Isomorphic shut form valuation for quantum field theory and biological population models

Maha S. M. Shehata; Hijaz Ahmad; Emad H. M. Zahran; Sameh Askar; Dilber Uzun Ozsahin

doi:10.1515/phys-2022-0252

Article Open Access

Isomorphic shut form valuation for quantum field theory and biological population models

Maha S. M. Shehata , Hijaz Ahmad , Emad H. M. Zahran , Sameh Askar and Dilber Uzun Ozsahin

Published/Copyright: July 29, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

The fundamental objective of this work is focused to achieve a class of advanced and impressive exact estimations to the Zoomeron equation and the time-fraction biological population model through contrivance by a couple of important and magnificent techniques, namely, the modified extended tanh-function method which depend on the balance theory and the Ricatti–Bernoulli sub-ODE method which is independent of the balancing principle. The suggested model is one of the major concerns for studying population distribution dynamics as well as the quantum field theory which is an important discipline for the description of interactions between light and electrons. The two suggested reliable, effective techniques are considered famous among ths ansatz methods that have various visions to realize the exact solutions to the non linear partial differential equation that reduce the volume of calculations examined before and usually give good results. It is solicited for this contrivance finding new exact solutions for two models in terms of some variable. The models are significant in quantum field theory, description of interactions between light and electron, quantum electrodynamics, demographic model, important to bring it into line with the reasonable distribution of wealth, resources, income, etc. The achieved results predict many types of solutions as trigonometric functions, hyperbolic functions, perfect periodic soliton solutions, singular periodic soliton solutions, and other rational solitons solutions. The efficiency of the techniques is demonstrated by the satisfactory results obtained through the derivation of closed-form soliton solutions from the exact solution by assigning definite values to the variables present in it.

Keywords: modified extended tanh-function method; Ricatti–Bernoulli sub-ODE method; time-fraction biological population model; (2 + 1)-dimensional Zoomeron model; traveling wave solutions

1 Introduction

The biological population model is one of the major concerns for studying population dynamics and the quantum field theory is also an important discipline for the description of interactions between light and electrons. The position, time, and density are the principle axes that identfy this scheme and provides an important contribution to directive population process.

The population processes are studied through important biological models that have been reported by Shakeri and Dehghon [1]. One of these models is the (1 + 1)-dimensional model, which has been extensively investigated in the literature and describes solitons achieved in various branches of physics and applied mathematics. The other model is the time-fraction biological population model (TFBPM) [2], which is a well-known model for representing population processes. Gurney and Nisbet [3] provides a good model for the TFBPM in animals, which is a special case of the (2 + 1)-dimensional non-fractional Zoomeron model. In both models, the functions u and λ ( u 2 − r ) represent the population density and the principle support of the population, respectively, and depend on births and deaths, while λ and r are constants. Many researchers have developed different methods for studying nonlinear differential models, most of which are referred to in previous studies [4–22]. The modified extended tanh-function method (METFM) [21] and the Ricatti–Bernoulli sub-ODE method (RBSOM) [13] are vibrant and dominant techniques for extracting the analytic solutions to the equations of nonlinear waves. In the same connection, modern techniques have been employed to obtain both numerical and analytical solutions for certain non linear partial differential equation problems arising in various branches of science, for example, Wang and Si [23] who derived abundant optical solitons comprising the kinky-bright soliton, bright soliton, bright-like soliton, dark soliton, singular periodic soliton, double-bright soliton, perfect periodic soliton, and other solitons to the complex Ginzburg–Landau equation utilizing the sub-equation method, Sardar-sub-equation method, and variational direct method. According to a study conducted by Wang et al., they utilized the variational method and sub-equation method to analyze the nonlinear dynamics of magnets described by the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation. They were able to extract various solutions including dark soliton, bright soliton, bright-dark soliton, singular periodic soliton, and perfect periodic soliton. Wang and Liu [24] used two methods, Wang’s direct mapping and Bäcklund’s transformation-based method, to study the nonlinear Schrödinger equation that depicts pulse propagation in optical fiber. They found different soliton solutions represented by trigonometric, hyperbolic, exponential, and rational functions. Also, they used different analytical techniques such as the variational method, Bernoulli sub-equation function method, and Hamiltonian method to derive multiple wave solutions of the modified Benjamin–Bona–Mahony equation. Wang [25] introduced a new fractional Zakharov-Kuznetsov equation. They used Yang’s non-differentiable traveling wave transform and applied two novel techniques (the Yang’s special function method and Mittag-Leffler function-based method) to find exact solutions for the equation. Wang [26] worked further, and conducted three studies in which they derived new equations and models using fractional calculus. In the first study, they derived a fractional (2 + 1)-dimensional Boussinesq equation and obtained various solutions using the Hirota’s bilinear method and the variational method. In the second study, they proposed a new fractal active low-pass filter using local fractional calculus on the Cantor set and extracted its non-differentiable transfer function using the local fractional Laplace transform. In the third study, they developed a new fractional pulse narrowing nonlinear transmission lines model using local fractional calculus and obtained non-differentiable exact solutions using the direct mapping method.

Analyzing the existing literature, it is found that the biological population model and Zoomeron equation have not been thoroughly studied using previously introduced techniques. To fill this gap, the current article utilizes the METFM and RBSOM to establish general and archetypal solutions to these models. The obtained results reveal various physical structures such as kink solitons, bell shape solitons, and flat shape solitons that may help to understand population dynamics and quantum field theory. These solitary wave solutions can be realized easily under specific variable values.

2 Description of the METFM

To demonstrate the general formalism of the METFM, we can define the function R in terms of h ( x , t ) and its partial derivatives as follows

(1) R ( h , h t , h x , h tt , h xx , . . . . ) = 0 ,

which consist of terms that are nonlinear and have the highest order of derivatives.

One can reduce Eq. (1) to the following ordinary differential equation (ODE) by utilizing the transformation h ( x , t ) = h ( ξ ) , where ξ = x − ct .

(2) Q ( h , h ′ , h '' , h ''' , . . . . . ) = 0 ,

where Q is a function in h ( ξ ) and its total derivatives, while ′ = d d ξ .

The solution formula of Eq. (2) in conformity with this approach is

(3) h ( ξ ) = a 0 + ∑ i = 1 m ( a i ψ i + b i ψ − i ) .

According to Lu [18], the balance rule is employed for assessing the integer m 's increase in Eq. (3). The evaluation is performed based on the condition that either a m ≠ 0 or b m ≠ 0 , while a i and b i are computed. Additionally, the Riccati equation ψ ′ = b + ψ 2 must be satisfied by ψ . Based on the value of b , one of the three commonly known solutions is admitted.

(4) ψ ( ξ ) = − − b tanh ( − b ξ ) or ψ ( ξ ) = − − b coth ( − b ξ ) , b < 0 ,

(5) ψ ( ξ ) = b tan ( b ξ ) or ψ ( ξ ) = − b cot ( b ξ ) , b > 0 ,

(6) ψ ( ξ ) = − 1 ξ , b = 0 .

To obtain the values of the involved constants, we can create an algebraic system of equations by setting the coefficients of various powers of ψ i to zero. This system can then be analyzed using a computer program.

2.1 The Zoomeron equation

In this section, the formerly put forward technique has been contrivance to the (2 + 1)-dimensional Zoomeron Eq. (1).

(7) h xy h tt − h xy h xx + 2 ( h 2 ) xt = 0 ,

where h ( x , y , t ) = h ( ξ ) and ξ = x − y − kt indicate the relative wave mode. Under these transformations, Eq. (7) became

(8) ( k 2 − 1 ) h'' − 2 k h 3 + k 1 h = 0 .

Utilizing the balance rule between u ″ , u 3 ⇒ m = 1 , consequently, the solution according to Eq. (3) is as follows:

(9) h ( ξ ) = a 0 + a 1 ψ ( ξ ) + b 1 ψ ( ξ ) .

Substituting h , h 3 , and h '' in Eq. (7) and by setting the coefficients of various powers of ψ ( ξ ) to zero, the following equations can be obtained:

(10) k 2 − 1 − k a 1 2 = 0 , − 6 k a 0 a 1 2 = 0 , 2 ( k 2 − 1 ) b a 1 − 6 k a 0 a 1 2 + k 1 a 1 = 0 , ( k 2 − 1 ) b 2 − k b 1 2 = 0 , − 6 k a 0 b 1 2 = 0 , 2 ( k 2 − 1 ) b b 1 − 6 k a 0 b 1 + k 1 b 1 = 0 , − 12 k a 1 b 1 + k 1 = 0 .

Unraveling this system of equations by computer algebra program, we obtain

(11) ( 1 ) a 0 = 0 , a 1 = k 2 − 1 k , b 1 = k 2 − 1 k b , b = k 1 2 ( 1 − k 2 ) , ( 2 ) a 0 = 0 , a 1 = k 2 − 1 k , b 1 = − k 2 − 1 k b , b = k 1 2 ( 1 − k 2 ) , ( 3 ) a 0 = 0 , a 1 = − k 2 − 1 k , b 1 = k 2 − 1 k b , b = k 1 2 ( 1 − k 2 ) , ( 4 ) a 0 = 0 , a 1 = − k 2 − 1 k , b 1 = − k 2 − 1 k b , b = k 1 2 ( 1 − k 2 ) .

Several solutions to the Zoomeron equation are obtainable subject to the value of b ( b < 0 , b > 0 , and b = 0 ) and the values of the parameters. However, in order to evade harmonious repetition, we recommend the solutions only for the values of the parameters assorted in set (1).

a 0 = 0 , a 1 = k 2 − 1 k , b 1 = k 2 − 1 k b , b = k 1 2 ( 1 − k 2 ) .

For b < 0 , embedding above values into Eq. (9), we achieve the following realistic solutions to the Zoomeron equation:

(12) h ( x , y , t ) = − k 2 − 1 k − k 1 2 1 − k 2 × tanh − k 1 2 1 − k 2 x + y − kt − k 1 2 ( 1 − k 2 ) k 2 − 1 k 2 1 − k 2 − k 1 × coth − k 1 2 1 − k 2 x + y − kt .

A closed form wave solution and its sketch is significantly important to describe and understand a phenomena rigorously. Therefore, the 3D profile of the solution is portrayed for t = 0.1 , k = 2 , k 1 = 1 , 0 ≤ x , y ≤ 1 and 2D shape is sketched for t = 0.1 , y = 0 , k = 2 , k 1 = 1 , 0 ≤ x ≤ 1 and labeled in Figure 1.

Figure 1

The plot of solution (12) in 3D and 2D for the prescribed values.

On the other hand, when b > 0 by means of the values of the constraints gathered in set (1) from solution (9), we attain the useful solutions as follows:

(13) h x , y , t = − k 2 − 1 k − k 1 2 ( 1 − k 2 ) × tan − k 1 2 ( 1 − k 2 ) ( x + y − kt ) − k 2 − 1 k k 1 2 ( 1 − k 2 ) 2 ( 1 − k 2 ) − k 1 × cot − k 1 2 ( 1 − k 2 ) ( x + y − kt ) .

Consequently, for t = 0.1 , y = 0 , k = 2 , k 1 = 1 , − 10 ≤ x ≤ 10 , the 2D shape and for t = 0.1 , k = 2 , k 1 = 1 , − 10 ≤ x , y ≤ 10 , the 3D shape of solution (13) is depicted and marked in Figure 2.

Figure 2

The plot of solution (13) in 2D and 3D for prescribed values.

Furthermore, when b = 0 , with the help of the argument values collected in set (1) from solution (9), we arrive at the following strategic solution:

(14) h ( x , y , t ) = − 1 x + y − kt .

For t = 0.1 , y = 0 , k = 2 , k 1 = 1 , − 10 ≤ x ≤ 10 , the 2D structure and for t = 0.1 , k = 2 , k 1 = 1 , − 10 ≤ x , y ≤ 10 , the 3D layout of the solution (14) is interpreted and arranged in Figure 3.

Figure 3

The plot of solution (14) in 2D and 3D for suggested values.

Three sets of compatible solutions are ascertained for one set of values of the parameters and thus futher nine solutions can be found for the other three sets. The other solutions are not noted down, however, for terse.

2.2 TFBPM

In this paragraph, we will put through the technique to the time fractional biological population model TFBPM [2]. We first bring in some notes on the fractional calculus before materializing the erstwhile approach to the TFBPM.

D t ∝ t r = Г ( 1 + r ) Г ( 1 + r − ∝ ) t r − 1 ,

D t ∝ ( f ( t ) g ( t ) ) = f ( t ) D t ∝ g ( t ) + g ( t ) D t ∝ f ( t ) ,

D t ∝ f [ g ( t ) ] = f ′ [ g ( t ) ] D t ∝ g ( t ) = D g ∝ [ g ( t ) ] ( g ′ ( t ) ) ∝ ,

and the two well-known forms

D ∝ J ∝ f ( x ) = f ( x ) , x > 0 ,

D ∝ J ∝ f ( x ) = f ( x ) − ∑ k = 0 m f ( k ) ( 0 + ) x k k ! , x > 0 .

These two properties represent the derivative of Caputo operator.

Now, we introduce the general form of the NLFPDE as follows:

(15) L ( u , D t α u , D x α u , … ) = 0 , 0 < α ≤ 1 ,

where D t α u and D x α u are the modified Riemann–Liouville derivatives, and using the wave transformation

(16) u ( x , t ) = u ( ξ ) , ξ = K x α Г ( 1 + α ) + C t α Г ( 1 + α ) + ξ 0 ,

while K , C , and ξ 0 are the constants with K , C ≠ 0 will convert Eq. (15) to the following nonlinar ODE with integer order

(17) M ( u , u ′ , u '' , u ''' , . . . . . ) = 0 , where ′ = d d ξ .

Conformable fractional derivative:

For a function u , [ 0 , ∞ ) → R , the conformable fractional derivative of u ( x ) , x > 0 is defined as follows:

(18) D x γ ( u ( x ) ) = lim h → 0 u ( x + h x 1 − γ ) − u ( x ) h , γ ϵ [ 0 , 1 ] .

The TFBPM is [2] as follows:

(19) ∂ α u ∂ t α = ∂ 2 ∂ x 2 ( u 2 ) + ∂ 2 ∂ y 2 ( u 2 ) + λ ( u 2 − r ) ,

t > 0 , 0 < α ≤ 1 , x , y ∈ R , while u and λ ( u 2 − r ) represent the density and the princple support of population, respictvely, which are built on births and deaths while λ and r are constants.

Let us consider this complex wave transformation.

(20) u ( x , y , t ) = u ( ξ ) and ξ = x + iy − c t α Γ ( 1 + α ) , c is a costant and i 2 = − 1 ,

In conformity with Wang [26], it realizes all the classical derivative rules that are also effective for functions for fractional derivative and according to Bekir and Ozkan [2], the TFBPM is converted into the following:

(21) c u ′ + λ u 2 − λ r = 0 , u ′ = d u d ξ .

Balancing linear and nonlinear terms u ′ and u 2 of maximum exponents implie m = 1 . Thus, the proposed approach allows for the solution to be expressed as follows:

(22) u ( ξ ) = a 0 + a 1 ψ ( ξ ) + b 1 ψ ( ξ ) .

Injecting the solution (22) into Eq. (21) and comparing the cohorts of diverse powers of ψ ( ξ ) to zero, we have

(23) c + a 1 λ = 0 , 2 a 0 a 1 λ = 0 , − c b + λ b 1 = 0 , 2 a 0 b 1 λ = 0 , c ( a 1 b − b 1 ) + λ a 0 2 − λ r = 0 .

Resolving this system of algebraic equations, we found

(24) a 0 = 0 , a 1 = − c λ , b 1 = − cb λ , b = − λ 2 r 2 c 2 .

Hence, for the above assessment of the parameters, the solution developed into

(25) u ( ξ ) = − c λ ψ ( ξ ) − cb λ 1 ψ ( ξ ) .

Moreover, we can construct three types of solutions allowing the distinct probabilty of b .

Case 1. When b < 0 , the solution (25) is simplified as follows:

(26) u ( x , y , t ) = − c λ λ 2 r 2 c 2 tanh λ 2 r 2 c 2 x + iy − c t α Γ ( 1 + α ) − cb λ λ 2 r 2 c 2 coth λ 2 r 2 c 2 x + iy − c t α Γ ( 1 + α ) .

Graphs act upon a key foreword in determining the realistic environment of wave propagation. Therefore, the phenomenon is analyzed by portraying 2D and 3D depictions of the solution obtained here. Therefore, the solution is presented in the underlying 3D profile t = 0.1 , α = 1 / 2 , λ = 2 , r = 2 , c = 2 , 0 ≤ x , y ≤ 1 and the 2D shape is sketched for t = 0.1 , y = 0 , α = 1 / 2 , λ = 2 , r = 2 , c = 2 , 0 ≤ x ≤ 1 and characterized in Figure 4.

Figure 4

The plot of solution (26) in 2D and 3D with above mentioned values.

Case 2: On the other hand, when b > 0 , from solution (25), we achieve the valuable solutions as follows:

(27) u ξ = − c λ λ 2 r 2 c 2 tan − λ 2 r 2 c 2 x + iy − c t α Γ ( 1 + α ) − cb λ λ 2 r 2 c 2 cot − λ 2 r 2 c 2 x + iy − c t α Γ ( 1 + α ) .

The 3D profile of solution (27) is depicted for t = 0.1 , α = 1 / 2 , λ = 2 , r = − 2 , c = 2 , 0 ≤ x , y ≤ 1 and 2D Figure is outlined for t = 0.1 , y = 0 , α = 1 / 2 , λ = 2 , r = − 2 , c = 2 , 0 ≤ x ≤ 1 and categorized in Figure 5.

Figure 5

The plot of solution (27) in 2D and 3D for the above values.

Case 3: Alternatively, when b = 0 , from solution (25), we accomplish the esteemed results provided below

(28) u ( ξ ) = − 1 x + iy − c t α Γ ( 1 + α ) .

The 3D outline of the solution (28) is portrayed for t = 0.1 , α = 1 / 2 , c = 2 , 0 ≤ x , y ≤ 1 and 2D Figure is outlined for t = 0.1 , y = 0 , α = 1 / 2 , c = 2 , 0 ≤ x ≤ 1 and categorized in Figure 6.

Figure 6

The plot of solution (28) in 2D and 3D.

3 The RBSOM

The solution developed for RBSOM [13] can be expressed based on the standard formalism of the nonlinear evolution equation outlined in Eqs. (1) and (2).

(29) h ′ = A h 2 − β + Bh + C h β .

When AC ≠ 0 and β = 0 , then Eq. (29) becomes the Riccati equation. On the other hand, if A ≠ 0 , C ≠ 0 , and β ≠ 1 , then the equation represents the Bernoulli equation. The values of A , B , C , and β will be determined at a later stage.

Upon plugging in expressions for u and its derivatives into Eq. (29), a set of equations in terms of these unknowns can be derived by utilizing an appropriate parameter β and equating the exponential powers of u to zero. As a result, the proposed method produces diverse solutions depending on the constants’ values and the transformation ξ = x − ct , which are determined by the aforementioned system of equations.

(30) h ( ξ ) = C 1 e ( A + B + C ) ξ , for β = 1 .

(31) h ( ξ ) = { A ( β − 1 ) ( ξ + C 1 ) } 1 / ( 1 − β ) , for β ≠ 1 , B = 0 , C = 0 .

(32) h ( ξ ) = − A B + C 1 e B ( β − 1 ) ξ 1 / ( β − 1 ) , for β ≠ 1 , B ≠ 0 , C = 0 .

(33) h ξ = − B 2 A + 4 AC − B 2 2 A × tan 1 − β 4 AC − B 2 2 ξ + C 1 1 / ( 1 − β ) , h ξ = − B 2 A + 4 AC − B 2 2 A × cot 1 − β 4 AC − B 2 2 ξ + C 1 1 / ( 1 − β ) ,

for β ≠ 1 , A ≠ 0 , and B 2 − 4 A C < 0 .

(34) h ξ = − B 2 A + B 2 − 4 AC 2 A tanh ξ + C 1 1 / ( 1 − β ) , h ξ = − B 2 A + B 2 − 4 AC 2 A × coth 1 − β B 2 − 4 AC 2 ξ + C 1 1 / ( 1 − β ) ,

for β ≠ 1 , A ≠ 0 , and B 2 − 4 AC > 0 .

(35) h ( ξ ) = 1 A ( β − 1 ) ( ξ + C 1 ) − B 2 A 1 / ( 1 − β ) ,

for β ≠ 1 , A ≠ 0 , and B 2 − 4 AC = 0 , where C 1 is an arbitrary constant.

3.1 Application

For Eq. (9) mentioned earlier and for the transformation ξ = x − ct , we found

(36) ( k 2 − 1 ) h ″ − 2 k h 3 + k 1 h = 0 .

Now, we put into effect the steps of the sum up in the method [13] mentioned above.

From Eq. (29), by direct differentiation, we attain

(37) h ″ = ab 3 − β h ( 2 − β ) + a 2 2 − β h ( 3 − 2 β ) + β c 2 h ( 2 β − 1 ) + bc β + 1 h 2 + 2 ac + b 2 h .

Injecting (37) into Eq. (36) for h ″ , with suitable chosen β by setting the coefficients of various powers of h to zero, we can derive a set of algebraic equations:

(38) A 2 − k = 0 , 3 AB = 0 , 2 AC + B 2 + k 1 = 0 , BC = 0 .

The solution of the above system of algebraic equations is as follows:

(39) A = ± k , B = 0 , C = k 1 2 A = ± k 1 2 k .

According to these values of the costants and the constructed method we will take only the solution forms (33) and (34).

Case 1. For

β = 0 , A = k , B = 0 , C = k 1 2 k or

(40) β = 0 , A = − k , B = 0 , C = − k 1 2 k .

In these two cases, the solutions are as follows:

(41) h x , t = 2 k 1 tan 2 k 1 ( x − ct + C 1 ) 2 k , h x , t = − 2 k 1 cot 2 k 1 ( x − ct + C 1 ) 2 k .

In order to rigorously explain and interpret a phenomena, a closed form wave solution and its sketch are significantly important. Therefore, the solutions are presented in the underlying 2D and 3D profiles for t = 0.1 , k = 2 , k 1 = 1 , c = 1 , C 1 = 0 , 0 ≤ x , y ≤ 1 and 2D shape is sketched for t = 0.1 , y = 0 , k = 2 , k 1 = 1 , c = 1 , C 1 = 0 , 0 ≤ x ≤ 1 and labeled in Figure 7.

Figure 7

The plot of solution (41) in 2D and 3D for the above values.

Case 2. For

A = − k , B = 0 , C = k 1 2 k ,

(42) A = k , B = 0 , C = − k 1 2 k .

In these two cases, the solutions are as follows:

(43) h ( x , t ) = − 2 k 1 tanh 2 k 1 ( x − ct + C 1 ) 2 k , h ( x , t ) = − 2 k 1 coth 2 k 1 ( x − ct + C 1 ) 2 k .

The solutions assigned in (43) are sketched in 2D and 3D profiles for t = 0.1 , k = 2 , k 1 = 1 , c = 1 , C 1 = 1 , 0 ≤ x , y ≤ 1 and 2D shape is sketched for t = 0.1 , y = 0 , k = 2 , k 1 = 1 , c = 1 , C 1 = 1 , 0 ≤ x ≤ 1 and labeled in Figure 8.

Figure 8

The plot of solution (43) in 2D and 3D figure.

3.2 Application of the TFBPM

For the wave transformation (20), the TFBPM (19) transmute to

(44) c u ′ + λ u 2 − λ r = 0 ,

From Eq. (29), after replacing h by u , we derive

(45) u ′ = A u 2 − β + Bu + C u β .

Embedding (45) in Eq. (44) for u' , with suitable assumption of β and equating the coefficients of different powers of u to zero, we attain

(46) c 1 A + λ = 0 , c 1 B = 0 , c 1 C − λ r = 0 .

The solutions of these algebraic equations are as follows:

(47) A = − λ c 1 , B = 0 , C = λ r c 1 .

According to these values of costants and the raised method, we will take only the solution forms (33) and (34).

Case 1: For β ≠ 1 , A ≠ 0 , and B 2 − 4 AC < 0 , c 1 is positive, then

(48) β = 0 , A = − λ c 1 , B = 0 , C = λ r c 1 .

Consequently, the solutions are as follows:

(49) u ξ = − − r tan − r 2 ( ξ + C ) , u ξ = − r cot − r 2 ( ξ + C ) ,

where ξ = x + iy − c 1 t α Γ ( 1 + α ) , c 1 is constant.

The solutions allocated in (49) are delineated in 2D and 3D profiles for t = 0.1 , α = 1 / 2 , r = − 2 , c = 2 , C = 1 , 0 ≤ x , y ≤ 1 and 2D shape is sketched for t = 0.1 , y = 0 , α = 1 / 2 , r = − 2 , c = 2 , C = 1 , 0 ≤ x ≤ 1 and labeled in Figure 9.

Figure 9

The plot of solution (49) in 2D and 3D for the above values.

Case 1. When β ≠ 1 , A ≠ 0 , and B 2 − 4 AC > 0 , c 1 is negative, then

(50) β = 0 , A = − λ c 1 , B = 0 , C = λ r c 1 ,

Consequently, the solutions are as follows:

(51) u ξ = r tanh r 2 ( ξ + C ) , u ξ = r coth r 2 ξ + C .

The solutions consigned in (51) are explained in 2D and 3D contours for t = 0.1 , α = 1 / 2 , r = − 2 , c = 2 , C = 1 , 0 ≤ x , y ≤ 1 and 2D shape is sketched for t = 0.1 , y = 0 , α = 1 / 2 , r = − 2 , c = 2 , C = 1 , 0 ≤ x ≤ 1 and labeled in Figure 10.

Figure 10

The plot of solution (51) in 2D and 3D shapes.

4 Bäcklund transformation of the investigated equation

If h n − 1 ( ξ ) and h n are the solutions of the investigated equation [13], then these solutions will satisfy this equation and thus it becomes

(52) h n ′ = a h n 2 − m + b h n + c h n m ,

(53) h n − 1 ′ = a h n − 1 2 − m + b h n − 1 + c h n − 1 m ,

(54) h n ′ = d h n ( ξ ) d ξ = d h n ( ξ ) d h n − 1 ( ξ ) d h n − 1 ( ξ ) d ξ = d h n ( ξ ) d h n − 1 ( ξ ) [ a h n − 1 2 − m + b h n − 1 + c h n − 1 m ] .

Now, from Eqs. (52)–(54), we obtain

(55) a h n 2 − m + b h n + c h n m = d h n ( ξ ) d h n − 1 ( ξ ) [ a h n − 1 2 − m + b h n − 1 + c h n − 1 m ] ,

i.e.,

(56) d h n ( ξ ) a h n 2 − m + b h n + c u n m = d h n − 1 ( ξ ) a h n − 1 2 − m + b h n − 1 + c h n − 1 m .

Therefore, we can easly obtain

(57) h n ( ξ ) = − c A 1 + a A 2 ( h n − 1 ( ξ ) ) 1 − m b A 1 + a A 2 + a A 1 ( h n − 1 ( ξ ) ) 1 − m 1 1 − m ,

which genrate indefinite series of solutions of Eq. (29) in terms of some arbitrary constants.

5 Conclusion

The METFM and the RBSOM have been implemented effectively to establish advanced, accurate, and wide-ranging perception of the TFBPM and the (2+1)-dimensional Zoomeron models. The achieved exact wave solutions are depicted in Figures 1–6 for the METFM and Figures 7–10 for the RBSOM. Through these figures many types of solutions such as trigonometric functions, hyperbolic functions, perfect periodic soliton solutions, singular periodic soliton solutions, and other rational soliton solutions have been detected. The established solutions are new, accurate, and impressive, and they inspire positive motivation for populations to strike a balance within societies and at the international level, ultimately contributing to peace among people. Some of these new accurate solutions agree with the previously obtained results by other authors [16,17,18] and some are better than the results available in the literature. Furthermore, the RBSOM is new technique that treat the problem in which the balance rule fails, we can locate infinite series of solutions of solitary solutions that achieved by this technique according to the Bäcklund transformation.

Acknowledgments

Research Supporting Project number (RSP2023167), King Saud University, Riyadh, Saudi Arabia.

Funding information: This Project is funded by King Saud University, Riyadh, Saudi Arabia.
Author contributions: 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.

References

[1] Shakeri E, Dehghon M. Numerical solution of a biological population model using He’s variatiional iteration method. Comput Math Appl. 2007;54:1197–207.10.1016/j.camwa.2006.12.076Search in Google Scholar

[2] Bekir A, Ozkan G. Exact solutions of nonlinear fractional differential equations by (G′/G)-expansion method. Chin Phys B. 2013;22(11).10.1088/1674-1056/22/11/110202Search in Google Scholar

[3] Gurney WS, Nisbet RM. The regulation of inhomogeneous populations. J Theor Biol. 1975;52(2):441–57.10.1016/0022-5193(75)90011-9Search in Google Scholar PubMed

[4] Khan MN, Hussain I, Ahmad I, Ahmad H. A local meshless method for the numerical solution of space‐dependent inverse heat problems. Math Methods Appl Sci. 2020. 10.1002/mma.6439.Search in Google Scholar

[5] Hashemi MS, Mirzazadeh M, Ahmad H. A reduction technique to solve the (2 + 1)-dimensional KdV equations with time local fractional derivatives. Opt Quantum Electron. 2023 Aug;55(8):721.10.1007/s11082-023-04917-3Search in Google Scholar

[6] Muhammad T, Ahmad H, Farooq U, Akgül A. Computational investigation of magnetohydrodynamics boundary of Maxwell fluid across nanoparticle-filled sheet. Al-Salam J Eng Technol. 2023 Apr 7;2(2):88–97.10.55145/ajest.2023.02.02.011Search in Google Scholar

[7] Yilmaz EU, Khodad FS, Ozkan YS, Abazari R, Abouelregal AE, Shaayesteh MT, et al. Manakov model of coupled NLS equation and its optical soliton solutions. J Ocean Eng Sci. 2022 Mar 17. 10.1016/j.joes.2022.03.005.Search in Google Scholar

[8] Akinyemi L, Rezazadeh H, Shi QH, Inc M, Khater MM, Ahmad H, et al. New optical solitons of perturbed nonlinear Schrödinger-Hirota equation with spatio-temporal dispersion. Results Phys. 2021;29:104656.10.1016/j.rinp.2021.104656Search in Google Scholar

[9] Ahmad I, Ahmad H, Inc M, Rezazadeh H, Akbar MA, Khater MM, et al. Solution of fractional-order Korteweg-de Vries and Burgers’ equations utilizing local meshless method. J Ocean Eng Sci. 2021. 10.1016/j.joes.2021.08.014.Search in Google Scholar

[10] Rezazadeh H, Ullah N, Akinyemi L, Shah A, Mirhosseini SM, Chu YM, et al. Optical soliton solutions of the generalized non-autonomous nonlinear Schrödinger equations by the new Kudryashov’s method. Results Phys. 2021;24:104179.10.1016/j.rinp.2021.104179Search in Google Scholar

[11] Ahmad I, Seadawy AR, Ahmad H, Thounthong P, Wang F. Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method. Int J Nonlinear Sci Numer Simul. 2021;23(1):115–21.10.1515/ijnsns-2020-0166Search in Google Scholar

[12] Bekir A, Zahran EMH. Exact and numerical solutions for the nano-soliton of ionic wave propagating through microtubules in living cells. Pramana-J Phys. 2021;95:158.10.1007/s12043-021-02177-ySearch in Google Scholar

[13] Zahran EHM, Ahmed H, Askar S, Botmart T, Shehata MSM. Dark-soliton behaviors arising from a coupled nonlinear Schrödinger system. Results Phys. 2022;36:105459.10.1016/j.rinp.2022.105459Search in Google Scholar

[14] Wang F, Zhang J, Ahmad I, Farooq A, Ahmad H. A novel meshfree strategy for a viscous wave equation with variable coefficients. Front Phys. 2021;9:359.10.3389/fphy.2021.701512Search in Google Scholar

[15] Ahmad H, Khan TA, Ahmad I, Stanimirović PS, Chu Y-M. A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations. Results Phys. 2020;19:103462.10.1016/j.rinp.2020.103462Search in Google Scholar

[16] Liu Y, Li Z, Zhang Y. Homotopy perturbation method to fractional biological population equation. Fract Diff Cal. 2011;1:117–24.10.7153/fdc-01-07Search in Google Scholar

[17] Arafa AA, Rida SZ, Mohamed H. Homotopy analysis method for solving biological population model. Commun Theor Phys. 2011;56:7972800.10.1088/0253-6102/56/5/01Search in Google Scholar

[18] Lu YGH. Ider estimate of solutions of biological population equations. Appl Math Lett. 2000;13:123e6.10.1016/S0893-9659(00)00066-5Search in Google Scholar

[19] Wang F, Zheng K, Ahmad I, Ahmad H. Gaussian radial basis functions method for linear and nonlinear convection–diffusion models in physical phenomena. Open Phys. 2021;19(1):69–76.10.1515/phys-2021-0011Search in Google Scholar

[20] Shah NA, Ahmad I, Omar B, Abouelregal AE, Ahmad H. Multistage optimal homotopy asymptotic method for the nonlinear Riccati ordinary differential equation in nonlinear physics. Appl Math Inf Sci. 2020;14(6):1–7.10.18576/amis/140608Search in Google Scholar

[21] Shakeel M, Hussain I, Ahmad H, Ahmad I, Thounthong P, Zhang YF. Meshless technique for the solution of time-fractional partial differential equations having real-world applications. J Funct Spaces. 2020;1–17. 10.1155/2020/8898309.Search in Google Scholar

[22] Khan MN, Ahmad I, Akgül A, Ahmad H, Thounthong P. Numerical solution of time-fractional coupled Korteweg–de Vries and Klein–Gordon equations by local meshless method. Pramana-J Phys. 2021;95(1):1–3.10.1007/s12043-020-02025-5Search in Google Scholar

[23] Wang KJ, Si J. Diverse optical solitons to the complex Ginzburg–Landau equation with Kerr law nonlinearity in the nonlinear optical fiber. Eur Phys J Plus. 2023;138(3):187.10.1140/epjp/s13360-023-03804-wSearch in Google Scholar

[24] Wang KJ, Liu JH. Diverse optical solitons to the nonlinear Schrödinger equation via two novel techniques. Eur Phys J Plus. 2023;138:74.10.1140/epjp/s13360-023-03710-1Search in Google Scholar

[25] Wang KJ. Diverse wave structures to the modified Benjamin–Bona–Mahony equation in the optical illusions field. Mod Phys Lett B. 2023;37(11):2350012.10.1142/S0217984923500124Search in Google Scholar

[26] Wang KJ. The fractal active low-pass filter within the local fractional derivative on the Cantor set. COMPEL - Int J Comput Math Electr Electron Eng. 2023.10.1108/COMPEL-09-2022-0326Search in Google Scholar

Received: 2023-02-09

Revised: 2023-05-07

Accepted: 2023-05-07

Published Online: 2023-07-29

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

Articles in the same Issue

https://doi.org/10.1515/phys-2022-0252

Keywords for this article

modified extended tanh-function method; Ricatti–Bernoulli sub-ODE method; time-fraction biological population model; (2 + 1)-dimensional Zoomeron model; traveling wave solutions

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