Exploring New Traveling Wave Solutions for the Spatiotemporal Evolution of a Special Reaction–Diffusion Equation by Extended Riccati Equation Method

Abstract

In this work, we aim to explore new exact traveling wave solutions for the reaction–diffusion equation, which describes complex nonlinear phenomena such as cell growth and chemical reactions in nature. Obtaining exact solutions to this equation is crucial for understanding aspects such as reaction activity and the diffusion coefficient. We solve the reaction–diffusion equation by using the Riccati equation as an auxiliary equation. By controlling the parameters in the Riccati equation, we obtained a large number of traveling wave solutions, many of which were not formerly recorded in other documents. Numerical simulations demonstrate the evolution of various traveling waves of the reaction–diffusion equation in time and space. These rich exact solutions and wave phenomena help to expand our knowledge of this equation.

1. Introduction

At present, the study of the nonlinear wave phenomenon in dynamic systems is a topic of high concern, covering multiple fields such as chemical dynamics, biological sciences, population dynamics, plasmas, and optics [1,2,3,4,5,6]. In specific research fields, under reasonable assumptions, various mathematical models have been constructed to expound the control laws of related nonlinear wave phenomena. Among the various mathematical models, nonlinear evolution equations (NLEEs) play an essential role in understanding nonlinear dynamic characteristics. Therefore, finding accurate solutions to NLEEs, such as solitary wave solutions, breather solutions, lump wave solutions, etc., is particularly important for understanding related nonlinear wave phenomena and dynamic behaviors.

Over the past few decades, many scientists have worked to obtain exact solutions to NLEEs. In previous efforts, many powerful research methods have been created [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. However, despite the success of these methods in their application, there are still many new methods and new exact solutions to be explored. In this paper, we consider revealing the new traveling solutions for the following special reaction–diffusion equation [24,25]:

$$u_t-{{(u}^2)}_{xx}-u\left(1-u\right)=0,$$

(1)

which was originally used to describe the diffusion process of matter in space, but with research by scientists, its application fields have become increasingly broad. It can also be used to describe chemical reactions, the development process of various coastal shallow water biological communities, the diffusion process of air pollution, and the diffusion process of drugs in the human body. This type of reaction–diffusion equation is a special case of the general reaction–diffusion equation [26] under the conditions of diffusion coefficient D = 2u and other terms = u(1 − u). In Ref. [24], an extended tanh method was used to deal with Equation (1), and several new traveling wave solutions were discovered. Reference [25] obtained some new solitary and traveling wave solutions for the reaction–diffusion equation with the quadratic and cubic nonlinearities by using the auxiliary equation method. The method described in Refs [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23] is also effective and convenient for solving Equation (1). However, these methods still need improvement in constructing new exact solutions for complex structures. The Riccati equation is widely used as an auxiliary equation to solve NLEEs due to its very simple structure, symmetry, and constructability [27,28,29,30]. Therefore, this article proposes using a Riccati equation as an auxiliary equation to manipulate Equation (1), providing the possibility for novel discoveries. The method proposed in this article can not only simplify the solution process of NLEEs, but also reveal many new traveling wave phenomena with complex structures.

The structure of this paper is as follows: Section 2 introduces how to construct exact solutions to the Ricatti equation using two kinds of new methods. Section 3 expounds on how to apply this method to deal with the reaction–diffusion equation, obtain a large number of new traveling wave solutions, and provide intuitive images of nonlinear wave evolution. Finally, we present the conclusion of this article.

2. Processing Techniques of the Riccati Equation

Due to its very simple structure, the Riccati equation is an ideal auxiliary equation for solving NLEEs. Therefore, it is often used to solve the constant coefficient, variable coefficient, and high-dimensional nonlinear evolution equations. In this paper, we first use the Ricatti equation in the following form [27,28,29,30]:

$$f^{\prime }\left(\xi \right)={p_{1}f}^2\left(\xi \right)+q_1,$$

(2)

where p₁ and q₁ are the coefficients of the equation. Equation (2) has well-known solutions tanh(ξ) and coth(ξ) when p₁ = −1 and q₁ = 1. Directly applying this equation or its deformation as an auxiliary equation to solve NLEEs is widely used, and relevant research can be found in Refs [17,27,28,29,30]. The solution of Equation (2) has also been extended to a series of trigonometric and hyperbolic functions. In fact, as a nonlinear equation, the solution to Equation (2) is much better than that. In order to solve Equation (2) and explore its new exact solutions, we introduce another auxiliary equation that satisfies the following form

$${{[g}^{\prime }\left(\xi \right)]}^2={p_{2}g}^2\left(\xi \right)+q_2,$$

(3)

where p₂ and q₂ are the coefficients of the equation. Equation (3) has the following hyperbolic function solutions:

$$g_1\left(\xi \right)=sinh(\xi ), (p_2=1,q_2=1),$$

(4)

$$g_2\left(\xi \right)=cosh(\xi ), (p_2=1,q_2=-1),$$

(5)

$$g_3\left(\xi \right)={\mathit{cosh}}^2\left(\xi \right)-{\displaystyle \frac{1}{2}}={\mathit{sinh}}^2\left(\xi \right)+{\displaystyle \frac{1}{2}}, (p_2=4,q_2=-1),$$

(6)

$${\mathrm{g}}_4\left(\mathsf{\xi }\right)=\sqrt{cosh(\xi )+\epsilon }, (p_2={\displaystyle \frac{1}{4}},q_2=-{\displaystyle \frac{1}{2}}\epsilon ,{\epsilon }^2=1),$$

(7)

Among these solutions, Equations (6) and (7) have rarely been reported in other studies. Equation (3) cannot be directly used as an auxiliary equation to solve Equation (1), as the order of g(ξ) is the same as that of g^′(ξ), which would result in Equation (1) having no solution. To apply Equation (3) to solve Equation (1), it is necessary to make certain transformations. In this article, we first apply the known solutions of Equation (3) to solve Equation (2), and then use Equation (2) as an auxiliary equation to solve Equation (1). We first assume that $f\left(\xi \right)$ and $g\left(\xi \right)$ satisfy the following form of relationship

$$f\left(\xi \right)={\displaystyle \frac{[g\left(\xi \right)+r]g^{\prime }\left(\xi \right)}{g^2\left(\xi \right)+kg\left(\xi \right)+\delta }},$$

(8)

where r, k, and δ are pending constants. If r = 0, k = 0, and δ = 0, Equation (8) can be transformed into the $G^{\prime }$/$G$ expansion method described in Ref. [23]. By substituting Equation (8) into Equation (2) and using Equation (3) and its derivation, we can obtain a series of algebraic equations about $g\left(\xi \right)$, k, δ, r, p₁, and q₁. Then, we collect all terms with the same power of $g\left(\xi \right)$ and set each coefficient to zero. We can obtain the following solutions

• Group 1

$$\left\{\begin{array}{l}r=0,\\ k=0,\\ \delta =0,\\ p_1=-1,\\ q_1{=p}_2.\end{array}\right.$$

(9)

• Group 2

$$\left\{\begin{array}{l}r=0,\\ k=\pm \sqrt{-{\displaystyle \frac{q_2}{p_2}}},\\ \delta =0,\\ p_1=-{\displaystyle \frac{1}{2}},\\ q_1={\displaystyle \frac{p_2}{2}}.\end{array}\right.$$

(10)

• Group 3

$$\left\{\begin{array}{l}r=0,\\ k=0,\\ \delta ={\displaystyle \frac{q_2}{p_2}},\\ p_1=-1,\\ q_1=p_2.\end{array}\right.$$

(11)

• Group 4

$$\left\{\begin{array}{l}r=\pm {\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{q_2}{p_2}}},\\ k=\mp {\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{q_2}{p_2}}},\\ \delta ={\displaystyle \frac{q_2}{{2p}_2}},\\ p_1=-{\displaystyle \frac{1}{2}},\\ q_1={\displaystyle \frac{p_2}{2}}.\end{array}\right.$$

(12)

Therefore, we have the exact solutions for Equation (2) as follows:

$$f_1\left(\xi \right)={\displaystyle \frac{cosh(\xi )}{sinh(\xi )}}, (p_1=-1,q_1=1),$$

(13)

$$f_2\left(\xi \right)={\displaystyle \frac{sinh(\xi )}{cosh(\xi )}}, (p_1=-1,q_1=1),$$

(14)

$$f_3\left(\xi \right)={\displaystyle \frac{2sinh(\xi )cosh(\xi )}{{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}}}, (p_1=-1,q_1=4),$$

(15)

$$f_4\left(\xi \right)={\displaystyle \frac{sinh(\xi )}{cosh(\xi )+\epsilon }}, (p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{2}}, {\epsilon }^2=1),$$

(16)

$$f_5\left(\xi \right)={\displaystyle \frac{\frac{1}{2}sinh(\xi )}{cosh(\xi )+1\pm \sqrt{2}\sqrt{cosh(\xi )+1}}}, (p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{8}}),$$

(17)

$$f_6\left(\xi \right)={\displaystyle \frac{2sinh\left(\xi \right){cosh}^3\left(\xi \right)-sinh(\xi )cosh(\xi )}{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2-\frac{1}{8}}}, (p_1=-2,q_1=8),$$

(18)

$$f_7\left(\xi \right)={\displaystyle \frac{[cosh(\xi )\pm \frac{1}{2}]sinh(\xi )}{{cosh}^2\left(\xi \right)\mp \frac{1}{2}cosh(\xi )-\frac{1}{2}}}, (p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{2}}),$$

(19)

$$f_8\left(\xi \right)={\displaystyle \frac{2[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}\pm \frac{1}{4}]sinh(\xi )cosh(\xi )}{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2\mp \frac{1}{4}[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]-\frac{1}{8}}}, (p_1=-{\displaystyle \frac{1}{2}},q_1=2),$$

(20)

$$f_9\left(\xi \right)={\displaystyle \frac{\frac{1}{2}[\sqrt{cosh(\xi )+1}\pm \frac{1}{2}\sqrt{2}]\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}{\sqrt{cosh(\xi )+1}}}{cosh(\xi )\mp \frac{1}{2}\sqrt{2}\sqrt{cosh(\xi )+1}}}, (p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{8}}).$$

(21)

In these solutions, we discarded the meaningless solutions of Equation (8) in the imaginary space. Next, we use another form of the relationship between $f\left(\xi \right)$ and $g\left(\xi \right)$ to solve Equation (2) as follows:

$$f\left(\xi \right)={\displaystyle \frac{[g\left(\xi \right)+r]g^{\prime }\left(\xi \right)}{g^2\left(\xi \right)+k{g}^{\prime }\left(\xi \right)+\delta }},$$

(22)

where r, k, and δ are pending constants. If k = 0, Equation (22) degenerates into a part of Equation (8). Therefore, we consider the case where k $\ne$ 0. By applying the same method as described above and solving Equations (2), (3) and (22), we have

$$\left\{\begin{array}{l}k=\pm \sqrt{{\displaystyle \frac{{r^2{p}_2+q}_2}{{p_2}^2}}},\\ \delta ={\displaystyle \frac{q_2}{p_2}},\\ p_1=-{\displaystyle \frac{1}{2}},\\ q_1={\displaystyle \frac{p_2}{2}}.\end{array}\right.$$

(23)

where r is an arbitrary constant. However, to make them meaningful, there must be ${r^2{p}_2+q}_2$ > 0. Therefore, the following exact solutions of Equation (2) are obtained:

$$f_{10}\left(\xi \right)={\displaystyle \frac{sinh(\xi )+r}{cosh\left(\xi \right)\pm \sqrt{r^2+1}}},(p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{2}}),$$

(24)

where r is an arbitrary constant. If r = 0, Equation (24) degenerates into Equation (16).

$$f_{11}\left(\xi \right)={\displaystyle \frac{cosh(\xi )+r}{sinh\left(\xi \right)\pm \sqrt{r^2-1}}}, (p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{2}}),$$

(25)

where r < −1 or r > 1.

$$f_{12}\left(\xi \right)={\displaystyle \frac{2[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}+r]sinh(\xi )cosh(\xi )}{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2\pm \frac{1}{2}\sqrt{4r^2-1}sinh(\xi )cosh(\xi )-\frac{1}{4}}}, (p_1=-{\displaystyle \frac{1}{2}},q_1=2),$$

(26)

where r < −${\displaystyle \frac{1}{2}}$ or r > ${\displaystyle \frac{1}{2}}$.

$$f_{13}\left(\xi \right)={\displaystyle \frac{\frac{1}{2}[\sqrt{cosh(\xi )+\epsilon }+r]\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}}{cosh(\xi )\pm \frac{1}{8}\sqrt{{\frac{1}{4}r}^2-\frac{1}{2}\epsilon }\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}-\epsilon }}, (p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{8}},{\mathsf{\epsilon }}^2=1),$$

(27)

among them, if $\epsilon$ = −1, r is an arbitrary constant, and if $\epsilon$ = 1, r < −$\sqrt{2}$ or r > $\sqrt{2}$.

According to the symmetry of the Riccati Equation (2), $h\left(\xi \right)=1/f\left(\xi \right)$ is also its solution when $p_1^{\prime }=-q_1, q_1^{\prime }=-p_1$, and the following equations are also the solutions of Equation (2)

$$f_{14}\left(\xi \right)={\displaystyle \frac{{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}}{2sinh(\xi )cosh(\xi )}}, (p_1=-4,q_1=1),$$

(28)

$$f_{15}\left(\xi \right)={\displaystyle \frac{cosh(\xi )+\epsilon }{sinh(\xi )}}, (p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{2}}, {\epsilon }^2=1),$$

(29)

$$f_{16}\left(\xi \right)={\displaystyle \frac{cosh(\xi )+1\pm \sqrt{2}\sqrt{cosh(\xi )+1}}{\frac{1}{2}sinh(\xi )}}, (p_1=-{\displaystyle \frac{1}{8}},q_1={\displaystyle \frac{1}{2}}),$$

(30)

$$f_{17}\left(\xi \right)={\displaystyle \frac{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2-\frac{1}{8}}{2sinh\left(\xi \right){cosh}^3\left(\xi \right)-sinh(\xi )cosh(\xi )}}, (p_1=-8,q_1=2),$$

(31)

$$f_{18}\left(\xi \right)={\displaystyle \frac{{cosh}^2\left(\xi \right)\mp \frac{1}{2}cosh(\xi )-\frac{1}{2}}{[cosh(\xi )\pm \frac{1}{2}]sinh(\xi )}},(p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{2}}),$$

(32)

$$f_{19}\left(\xi \right)={\displaystyle \frac{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2\mp \frac{1}{4}[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]-\frac{1}{8}}{2[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}\pm \frac{1}{4}]sinh(\xi )cosh(\xi )}}, ({\mathrm{p}}_1=-2,{\mathrm{q}}_1={\displaystyle \frac{1}{2}}),$$

(33)

$$f_{20}\left(\xi \right)={\displaystyle \frac{cosh(\xi )\mp \frac{1}{2}\sqrt{2}\sqrt{cosh(\xi )+1}}{\frac{1}{2}[\sqrt{cosh(\xi )+1}\pm \frac{1}{2}\sqrt{2}]\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}{\sqrt{cosh(\xi )+1}}}}, ({\mathrm{p}}_1=-{\displaystyle \frac{1}{8}},{\mathrm{q}}_1={\displaystyle \frac{1}{2}}),$$

(34)

$$f_{21}\left(\xi \right)={\displaystyle \frac{cosh\left(\xi \right)\pm \sqrt{r^2+1}}{sinh(\xi )+r}},(p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{2}}),$$

(35)

where r is an arbitrary constant. If r = 0, Equation (35) degenerates into Equation (29).

$$f_{22}\left(\xi \right)={\displaystyle \frac{sinh\left(\xi \right)\pm \sqrt{r^2-1}}{cosh(\xi )+r}}, (p_1=-{\displaystyle \frac{1}{2}},q_1={\displaystyle \frac{1}{2}}),$$

(36)

where r < −1 or r > 1.

$$f_{23}\left(\xi \right)={\displaystyle \frac{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2\pm \frac{1}{2}\sqrt{4r^2-1}sinh(\xi )cosh(\xi )-\frac{1}{4}}{2[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}+r]sinh(\xi )cosh(\xi )}}, (p_1=-2,q_1={\displaystyle \frac{1}{2}}),$$

(37)

where r < −${\displaystyle \frac{1}{2}}$ or r > ${\displaystyle \frac{1}{2}}$.

$$f_{24}\left(\xi \right)={\displaystyle \frac{cosh(\xi )\pm \frac{1}{8}\sqrt{{\frac{1}{4}r}^2-\frac{1}{2}\epsilon }\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}-\epsilon }{\frac{1}{2}[\sqrt{cosh(\xi )+\epsilon }+r]\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}}}, (p_1=-{\displaystyle \frac{1}{8}},q_1={\displaystyle \frac{1}{2}}, {\epsilon }^2=1),$$

(38)

among them, if $\epsilon$ = −1, r is an arbitrary constant, and if $\epsilon$ = 1, r < −$\sqrt{2}$ or r > $\sqrt{2}$.

In these solutions $f_5\left(\xi \right)$–$f_{13}\left(\xi \right) \mathrm{a}\mathrm{n}\mathrm{d} f_{16}\left(\xi \right)$–$f_{24}\left(\xi \right)$ are the new exact solutions of Equation (2), rarely reported in the other literature. Due to the very simple structure of Equation (2), using it as an auxiliary equation to solve Equation (1) can greatly simplify the solution process.

3. Application to Reaction–Diffusion Equation

We assume that Equation (1) has the following traveling wave solution

$$u\left(x,t\right)=u(\xi ), \xi =\mu x+ct.$$

(39)

where μ and c are wave parameters to be determined. Substituting Equation (39) into Equation (1) yields

$${\mathrm{c}u}^{\prime }-2{{\mu }^2{u^{\prime }}^2-2\mu }^2{uu}^{″}-u\left(1-u\right)=0.$$

(40)

Making $u\left(\xi \right)=1/v(\xi )$, the above equation is transformed into

$${\mathrm{c}{v}^{2}v}^{\prime }+6{{\mu }^2{v^{\prime }}^2-2\mu }^2{vv}^{″}-v^2+v^3=0.$$

(41)

Balancing ${v^{2}v}^{\prime }$ with ${vv}^{″}$ gives

$$3n+1=2n+2,$$

(42)

so that n = 1.

Therefore, we assume that Equation (45) has the formal solution as follows

$$v\left(\xi \right)=a_0{+a}_{1}f\left(\xi \right){+b}_1{f}^{-1}\left(\xi \right).$$

(43)

Substituting Equation (43) into Equation (41) and setting the coefficients of $f^i\left(\xi \right)$ to zero, then solving the resulting equations about $a_0$, $a_1$, $b_1$, $\mu$, and $c$, we can obtain

(1) 

The first set:

$$a_0={\displaystyle \frac{1}{2}}, a_1=\pm {\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{p_1}{q_1}}}{, b}_1=0,{\mu }^2=-{\displaystyle \frac{1}{16p_1{q}_1}},c^2=-{\displaystyle \frac{1}{16p_1{q}_1}},$$

(44)

(2) 

The second set:

$$a_0={\displaystyle \frac{1}{2}}, a_1=0{, b}_1=\pm {\displaystyle \frac{1}{2}}\sqrt{-{\displaystyle \frac{q_1}{p_1}}}, {\mu }^2=-{\displaystyle \frac{1}{16p_1{q}_1}}, c^2=-{\displaystyle \frac{1}{16p_1{q}_1}},$$

(45)

(3) 

The third set:

$$a_0={\displaystyle \frac{1}{2}}, a_1=\pm {\displaystyle \frac{1}{4}}\sqrt{-{\displaystyle \frac{p_1}{q_1}}}{, b}_1=\pm {\displaystyle \frac{1}{4}}\sqrt{-{\displaystyle \frac{q_1}{p_1}}}, {\mu }^2=-{\displaystyle \frac{1}{64p_1{q}_1}},c^2=-{\displaystyle \frac{1}{64p_1{q}_1}}.$$

(46)

According to the first set, we can obtain the following solutions for the reaction–diffusion system

$$u_1\left(\xi \right)={1/v}_1\left(\xi \right)={\displaystyle \frac{2}{1\pm \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\mathsf{\xi }\right)}},$$

(47)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{4}}$. This is a set of traveling wave solutions for the reaction–diffusion equation. It is singular and blows up at $\xi \to -\infty$, indicating the explosive instability of non-trivial diffusion coefficient problems. Figure 1a provides a three-dimensional diagram for Equation (47), which represents the traveling wave of Equation (41) with singularities when $\xi \to -\infty$. Figure 1b shows the two-dimensional image of the traveling wave, representing the evolution of the traveling wave from having singularities to having no singularities.

$$u_2\left(\xi \right)={1/v}_2\left(\xi \right)={\displaystyle \frac{2}{1\pm \mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\mathsf{\xi }\right)}},$$

(48)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{4}}$. This is a set of traveling wave solutions for the reaction–diffusion equation. It is singular and blows up at $\xi \to \infty$, indicating the explosive instability of non-trivial diffusion coefficient problems. Figure 2a provides the three-dimensional diagram for Equation (48), representing the traveling wave of Equation (41) with singularities when $\xi \to \infty$. Figure 2b shows a two-dimensional image of the traveling wave, representing the evolution of the traveling wave from having no singularities to having singularities.

$$u_3\left(\xi \right)={1/v}_3\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{sinh(\xi )cosh(\xi )}{{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}}}},$$

(49)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{8}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{8}}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 3a provides a three-dimensional diagram for Equation (49), representing the traveling wave solution of Equation (41) with singularities when $\xi \to -\infty$. Figure 3b shows a two-dimensional image of the traveling wave solution, representing the evolution of the traveling wave from having singularities to having no singularities.

$$u_4\left(\xi \right)={1/v}_4\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{sinh(\xi )}{cosh(\xi )+\epsilon }}},$$

(50)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}}, \mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\epsilon }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 4a provides the three-dimensional diagram for Equation (50), representing the traveling wave of Equation (41). Figure 4b shows a two-dimensional image of the traveling wave, which is the same as that in Figure 3.

$$u_5\left(\xi \right)={1/v}_5\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{sinh(\xi )}{cosh(\xi )+1+\sqrt{2}\mathsf{\lambda }\sqrt{cosh(\xi )+1}}}},$$

(51)

where $\xi =\mu x+ct, \mu =\pm 1, c=\pm 1, \mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 5a provides a three-dimensional diagram for Equation (51), representing the traveling wave of Equation (41). Figure 5b shows a two-dimensional image of the traveling wave, which is the same as that in Figure 1.

$$u_6\left(\xi \right)={1/v}_6\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{2sinh\left(\xi \right){cosh}^3\left(\xi \right)-sinh(\xi )cosh(\xi )}{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2-\frac{1}{8}}}},$$

(52)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{16}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{16}}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 6a provides a three-dimensional diagram for Equation (52), representing the traveling wave of Equation (41). Figure 6b shows a two-dimensional image of the traveling wave, which is the same as that in Figure 3.

$$u_7\left(\xi \right)={1/v}_7\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{[cosh(\xi )-\frac{1}{2}\lambda ]sinh(\xi )}{{cosh}^2\left(\xi \right)+\frac{1}{2}\lambda cosh(\xi )-\frac{1}{2}}}},$$

(53)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}},$ and ${\lambda }^2$ = 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 7a,b show three-dimensional and two-dimensional diagrams for Equation (53), representing the traveling wave of Equation (41), which is the same as that in Figure 3.

$$u_8\left(\xi \right)={1/v}_8\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}-\frac{1}{4}\lambda ]sinh(\xi )cosh(\xi )}{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2+\frac{1}{4}\lambda [{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]-\frac{1}{8}}}},$$

(54)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, c=\pm {\displaystyle \frac{1}{4}},$ and ${\lambda }^2$ = 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 8a,b show three-dimensional and two-dimensional diagrams for Equation (54), representing the traveling wave of Equation (41). It can be seen that the traveling wave shown in Figure 8 is the same as that shown in Figure 3.

$$u_9\left(\xi \right)={1/v}_9\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{[\sqrt{cosh(\xi )+1}-\frac{1}{2}\sqrt{2}\lambda ]\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}{\sqrt{cosh(\xi )+1}}}{cosh(\xi )+\frac{1}{2}\sqrt{2}\lambda \sqrt{cosh(\xi )+1}}}},$$

(55)

where $\xi =\mu x+ct, \mu =\pm 1, c=\pm 1,$ and ${\lambda }^2$ = 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 9a,b show a three-dimensional diagram and two-dimensional image of Equation (55), representing the traveling wave of Equation (41), which is the same as that in Figure 3.

$$u_{10}\left(\xi \right)={1/v}_{10}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{sinh(\xi )+r}{cosh\left(\xi \right)+\lambda \sqrt{r^2+1}}}},$$

(56)

where r is an arbitrary constant, $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}},$ and ${\lambda }^2$ = 1.This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 10a,b show three-dimensional and two-dimensional diagrams for Equation (56), representing the traveling wave of Equation (41). In this set of solutions, different values of r can cause changes in the shape of the wave.

$$u_{11}\left(\xi \right)={1/v}_{11}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{cosh(\xi )+r}{sinh\left(\xi \right)+\lambda \sqrt{r^2-1}}}},$$

(57)

where r < −1 or r > 1, $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}}$, and ${\lambda }^2$ = 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 11a,b show three-dimensional and two-dimensional diagrams for Equation (57), representing the traveling wave of Equation (41). In this set of solutions, different values of r can cause changes in the shape of the wave.

$$u_{12}\left(\xi \right)={1/v}_{12}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}+r]sinh(\xi )cosh(\xi )}{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2+\frac{1}{2}\lambda \sqrt{4r^2-1}sinh(\xi )cosh(\xi )-\frac{1}{4}}}},$$

(58)

where r < −${\displaystyle \frac{1}{2}}$ or r > ${\displaystyle \frac{1}{2}}$, $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, c=\pm {\displaystyle \frac{1}{4}}$, and ${\lambda }^2$ = 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 12a,b show three-dimensional and two-dimensional diagrams for Equation (58), representing the traveling wave of Equation (41). In this set of solutions, different values of r can cause changes in the shape of the wave.

$$u_{13}\left(\xi \right)={1/v}_{13}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{[\sqrt{cosh(\xi )+\epsilon }+r]\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}}{cosh(\xi )+\frac{1}{8}\lambda \sqrt{{\frac{1}{4}r}^2-\frac{1}{2}\epsilon }\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}-\epsilon }}},$$

(59)

where $\xi =\mu x+ct, \mu =\pm 1, c=\pm 1$, ${\lambda }^2$=1$, {\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\epsilon }}^2=$ 1; if $\epsilon$ = −1, r is an arbitrary constant, and if $\epsilon$ = 1, r < −$\sqrt{2}$ or r > $\sqrt{2}$.This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 13a,b show three-dimensional and two-dimensional diagrams for Equation (59), representing the traveling wave of Equation (41).

$$u_{14}\left(\xi \right)={1/v}_{14}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}}{sinh(\xi )cosh(\xi )}}},$$

(60)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{8}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{8}}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 14a,b show three-dimensional and two-dimensional diagrams for Equation (60), representing the traveling wave of Equation (41), which is the same as that in Figure 1.

$$u_{15}\left(\xi \right)={1/v}_{15}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{cosh(\xi )+\epsilon }{sinh(\xi )}}},$$

(61)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\epsilon }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 15a,b show three-dimensional and two-dimensional diagrams for Equation (61), representing the traveling wave of Equation (41), which is the same as that in Figure 1.

$$u_{16}\left(\xi \right)={1/v}_{16}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{cosh(\xi )+1+\sqrt{2}\mathsf{\lambda }\sqrt{cosh(\xi )+1}}{sinh(\xi )}}},$$

(62)

where $\xi =\mu x+ct, \mu =\pm 1, c=\pm 1,\mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 16a,b show three-dimensional and two-dimensional diagrams for Equation (62), representing the traveling wave of Equation (41), which is the same as that in Figure 3.

$$u_{17}\left(\xi \right)={1/v}_{17}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{2{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2-\frac{1}{4}}{2sinh\left(\xi \right){cosh}^3\left(\xi \right)-sinh(\xi )cosh(\xi )}}},$$

(63)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{16}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{16}}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 17a,b show three-dimensional and two-dimensional diagrams for Equation (63), representing the traveling wave of Equation (41), which is the same as that in Figure 1.

$$u_{18}\left(\xi \right)={1/v}_{18}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{{cosh}^2\left(\xi \right)-\frac{1}{2}\lambda cosh(\xi )-\frac{1}{2}}{[cosh(\xi )+\frac{1}{2}\lambda ]sinh(\xi )}}},$$

(64)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}},\mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\lambda }}^2=$ 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 18 a,b show three-dimensional and two-dimensional diagrams for Equation (64), representing the traveling wave of Equation (41), which is the same as that in Figure 3.

$$u_{19}\left(\xi \right)={1/v}_{19}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2-\frac{1}{4}\lambda [{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]-\frac{1}{8}}{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}\lambda ]sinh(\xi )cosh(\xi )}}},$$

(65)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, c=\pm {\displaystyle \frac{1}{4}},{\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 19a,b show three-dimensional and two-dimensional diagrams for Equation (65), representing the traveling wave of Equation (41), which is the same as that in Figure 1.

$$u_{20}\left(\xi \right)={1/v}_{20}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{cosh(\xi )-\frac{1}{2}\sqrt{2}\mathsf{\lambda }\sqrt{cosh(\xi )+1}}{[\sqrt{cosh(\xi )+1}+\frac{1}{2}\sqrt{2}\mathsf{\lambda }]\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}{\sqrt{cosh(\xi )+1}}}}},$$

(66)

where $\xi =\mu x+ct, \mu =\pm 1, c=\pm 1,{\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\lambda }}^2=$ 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 20a,b show three-dimensional and two-dimensional diagrams for Equation (66), representing the traveling wave of Equation (41), which is the same as that in Figure 1.

$$u_{21}\left(\xi \right)={1/v}_{21}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{cosh\left(\xi \right)+\mathsf{\lambda }\sqrt{r^2+1}}{sinh(\xi )+r}}},$$

(67)

where r is an arbitrary constant, $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}},{\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 21a,b show three-dimensional and two-dimensional diagrams for Equation (67), representing the traveling wave of Equation (41).

$$u_{22}\left(\xi \right)={1/v}_{22}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{sinh\left(\xi \right)\pm \sqrt{r^2-1}}{cosh(\xi )+r}}},$$

(68)

where r < −1 or r > 1, $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}},{\mathsf{\lambda }}^2=$ 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 22a,b show three-dimensional and two-dimensional diagrams for Equation (68), representing the anti-kink wave of Equation (41).

$$u_{23}\left(\xi \right)={1/v}_{23}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2+\frac{1}{2}\mathsf{\lambda }\sqrt{4r^2-1}sinh(\xi )cosh(\xi )-\frac{1}{4}}{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}+r]sinh(\xi )cosh(\xi )}}},$$

(69)

where r < −${\displaystyle \frac{1}{2}}$ or r > ${\displaystyle \frac{1}{2}}, \xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, c=\pm {\displaystyle \frac{1}{4}},{\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 23a,b show three-dimensional and two-dimensional diagrams for Equation (69), representing the traveling wave of Equation (41).

$$u_{24}\left(\xi \right)={1/v}_{24}\left(\xi \right)={\displaystyle \frac{2}{1\pm \frac{cosh(\xi )+\frac{1}{8}\mathsf{\lambda }\sqrt{{\frac{1}{4}r}^2-\frac{1}{2}\epsilon }\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}-\epsilon }{[\sqrt{cosh(\xi )+\epsilon }+r]\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}}}},$$

(70)

where r < −${\displaystyle \frac{1}{2}}$ or r > ${\displaystyle \frac{1}{2}}, \xi =\mu x+ct, \mu =\pm 1, c=\pm 1,{\epsilon }^2=1,{\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\lambda }}^2=$1; if $\epsilon$ = −1, r is an arbitrary constant, and if $\epsilon$ = 1, r < −$\sqrt{2}$ or r > $\sqrt{2}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 24a,b show three-dimensional and two-dimensional diagrams for Equation (70), representing the traveling wave of Equation (41).

The solutions of the reaction–diffusion equation in the second set are the same as those in the first set, since $h\left(\xi \right)=1/f\left(\xi \right)$ is already included in the solutions. According to the third set, we can obtain the following solutions of the reaction–diffusion equation

$$u_{25}\left(\xi \right)={1/v}_{25}\left(\xi \right)={\displaystyle \frac{4}{2\pm \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}\left(\mathsf{\xi }\right)\pm tanh\left(\mathsf{\xi }\right)}},$$

(71)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{8}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{8}}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 25a,b show three-dimensional and two-dimensional diagrams for Equation (71), representing the traveling wave of Equation (41).

$$u_{26}\left(\xi \right)={1/v}_{26}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\mathsf{\xi })}{{\cosh }^2\left(\mathsf{\xi }\right)-\frac{1}{2}}\pm \frac{{[\cosh }^2\left(\mathsf{\xi }\right)-\frac{1}{2}]}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\mathsf{\xi })}}},$$

(72)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{16}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{16}}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 26a,b show three-dimensional and two-dimensional diagrams for Equation (72), representing the traveling wave of Equation (41).

$$u_{27}\left(\xi \right)={1/v}_{27}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\mathsf{\xi })+\mathsf{\epsilon }}\pm \frac{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\mathsf{\xi })+\mathsf{\epsilon }}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}}},$$

(73)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, c=\pm {\displaystyle \frac{1}{4}}, {\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\epsilon }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 27a,b show three-dimensional and two-dimensional diagrams for Equation (72), which represent the traveling wave of Equation (41).

$$u_{28}\left(\xi \right)={1/v}_{28}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}{cosh(\xi )+1+\sqrt{2}\mathsf{\lambda }\sqrt{cosh(\xi )+1}}\pm \frac{cosh(\xi )+1+\sqrt{2}\mathsf{\lambda }\sqrt{cosh(\xi )+1}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}}},$$

(74)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}}, \mathrm{a}\mathrm{n}\mathrm{d} {\mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 28a,b show three-dimensional and two-dimensional diagrams for Equation (74), representing the traveling wave of Equation (41).

$$u_{29}\left(\xi \right)={1/v}_{29}\left(\xi \right)={\displaystyle \frac{8}{4\pm \frac{2\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\mathsf{\xi }\right){\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}^3\left(\mathsf{\xi }\right)-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\mathsf{\xi })}{{[{\cosh }^2\left(\mathsf{\xi }\right)-\frac{1}{2}]}^2-\frac{1}{8}}\pm \frac{{4[{\cosh }^2\left(\mathsf{\xi }\right)-\frac{1}{2}]}^2-\frac{1}{2}}{2\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\mathsf{\xi }\right){\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}}^3\left(\mathsf{\xi }\right)-\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(\mathsf{\xi })}}},$$

(75)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{32}}, \mathrm{a}\mathrm{n}\mathrm{d} c=\pm {\displaystyle \frac{1}{32}}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 29a,b show three-dimensional and two-dimensional diagrams for Equation (75), representing the kink solitary wave of Equation (41).

$$u_{30}\left(\xi \right)={1/v}_{30}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{[cosh(\xi )-\frac{1}{2}\mathsf{\lambda }]sinh(\xi )}{{cosh}^2\left(\xi \right)+\frac{1}{2}\mathsf{\lambda }cosh(\xi )-\frac{1}{2}}\pm \frac{{cosh}^2\left(\xi \right)+\frac{1}{2}\mathsf{\lambda }cosh(\xi )-\frac{1}{2}}{[cosh(\xi )-\frac{1}{2}\mathsf{\lambda }]sinh(\xi )}}},$$

(76)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, c=\pm {\displaystyle \frac{1}{4}}, {\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 30a,b show three-dimensional and two-dimensional diagrams for Equation (76), representing the traveling wave of Equation (41).

$$u_{31}\left(\xi \right)={1/v}_{31}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}-\frac{1}{4}\mathsf{\lambda }]sinh(\xi )cosh(\xi )}{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2+\frac{1}{4}\mathsf{\lambda }[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]-\frac{1}{8}}\pm \frac{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2+\frac{1}{4}\mathsf{\lambda }[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]-\frac{1}{8}}{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}-\frac{1}{4}\mathsf{\lambda }]sinh(\xi )cosh(\xi )}}},$$

(77)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{8}}, c=\pm {\displaystyle \frac{1}{8}}, {\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 31a,b represent three-dimensional and two-dimensional diagrams for Equation (77), representing the traveling wave of Equation (41).

$$u_{32}\left(\xi \right)={1/v}_{32}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{[\sqrt{cosh(\xi )+1}-\frac{1}{2}\sqrt{2}\mathsf{\lambda }]\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}{\sqrt{cosh(\xi )+1}}}{cosh(\xi )+\frac{1}{2}\sqrt{2}\mathsf{\lambda }\sqrt{cosh(\xi )+1}}\pm \frac{cosh(\xi )+\frac{1}{2}\sqrt{2}\mathsf{\lambda }\sqrt{cosh(\xi )+1}}{[\sqrt{cosh(\xi )+1}-\frac{1}{2}\sqrt{2}\mathsf{\lambda }]\frac{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathsf{\xi })}{\sqrt{cosh(\xi )+1}}}}},$$

(78)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}}, {\mathrm{a}\mathrm{n}\mathrm{d} \mathsf{\lambda }}^2=1$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 32a,b show three-dimensional and two-dimensional diagrams for Equation (78), representing the traveling wave of Equation (41).

$$u_{33}\left(\xi \right)={1/v}_{33}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{sinh(\xi )+r}{cosh\left(\xi \right)+\mathsf{\lambda }\sqrt{r^2+1}}\pm \frac{cosh\left(\xi \right)+\mathsf{\lambda }\sqrt{r^2+1}}{sinh(\xi )+r}}},$$

(79)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, c=\pm {\displaystyle \frac{1}{4}}, {\mathsf{\lambda }}^2=1$, and r is an arbitrary constant. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 33a,b represents three-dimensional and two-dimensional diagrams for Equation (79), which represents the traveling wave of Equation (41).

$$u_{34}\left(\xi \right)={1/v}_{34}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{cosh(\xi )+r}{sinh\left(\xi \right)+\mathsf{\lambda }\sqrt{r^2-1}}\pm \frac{sinh\left(\xi \right)+\mathsf{\lambda }\sqrt{r^2-1}}{cosh(\xi )+r}}},$$

(80)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{4}}, c=\pm {\displaystyle \frac{1}{4}}, {\mathsf{\lambda }}^2=1$, and r < −1 or r > 1. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 34a,b presents three-dimensional and two-dimensional diagrams for Equation (80), which represents the traveling wave of Equation (41).

$$u_{35}\left(\xi \right)={1/v}_{35}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}+r]sinh(\xi )cosh(\xi )}{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2+\frac{1}{2}\mathsf{\lambda }\sqrt{4r^2-1}sinh(\xi )cosh(\xi )-\frac{1}{4}}\pm \frac{{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}]}^2+\frac{1}{2}\mathsf{\lambda }\sqrt{4r^2-1}sinh(\xi )cosh(\xi )-\frac{1}{4}}{[{\mathit{cosh}}^2\left(\xi \right)-\frac{1}{2}+r]sinh(\xi )cosh(\xi )}}},$$

(81)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{8}}, c=\pm {\displaystyle \frac{1}{8}}, {\mathsf{\lambda }}^2=1$, and r < −${\displaystyle \frac{1}{2}}$ or r > ${\displaystyle \frac{1}{2}}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 35a,b represent three-dimensional and two-dimensional diagrams for Equation (81), representing the traveling wave of Equation (41).

$$u_{36}\left(\xi \right)={1/v}_{36}\left(\xi \right)={\displaystyle \frac{4}{2\pm \frac{[\sqrt{cosh(\xi )+\epsilon }+r]\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}}{cosh(\xi )+\frac{1}{8}\mathsf{\lambda }\sqrt{{\frac{1}{4}r}^2-\frac{1}{2}\epsilon }\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}-\epsilon }\pm \frac{cosh(\xi )+\frac{1}{8}\mathsf{\lambda }\sqrt{{\frac{1}{4}r}^2-\frac{1}{2}\epsilon }\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}-\epsilon }{[\sqrt{cosh(\xi )+\epsilon }+r]\frac{sinh(\xi )}{\sqrt{cosh(\xi )+\epsilon }}}}},$$

(82)

where $\xi =\mu x+ct, \mu =\pm {\displaystyle \frac{1}{2}}, c=\pm {\displaystyle \frac{1}{2}}, {\mathsf{\lambda }}^2=1$, and ${\mathsf{\epsilon }}^2=1$; if $\epsilon$ = −1, r is an arbitrary constant and if $\epsilon$ = 1, r < −$\sqrt{2}$ or r > $\sqrt{2}$. This is a set of traveling wave solutions for the reaction–diffusion equation, and it is singular and blows up at $\xi \to -\infty$. Figure 36a,b show three-dimensional and two-dimensional diagrams for Equation (82), representing the traveling wave of Equation (41) with singularities in certain spatio-temporal positions.

4. Conclusions

This article uses the Riccati equation to process the reaction–diffusion equation in order to find more traveling wave solutions. We first applied two different methods to solve the Riccati equation and obtained a large number of hyperbolic function solutions. Then, using this Riccati equation as an auxiliary equation to solve the reaction–diffusion equation, we obtained a large number of new traveling wave solutions. These obtained solutions are all singular and blow up at $\xi \to -\infty$ or $\xi \to \infty$, indicating the explosive instability of non-trivial diffusion coefficient problems. In these solutions, Equations (47), (48), and (71) are the same as those in Refs [24,26]. These traveling wave solutions for Equation (1) represented by Equations (51)–(59), (62)–(70), (72), and (74)–(82) have rarely been found in other studies. The numerical images show that these traveling wave solutions exhibited complex nonlinear wave phenomena. Some solutions had the same physical image and spatiotemporal evolution, indicating that they had the same amplitude, wave velocity, wave number, and propagation direction, while those with different wave structures and spatiotemporal evolution indicated that these waves had different amplitudes, wave velocities, wave numbers, and propagation directions. Compared with the methods in Refs [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26], the method proposed in this study only requires changing the parameters in the Riccati equation to obtain a large number of new exact solutions for NLEEs. Moreover, due to the simple structure of the Riccati equation, the solution process can be greatly simplified when solving nonlinear equations. The construction of complex exact solutions only needs to be completed in the construction of exact solutions to the Riccardi equation. Therefore, the method proposed in this article is more suitable for obtaining more complex nonlinear wave solutions and nonlinear wave structures of more complex nonlinear systems using very simple techniques. In the next step, we will consider higher-order NLEEs. The solution of high-order NLEEs is very complex due to the extremely complicated derivation process, and it is difficult to obtain exact solutions for complex structures. The application of the method used in this article will greatly reduce the difficulty of solving and can obtain exact solutions for a large number of new and complex structures that have not been found in other literature.