New Solitary Wave Solutions of the Fisher Equation

In this paper, we use Riccati equation to find new solitary wave solutions of Fisher equation, which describes the process of interaction between diffusion and reaction. It is of great importance to comprehend the equation to solve the problems in chemical kinetics and population dynamics. We resolve the Ricatti equation through diverse function transformation and many types of exact solutions are obtained. Then it is used as an auxiliary equation to solve Fisher equation. In the process, we select different coefficients in the Racatti equation, as a result, abundant solitary wave solutions are obtained, some of which haven’t been found in other documents yet. Moreover, these solutions we got in this paper will be favorable for understanding the Fisher equation.

• Nonlinear Evolution Equations
• Solitary Wave
• Soliton
• Fisher Equation
• Riccati Equation

1. Introduction

At present, the study of solitary waves and solitons is a hot topic in nonlinear physics, from chemical kinetics, population dynamics, plasma, optics, and biology, etc [1] - [6]. Through physical mechanisms and facts from experiments, we learned that solitons exist. Under some reasonable conditions, we can establish mathematics models of most physics’ laws, and many of these nonlinear identification studies can give the credit to the nonlinear evolution equations eventually. Hence, finding the exact solutions, such as solitary-wave solutions, is very important for exploring the relevant nonlinear problems and understanding their characteristics, which can help scientists apply them into practical researches and studies to solve the problems in chemical kinetics and population dynamics. A large number of scientists have devoted themselves to the solutions in the latest centuries and as a result, many effective and forceful methods have been put forward in other documents to get the exact solutions of Fisher equation, for example, tanh-sech method and the extended tanh-coth method [7] [8], F-expansion method [9] [10], Jacobi elliptic function expansion method [11] [12], auxiliary equation method [13] [14] [15] [16], and so on. However, even some of them are powerful and effective, not all of them can be suitable for exploring the exact solutions of Fisher equation [16], which describes the interaction process between diffusion and reaction. This equation is encountered in chemical kinetics and population dynamics, including problems such as the nonlinear evolution of the population in the one-dimensional conventional neutron population in nuclear reactions. In Ref. [17], Fisher equation is treated with extended tanh method and many new solitary wave solutions were obtained. In this paper, we will use Riccati equation as an auxiliary equation to solve Fisher equation, so that many new results are obtained. The method used in this paper, on the one hand, can simplify the solving process of the nonlinear equation, on the other hand, obtains a new solitary wave solution of the Fisher equation.

The framework of the paper is as follows: Section 2 introduces the construction of abundant exact solutions of Riccati equation; Section 3 establishes how to operate this method for producing new solitary wave solutions of Fisher equation; Section 4 is the conclusion.

2. Abundant Exact Solutions of Riccati Equation

The Riccati equation method is very simple but very effective. Hence, it is an ideal method to solve constant coefficient, variable coefficient and high-dimensional nonlinear evolution equations. In the paper, it first comes to our mind that we can use Racatti equation to solve the problems in the following form:

f ′ ( ξ ) = p 1 f 2 ( ξ ) + q 1 (1)

where p₁ and q₁ are constants and can be determined later. To find out new exact solutions of Equation (1), a new auxiliary function g ( ξ ) is introduced, which satisfies the following form

[ g ′ ( ξ ) ] 2 = p 2 g 2 ( ξ ) + q 2 (2)

where p₂ and q₂ are constants. Equation (2) has the following hyperbolic function solution

g 1 ( ξ ) = sinh ( ξ ) , ( p 2 = 1 , q 2 = 1 ) (3)

g 2 ( ξ ) = cosh ( ξ ) , ( p 2 = 1 , q 2 = − 1 ) (4)

g 3 ( ξ ) = cosh 2 ( ξ ) − 1 2 = sinh 2 ( ξ ) + 1 2 , ( p 2 = 4 , q 2 = − 1 ) (5)

Then we assume f ( ξ ) and g ( ξ ) have the following formal solution

f ( ξ ) = g ′ ( ξ ) g ( ξ ) + r (6)

where r is a constant. Substituting Equation (6) into Equation (1) and using Equation (2), we can obtain

{ p 1 p 2 + q 1 = 0 , p 2 r = 2 q 1 r , − q 2 = p 1 q 2 + q 1 r 2 . (7)

Solving this system, we can obtain

{ r = 0 , p 1 = − 1 , q 1 = p 2 , or { r = ± − q 2 p 2 , p 1 = − 1 2 , q 1 = p 2 2 . (8)

So, we have the following exact solutions of Equation (1)

f 1 ( ξ ) = cosh ( ξ ) sinh ( ξ ) , ( p 1 = − 1 , q 1 = 1 ) (9)

f 2 ( ξ ) = sinh ( ξ ) cosh ( ξ ) , ( p 1 = − 1 , q 1 = 1 ) (10)

f 3 ( ξ ) = 2 sinh ( ξ ) cosh ( ξ ) cosh 2 ( ξ ) − 1 2 , ( p 1 = − 1 , q 1 = 4 ) (11)

f 4 ( ξ ) = sinh ( ξ ) cosh ( ξ ) + ε , ( p 1 = − 1 2 , q 1 = 1 2 , ε 2 = 1 ) (12)

f 5 ( ξ ) = cosh ( ξ ) sinh ( ξ ) + ε , ( p 1 = − 1 2 , q 1 = 1 2 , ε 2 = − 1 ) (13)

Next, we use the following another formal solution to solve Equation (1)

f ( ξ ) = g ( ξ ) g ′ ( ξ ) g 2 ( ξ ) + r (14)

where r ≠ 0 . Substituting Equation (14) into Equation (1) and using Equation (2), we can obtain

{ p 1 p 2 + q 1 = 0 , − q 2 + 2 p 2 r = p 1 q 2 + 2 q 1 r , r q 2 = q 1 r 2 . (15)

Solving this system, we can obtain

{ r = q 2 2 p 2 , p 1 = − 2 , q 1 = 2 p 2 . (16)

So, we can have the following exact solutions

f 6 ( ξ ) = 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ ) [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8 , ( p 1 = − 2 , q 1 = 8 ) (17)

It is easy to know that h ( ξ ) = 1 / f ( ξ ) can also satisfy Equation (1) in the condition of p ′ 1 = − q 1 , q ′ 1 = − p 1 . Equations (9) and (10) are a pair of solutions on this condition. Therefore, the following equations are also the solutions of Equation (1)

f 7 ( ξ ) = cosh 2 ( ξ ) − 1 2 2 sinh ( ξ ) cosh ( ξ ) , ( p 1 = − 4 , q 1 = 1 ) (18)

f 8 ( ξ ) = cosh ( ξ ) + ε sinh ( ξ ) , ( p 1 = − 1 2 , q 1 = 1 2 , ε 2 = 1 ) (19)

f 9 ( ξ ) = sinh ( ξ ) + ε cosh ( ξ ) , ( p 1 = − 1 2 , q 1 = 1 2 , ε 2 = − 1 ) (20)

f 10 ( ξ ) = [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ ) , ( p 1 = − 8 , q 1 = 2 ) (21)

Equations (6) and (10) are the new types of exact solutions of Equation (1), which are rarely found in the other documents. Then, we use the Equation (1) and its solutions (9)-(13), (17) and (18)-(21) to solve the Fisher equation, and the solving process can be greatly simplified.

3. Application of the Method

The following Fisher equation [16] [17] is considered

u t − u x x − u ( 1 − u ) = 0 (22)

Then suppose Equation (22) has the traveling wave solution

u ( x , t ) = u ( ξ ) , ξ = μ x + c t (23)

where μ and c are travelling wave parameters. Substituting the traveling wave equations into Equation (22), the following equation can be obtained

c u ′ − μ 2 u ″ − u ( 1 − u ) = 0 (24)

We assume that Equation (24) has the following formal solution

u ( ξ ) = ∑ i = 1 0 a i f i ( ξ ) + ∑ i = 1 n b i f − i ( ξ ) (25)

where a i and b i are constants to be determined and f i ( ξ ) is the solutions of Equation (1) and n can be determined by the homogeneous balance method. In the Equation (24), it is easy to know n = 2, so that the solution can be expressed as

u ( ξ ) = a 0 + a 1 f ( ξ ) + a 2 f 2 ( ξ ) + b 1 f − 1 ( ξ ) + b 2 f − 2 ( ξ ) (26)

We bring above equation into Equation (24) and use Equation (1), resulting in a series of equations a set of algebraic equations about a₀, a₁, a₂, b₁, b₂, μ and c. Then we collect all the terms with the same power of f ( ξ ) , and set each coefficient to zero. Finally, we can obtain

Case 1 a 0 = 1 4 , a 1 = ± 1 2 − p 1 q 1 , a 2 = − p 1 4 q 1 , b 1 = b 2 = 0 , k 2 = − 1 24 p 1 q 1 , c = ∓ 5 12 − p 1 q 1 (27)

Case 2 a 0 = 1 4 , a 1 = a 2 = 0 , b 1 = ± 1 2 − q 1 p 1 , b 2 = − q 1 4 p 1 , k 2 = − 1 24 p 1 q 1 , c = ± 5 12 − p 1 q 1 (28)

Case 3 a 0 = 3 8 , a 1 = ± 1 4 − p 1 q 1 , a 2 = − p 1 16 q 1 , b 1 = ∓ 1 4 − q 1 p 1 , b 2 = − q 1 16 p 1 , k 2 = − 1 96 p 1 q 1 , c = ∓ 5 24 − p 1 q 1 (29)

According to Case 1, we have the following solitary wave solutions of the Fisher equation

u 1 ( ξ ) = 1 4 ± 1 2 coth ( ξ ) + 1 4 coth 2 ( ξ ) (30)

where ξ = μ x + c t , μ = 1 2 6 ε , c = ∓ 5 12 , ε 2 = 1 . Figure 1(a) shows the three-dimensional diagrams of Equation (30), which represents the bright solitary wave solution. Figure 1(b) shows that the amplitude and velocity of this solitary wave remain unchanged during propagation.

u 2 ( ξ ) = 1 4 ± 1 2 tanh ( ξ ) + 1 4 tanh 2 ( ξ ) (31)

where ξ = μ x + c t , μ = 1 2 6 ε , c = ∓ 5 12 , ε 2 = 1 . Figure 2 shows the

three-dimensional and two-dimensional plots diagrams of Equation (31), which

represents the kink solitary wave solution.

u 3 ( ξ ) = 1 4 ± 1 2 sinh ( ξ ) cosh ( ξ ) cosh 2 ( ξ ) − 1 2 + 1 4 [ sinh ( ξ ) cosh ( ξ ) cosh 2 ( ξ ) − 1 2 ] 2 (32)

where ξ = μ x + c t , μ = 1 4 6 ε , c = ∓ 5 24 , ε 2 = 1 . Figure 3 shows the three-dimensional and two-dimensional plots diagrams of Equation (32), which also represents the kink solitary wave solution.

u 4 ( ξ ) = 1 4 ± 1 2 sinh ( ξ ) cosh ( ξ ) + ε + 1 4 [ sinh ( ξ ) cosh ( ξ ) + ε ] 2 (33)

where ξ = μ x + c t , μ = 1 6 λ , c = ∓ 5 6 , ε 2 = 1 , λ 2 = 1 . Figure 4 shows the three-dimensional and two-dimensional plots diagrams of Equation (33), which also represents the kink solitary wave solution.

u 5 ( ξ ) = 1 4 ± 1 2 cosh ( ξ ) sinh ( ξ ) + ε + 1 4 [ cosh ( ξ ) sinh ( ξ ) + ε ] 2 (34)

where ξ = μ x + c t , μ = 1 6 λ , c = ∓ 5 6 , ε 2 = − 1 , λ 2 = 1 . The solution of Equation (34) represents the traveling wave solutions of Equation (24) in complex space.

u 6 ( ξ ) = 1 4 ± 1 4 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ ) [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8     + 1 16 { 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ ) [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8 } 2 (35)

where ξ = μ x + c t , μ = 1 8 6 ε , c = ∓ 5 48 , ε 2 = 1 . Figure 5 shows the three-dimensional and two-dimensional plots diagrams of Equation (35), which represents the kink solitary wave solution.

u 7 ( ξ ) = 1 4 ± 1 2 cosh 2 ( ξ ) − 1 2 sinh ( ξ ) cosh ( ξ ) + 1 4 [ cosh 2 ( ξ ) − 1 2 sinh ( ξ ) cosh ( ξ ) ] 2 (36)

where ξ = μ x + c t , μ = 1 4 6 ε , c = ∓ 5 24 , ε 2 = 1 . Figure 6 shows the three-dimensional and two-dimensional plots diagrams of Equation (36), which represents the bright solitary wave solution.

u 8 ( ξ ) = 1 4 ± 1 2 cosh ( ξ ) + ε sinh ( ξ ) + 1 4 [ cosh ( ξ ) + ε sinh ( ξ ) ] 2 (37)

where ξ = μ x + c t , μ = 1 6 λ , c = ∓ 5 6 , ε 2 = 1 , λ 2 = 1 . Figure 7 shows the three-dimensional and two-dimensional plots diagrams of Equation (37), which represents the bright solitary wave solution.

u 9 ( ξ ) = 1 4 ± 1 2 sinh ( ξ ) + ε cosh ( ξ ) + 1 4 [ sinh ( ξ ) + ε cosh ( ξ ) ] 2 (38)

where ξ = μ x + c t , μ = 1 6 λ , c = ∓ 5 6 , ε 2 = − 1 , λ 2 = 1 . The solution of Equation (38) represents the traveling wave solutions of Equation (24) in complex space.

u 10 ( ξ ) = 1 4 ± [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ )     + { [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ ) } 2 (39)

where ξ = μ x + c t , μ = 1 8 6 ε , c = ∓ 5 48 , ε 2 = 1 . Figure 8 shows the three-dimensional and two-dimensional plots diagrams of Equation (39), which represents the bright solitary wave solution.

Because the solutions f 1 ( ξ ) − f 10 ( ξ ) of Equation (1) contain that corresponding to h ( ξ ) = 1 / f ( ξ ) , the solitary wave solutions of the Fisher equation in Case 2 are the same as in Case1. Corresponding to case 3, we express the solitary wave solution of Fisher equation as

u 11 ( ξ ) = 3 8 ± 1 4 coth ( ξ ) + 1 16 coth 2 ( ξ ) ∓ 1 4 tanh ( ξ ) + 1 16 tanh 2 ( ξ ) (40)

where ξ = μ x + c t , μ = 1 4 6 ε , c = ∓ 5 24 , ε 2 = 1 . Figure 9 shows the three-dimensional and two-dimensional plots diagrams of Equation (40), which represents the bright solitary wave solution.

u 12 ( ξ ) = 3 8 ± 1 4 sinh ( ξ ) cosh ( ξ ) cosh 2 ( ξ ) − 1 2 + 1 16 [ sinh ( ξ ) cosh ( ξ ) cosh 2 ( ξ ) − 1 2 ] 2     ∓ 1 4 cosh 2 ( ξ ) − 1 2 sinh ( ξ ) cosh ( ξ ) + 1 16 [ cosh 2 ( ξ ) − 1 2 sinh ( ξ ) cosh ( ξ ) ] 2 (41)

where ξ = μ x + c t , μ = 1 8 6 ε , c = ∓ 5 48 , ε 2 = 1 . Figure 10 shows the

three-dimensional and two-dimensional plots diagrams of Equation (41), which represents the bright solitary wave solution.

u 13 ( ξ ) = 3 8 ± 1 4 cosh ( ξ ) + ε sinh ( ξ ) + 1 16 [ cosh ( ξ ) + ε sinh ( ξ ) ] 2     ∓ 1 4 sinh ( ξ ) cosh ( ξ ) + ε + 1 16 [ sinh ( ξ ) cosh ( ξ ) + ε ] 2 (42)

where ξ = μ x + c t , μ = 1 2 6 λ , c = ∓ 5 12 , ε 2 = 1 , λ 2 = 1 . Figure 11 shows the three-dimensional and two-dimensional plots diagrams of Equation (42), which represents the bright solitary wave solution.

u 14 ( ξ ) = 3 8 ± 1 4 cosh ( ξ ) sinh ( ξ ) + ε + 1 16 [ cosh ( ξ ) sinh ( ξ ) + ε ] 2     ∓ 1 4 sinh ( ξ ) + ε cosh ( ξ ) + 1 16 [ sinh ( ξ ) + ε cosh ( ξ ) ] 2 (43)

where ξ = μ x + c t , μ = 1 2 6 ε , c = ∓ 5 12 , ε 2 = − 1 . The solution of Equation (43)

represents the traveling wave solutions of Equation (24) in complex space.

u 15 ( ξ ) = 3 8 ± 1 2 [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ )     + 1 4 [ [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ ) ] 2     ∓ 1 8 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ ) [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8     + 1 64 [ 2 sinh ( ξ ) cosh 3 ( ξ ) − sinh ( ξ ) cosh ( ξ ) [ cosh 2 ( ξ ) − 1 2 ] 2 − 1 8 ] 2 (44)

where ξ = μ x + c t , μ = 1 16 6 ε , c = ∓ 5 96 , ε 2 = − 1 . Figure 12 shows the three-dimensional and two-dimensional plots diagrams of Equation (44), which represents the bright solitary wave solution.

4. Conclusion

In this paper, we use Riccati equation to find new solitary wave solutions of Fisher equation. Riccati equation is solved by using of two types of function transformation Equation (6) and Equation (14), so that many new exact solutions are obtained. With the formal solution of Equation (25), we have constructed abundant and new solitary wave solutions for the Fisher equation. The solitary wave solutions expressed by Equations (39), (41)-(44) are rarely found in other documents. The numerical images show that although the new expressions of many solutions are different, the solitary waves represented by them, including amplitude, wave velocity and space-time width, are the same. This method can greatly simplify the calculation process, especially suitable for solving more complex nonlinear systems.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

1. Guo, H.D., Xia, T.C. and Hu, B.B. (2020) High-Order Lumps, High-Order Breathers and Hybrid Solutions for an Extended (3+1)-Dimensional Jimbo-Miwa Equation in Fluid Dynamics. Nonlinear Dynamics, 100, 601-614. https://doi.org/10.1007/s11071-020-05514-9

2. Lan, Z.Z. and Guo, B.L. (2020) Nonlinear Waves Behaviors for a Coupled Generalized Nonlinear Schrodinger-Boussinesq System in a Homogeneous Magnetized Plasma. Nonlinear Dynamics, 100, 3771-3784. https://doi.org/10.1007/s11071-020-05716-1

3. Biswas, A., Ekici, M., Sonmezoglu, A. and Belic, M.R. (2019) Solitons in Optical Fiber Bragg Gratings with Dispersive Reflectivity by Extended Trial Function Method. Optik, 182, 88-94. https://doi.org/10.1016/j.ijleo.2018.12.156

4. Seadawy, A.R., Lu, D.C., Nasreen, N. and Nasreen, S. (2019) Structure of Optical Solitons of Resonant Schrodinger Equation with Quadratic Cubic Nonlinearity and Modulation Instability Analysis. Physica A, 534, Article ID: 122155. https://doi.org/10.1016/j.physa.2019.122155

5. Abdoud, M.A., Owyed, S., Abdel-Aty, A., Raffan, B.M. and Abdel-Khalek, S. (2020) Optical Soliton Solutions for a Space-Time Fractional Perturbed Nonlinear Schrodinger Equation Arising in Quantum Physics. Results in Physics, 16, Article ID: 102895. https://doi.org/10.1016/j.rinp.2019.102895

6. Peng, W.Q., Tian, S.F. and Zhang, T.T. (2019) Dynamics of the Soliton Waves, Breather Waves, and Rogue Waves to the Cylindrical Kadomtsev-Petviashvili Equation in Pair-Ion-Electron Plasma. Physics of Fluids, 31, Article ID: 102107. https://doi.org/10.1063/1.5116231

7. Seadawy, A.R. (2014) Stability Analysis for Zakharov-Kuznetsov Equation of Weakly Nonlinear Ion-Acoustic Waves in a Plasma. Computers & Mathematics with Applications, 67, 172-180. https://doi.org/10.1016/j.camwa.2013.11.001

8. Seadawy, A.R. (2014) Stability Analysis for Two-Dimensional Ion-Acoustic Waves in Quantum Plasmas. Physics of Plasmas, 21, Article ID: 052107. https://doi.org/10.1063/1.4875987

9. Liu, J.B. and Yang, K.Q. (2004) The Extended F-Expansion Method and Exact Solutions of Nonlinear PDEs. Chaos Solitons Fractals, 22, 111-121. https://doi.org/10.1016/j.chaos.2003.12.069

10. Zhang, S. (2007) Application of Exp-Function Method to a KdV Equation with Variable Coefficients. Physics Letters A, 365, 448-453. https://doi.org/10.1016/j.physleta.2007.02.004

11. Liu, S.K., Fu, Z.T., Liu, S.D. and Zhao, Q. (2001) Jacobi Elliptic Function Expansion Method and Periodic Wave Solutions of Nonlinear Wave Equations. Physics Letters A, 289, 69-74. https://doi.org/10.1016/S0375-9601(01)00580-1

12. Fu, Z.T., Liu, S.K., Liu, S.D. and Zhao, Q. (2001) New Jacobi Elliptic Function Expansion and New Periodic Solutions of Nonlinear Wave Equations. Physics Letters A, 290, 72-76. https://doi.org/10.1016/S0375-9601(01)00644-2

13. Wu, G., Han, J., Zhang, W. and Zhang, M. (2007) New Periodic Wave Solutions to Nonlinear Evolution Equations by the Extended Mapping Method. Physica D—Nonlinear Phenomena, 229, 116-122. https://doi.org/10.1016/j.physd.2007.03.015

14. Sirendaoreji and Sun, J. (2003) Auxiliary Equation Method for Solving Nonlinear Partial Differential Equations. Physics Letters A, 309, 387-396. https://doi.org/10.1016/S0375-9601(03)00196-8

15. Sirendaoreji (2004) New Exact Travelling Wave Solutions for the Kawahara and Modified Kawahara Equations. Chaos Solitons Fractals, 19, 147-150. https://doi.org/10.1016/S0960-0779(03)00102-4

16. Liu, Y. and Wu, G. (2021) Using a New Auxiliary Equation to Construct Abundant Solutions for Nonlinear Evolution Equations. Journal of Applied Mathematics and Physics, 9, 3155-3164. https://doi.org/10.4236/jamp.2021.912206

17. Wazwaz, A.M. (2007) The Extended Tanh Method for Abundant Solitary Wave Solutions of Nonlinear Wave Equations. Applied Mathematics and Computation, 187, 1131-1142. https://doi.org/10.1016/j.amc.2006.09.013