their limited and stable profiles. The study of energy transfer mechanisms in nonlinear media, including shallow water systems and optical fibers, depends heavily on these waves. Kink and anti-kink solutions, which are frequently used to mimic interfaces or domain walls like those seen in magnetic phase transitions or crystal dislocations, represent monotonic transitions between two stable states. Understanding phase separation and symmetry-breaking dynamics in physical systems requires an understanding of these structures. Less frequently found, rational function solutions describe limited structures with distinctive singular behaviors. These are frequently linked to severe occurrences or anomaly-driven phenomena, such as localized bursts in reaction–diffusion systems or rogue waves in hydrodynamics. Many waveforms and transition patterns are possible due to the interaction between the quadratic and quartic nonlinearities in these equations, which enriches the solution space. Visualizing these solutions offers important information about their stability, dynamics over time and space, and possible uses in systems controlled by complicated pattern generation, phase transitions, and nonlinear energy transfer. In particular, Figure 1 depicts a lump-kink, while Figures 2, 4, and 6 are kink solitons. Figures 3, 5, and 8 provide solitary waves, and Figure 7 shows an anti-kink behavior.

$Figure 1 Lump-kink soliton behavior of the solution φ 1 ( x , t ) {\varphi }_{1}\left(x,t) for the parameter values n = 1.5 n=1.5 , m = 1.5 m=1.5 , ν = 1.5 \nu =1.5 , β = 1.9 \beta =1.9 , k = 1.4 k=1.4 , F = 1.5 F=1.5 , and D = 1.4 D=1.4 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.$

Figure 1

Lump-kink soliton behavior of the solution φ 1 ( x , t ) for the parameter values n = 1.5 , m = 1.5 , ν = 1.5 , β = 1.9 , k = 1.4 , F = 1.5 , and D = 1.4 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.

$Figure 2 Kink soliton behavior of the solution φ 6 ( x , t ) {\varphi }_{6}\left(x,t) for the parameter values n = 0.1 n=0.1 , m = ‒ 1 m=‒1 , ν = 0.2 \nu =0.2 , β = 1.2 \beta =1.2 , k = 0.1 k=0.1 , F = 1.5 F=1.5 , and D = 1.1 D=1.1 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.$

Figure 2

Kink soliton behavior of the solution φ 6 ( x , t ) for the parameter values n = 0.1 , m = ‒ 1 , ν = 0.2 , β = 1.2 , k = 0.1 , F = 1.5 , and D = 1.1 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.

$Figure 3 Solitary wave behavior of the solution φ 11 ( x , t ) {\varphi }_{11}\left(x,t) for the parameter values n = 0 n=0 , m = 0.1 m=0.1 , ν = 1.5 \nu =1.5 , β = 0.4 \beta =0.4 , k = 0.2 k=0.2 , F = 1.2 F=1.2 , and D = 2 D=2 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.$

Figure 3

Solitary wave behavior of the solution φ 11 ( x , t ) for the parameter values n = 0 , m = 0.1 , ν = 1.5 , β = 0.4 , k = 0.2 , F = 1.2 , and D = 2 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.

$Figure 4 Kink soliton behavior of the solution φ 16 ( x , t ) {\varphi }_{16}\left(x,t) for the parameter values n = 0 n=0 , m = ‒ 1 m=‒1 , ν = 1.5 \nu =1.5 , β = 0.5 \beta =0.5 , k = 0.2 k=0.2 , F = 1.5 F=1.5 , and D = 1 D=1 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.$

Figure 4

Kink soliton behavior of the solution φ 16 ( x , t ) for the parameter values n = 0 , m = ‒ 1 , ν = 1.5 , β = 0.5 , k = 0.2 , F = 1.5 , and D = 1 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.

$Figure 5 Solitary wave behavior of the solution φ 1 ( x , t ) {\varphi }_{1}\left(x,t) for the parameter values n = 1 n=1 , m = 1 m=1 , ν = 1.5 \nu =1.5 , β = 1 \beta =1 , k = 1.2 k=1.2 , and F = 1.5 F=1.5 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.$

Figure 5

Solitary wave behavior of the solution φ 1 ( x , t ) for the parameter values n = 1 , m = 1 , ν = 1.5 , β = 1 , k = 1.2 , and F = 1.5 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.

$Figure 6 Kink soliton behavior of the solution φ 6 ( x , t ) {\varphi }_{6}\left(x,t) for the parameter values n = 1 n=1 , m = 2 m=2 , ν = − 1.5 \nu =-1.5 , β = 1 \beta =1 , k = 1.7 k=1.7 , and F = 1.5 F=1.5 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.$

Figure 6

Kink soliton behavior of the solution φ 6 ( x , t ) for the parameter values n = 1 , m = 2 , ν = − 1.5 , β = 1 , k = 1.7 , and F = 1.5 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.

$Figure 7 Anti-Kink soliton behavior of the solution φ 16 ( x , t ) {\varphi }_{16}\left(x,t) for the parameter values n = 0 n=0 , m = 1 m=1 , ν = − 1.5 \nu =-1.5 , β = 1.5 \beta =1.5 , k = 1.5 k=1.5 , and F = 1.5 F=1.5 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.$

Figure 7

Anti-Kink soliton behavior of the solution φ 16 ( x , t ) for the parameter values n = 0 , m = 1 , ν = − 1.5 , β = 1.5 , k = 1.5 , and F = 1.5 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.

$Figure 8 Solitary wave behavior of the solution φ 21 ( x , t ) {\varphi }_{21}\left(x,t) for the parameter values n = 0 n=0 , m = 1.5 m=1.5 , ν = 1.5 \nu =1.5 , β = 1.5 \beta =1.5 , k = 1.1 k=1.1 , and F = 1.5 F=1.5 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.$

Figure 8

Solitary wave behavior of the solution φ 21 ( x , t ) for the parameter values n = 0 , m = 1.5 , ν = 1.5 , β = 1.5 , k = 1.1 , and F = 1.5 : (a) three-dimensional plot, (b) contour plot, and (c) two-dimensional plot.

5 Conclusions

In this work, the NEDAM has been used to obtain solitary wave solutions for general reaction–diffusion equations with quadratic and quartic nonlinearities. By applying this approach, we have established different types of soliton solutions, including solitary wave, kink, anti-kink, and rational function solutions. To illustrate the physical behavior of the solutions, some plots have been obtained using the Mathematica software. One can obtain various possible results by adjusting the parametric values appropriately. As a conclusion on the side, we have found out that the NEDAM is an appropriate and reliable method for locating precise soliton solutions for broad categories of nonlinear problems. The results obtained in this work were verified by substituting them into the original model equations with the help of Mathematica software. We verified in all cases that the functions derived in this work are actually solutions of our mathematical models. As a future direction of investigation, the models investigated in this work will be extended as NPDEs of higher-order partial differential equations, considering fractional orders of differentiation, and stochastic sources. In those cases, we expect to derive solitary wave solutions by employing the NEDAM, and we will confirm our result numerically through suitable mathematical software.

jorge-maciasdiaz@edu.uaa.mx, jemacias@correo.uaa.mx

Acknowledgments

The authors wish to thank the anonymous reviewers for their criticisms. All of their suggestions contributed to improving the quality of this work.

Funding information: One of the authors of this work (J.E.M.-D.) wishes to acknowledge the financial support from the National Council of Humanities, Science and Technology of Mexico (CONACYT) through grant A1-S-45928, associated to the research project “Conservative methods for fractional hyperbolic systems: analysis and applications.” In turn, M.G.M.-G. acknowledges the financial support from the program PROSNI of the University of Guadalajara, Mexico.
Author contributions: Conceptualization: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G.; data curation: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G.; formal analysis: N.A., J.E.M.-D., M.A., M.J., M.Z.B.; funding acquisition: J.E.M.-D., M.G.M.-G.; investigation: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G.; methodology: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G.; project administration: J.E.M.-D., M.G.M.-G.; resources: J.E.M.-D., M.G.M.-G.; software: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G.; supervision: N.A., J.E.M.-D.; validation: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G.; visualization: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G.; roles/writing – original draft: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G.; writing – review editing: N.A., J.E.M.-D., M.A., M.J., M.Z.B., M.G.M.-G. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2024-10-03

Revised: 2024-12-09

Accepted: 2025-01-08

Published Online: 2025-02-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0115

Keywords for this article

extended direct algebraic method; quadratic and quartic nonlinearities; soliton solutions; traveling-wave solutions

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