The Exact Traveling Wave Solutions of a KPP Equation

Abstract

We obtain the exact analytical traveling wave solutions of the Kolmogorov–Petrovskii–Piskunov equation, with the reaction term belonging to the class of functions, which includes that of the (generalized) Fisher equation, for the particular values of the wave’s speed. Additionally we obtain the exact analytical traveling wave solutions of the generalized Burgers–Huxley equation.

1. Introduction

Fisher [1] first introduced a nonlinear evolution equation to investigate the wave propagation of an advantageous gene in a population. His equation also describes the logistic growth–diffusion process and has the form

$$\begin{array}{c}\hfill u_t-\nu u_{xx}=ku\left(1-{\displaystyle \frac{u}{\kappa }}\right)\end{array}$$

(1)

where $\kappa (>0)$ is the carrying capacity of the environment. The term $f\left(u\right)=ku(1-u/\kappa )$ represents a nonlinear growth rate, which is proportional to u for small u, but decreases as u increases, and vanishes when $u=\kappa$. Equation (1) arises in many physical, biological, and chemical problems involving diffusion and nonlinear growth. It is convenient to introduce the nondimensional quantities

$$\begin{array}{cc}\hfill x^{*}& =x\sqrt{k/\nu },\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill t^{*}& =kt,\hfill \end{array}$$

(2b)

$$\begin{array}{cc}\hfill u^{*}& =u/\kappa ,\hfill \end{array}$$

(2c)

so after dropping the asterisks, Equation (1) takes the nondimensional form

$$\begin{array}{c}\hfill u_t-u_{xx}=u\left(1-u\right).\end{array}$$

(3)

In the same year (1937) as Fisher, Kolmogorov, Petrovsky, and Piskunov (KPP) [2] introduced the more general reaction–diffusion equation, which in the nondimensional form can be presented as

$$\begin{array}{c}\hfill u_t-u_{xx}=F\left(u\right),\end{array}$$

(4)

where F is a sufficiently smooth function with the properties that $F\left(0\right)=F\left(1\right)=0$, $F^{\prime }\left(0\right)>0$, and $F^{\prime }\left(1\right)<0$.

The Fisher–KPP equation belongs to the class of reaction–diffusion equations; in fact, it is one of the simplest semilinear reaction–diffusion equations, which can exhibit traveling wave solutions that switch between equilibrium states given by $F\left(u\right)=0$. Such equations occur, e.g., in ecology, where they describe, as was mentioned above, the population growth, physiology, combustion, crystallization, plasma physics, and in general phase transition problems.

Equation (4) admits traveling wave solutions

$$\begin{array}{c}\hfill u(x,t)=u\left(\xi \right),\xi =x-ct,\end{array}$$

(5)

where c is the wave speed (measured in the units of $\sqrt{\nu /k}$) and $u\left(\xi \right)$ satisfies the equation

$$\begin{array}{c}\hfill u_{\xi \xi }+c{u}_{\xi }=-F\left(u\right)\end{array}$$

(6)

with the boundary conditions

$$\begin{array}{cc}\hfill \underset{\xi \to -\infty }{lim}u\left(\xi \right)& =1,\hfill \end{array}$$

(7a)

$$\begin{array}{cc}\hfill \underset{\xi \to +\infty }{lim}u\left(\xi \right)& =0.\hfill \end{array}$$

(7b)

Equation (6) has a simple mechanical interpretation. It describes motion of a Newtonian particle with the mass 1 (u being the coordinate of the particle and $\xi$—the time) in the potential well $U\left(u\right)$ defined by the equation

$$\begin{array}{c}\hfill {\displaystyle \frac{dU}{du}}=F\left(u\right).\end{array}$$

(8)

The motion is with friction, and the coefficient of friction is equal to c. The point $u=0$ is the point of stable equilibrium, and the point $u=1$ unstable.

Equation (6) is a nonlinear nonintegrable differential equation. The Cauchy problem cannot be solved by the inverse scattering transform for this equation and the problem of finding some exact solutions for this equation is an important task. Several methods of obtaining such solutions have been developed. We will mention a few of them here: the tanh-function method [3,4,5], the Jacobi elliptic function expansion method [6], the simplest equation method [7,8], the Exp-function method [9], the $G^{\prime }/G$ expansion method [10,11,12,13], the Kudryashov method [14], and the attached flow method [15,16].

In this paper, we used the method of exclusion of the independent variable [17,18,19,20,21], which allows the problem of integration of the second-order ODE to be reduced to the problem of integration of the first-order ODE, to obtain the exact solutions of Equation (6). Thus, we were able to obtain the exact solutions of the KPP equation, which to the best of our knowledge were not known before. Additionally, we obtained the exact analytical traveling wave solutions of the generalized Burgers–Huxley equation.

2. The ODE That Does Not Explicitly Contain the Independent Variable

Equation (6) does not explicitly contain the independent variable $\xi$. This prompts the idea of considering u as the new independent variable and

$$\begin{array}{c}\hfill p=u_{\xi }\end{array}$$

(9)

as the new dependent variable. In the new variables, (6) takes the form of an Abel equation of the second kind [16].

$$\begin{array}{c}\hfill p{p}_u+cp=-F\left(u\right).\end{array}$$

(10)

The boundary conditions for (6) are (7a,b). The boundary conditions for (10) are (however, see the next Section)

$$\begin{array}{cc}\hfill p\left(1\right)& =0,\hfill \end{array}$$

(11a)

$$\begin{array}{cc}\hfill p\left(0\right)& =0.\hfill \end{array}$$

(11b)

In search of elementary solutions of (10), in accordance with the well-known principle in mathematics stating that the more general the problem is, the easier it is to solve it, let us generalize it to

$$\begin{array}{c}\hfill p{p}_u+\gamma \left(u\right)p=-F\left(u\right),\end{array}$$

(12)

where $\gamma \left(u\right)$ is some function. Being inspired by the method of factorization [22],

We present the r.h.s. of (12) as

$$\begin{array}{c}\hfill -F\left(u\right)=\Phi \left(u\right)\Psi \left(u\right).\end{array}$$

(13)

After that, we realize that there exists the particular solution of (12)

$$\begin{array}{c}\hfill p\left(u\right)=P\Phi \left(u\right),\end{array}$$

(14)

provided the unknown multiplier P and the function $\gamma \left(u\right)$ satisfy equation

$$\begin{array}{c}\hfill P^2{\Phi }_u+P\gamma \left(p\right)=\Psi \left(u\right).\end{array}$$

(15)

This includes the case of nonlinear $F\left(u\right)$, considered in Ref. [23]

$$\begin{array}{c}\hfill F\left(U\right)=\left[u-A\left(u\right)\right]\left[\chi A^{\prime }\left(u\right)+a\right],\end{array}$$

(16)

where $A\left(u\right)$ is some function, $\chi ,a$ are some numerical parameters, and $\gamma \left(p\right)=c=$ const. In fact, taking

$$\begin{array}{c}\hfill \Phi \left(u\right)=\left[A\left(u\right)-u\right]\end{array}$$

(17)

we obtain (15) in the form

$$\begin{array}{c}\hfill P^2\left[A^{\prime }\left(u\right)-1\right]+Pc=\chi A^{\prime }\left(u\right)+a.\end{array}$$

(18)

Choosing $P^2=\chi$ and $c=P+a/P$, we turn (18) into an identity.

Now, let us be more specific and consider

$$\begin{array}{c}\hfill F\left(u\right)=u\left(1-u^n\right)\left(u^n+a\right),\end{array}$$

(19)

where $n>0$ and $a\ge 0$ are some constants, so that (6) will take the form

$$\begin{array}{c}\hfill u_{\xi \xi }+c{u}_{\xi }=u\left(u^n-1\right)\left(u^n+a\right).\end{array}$$

(20)

Note that for $n=1/2$ and $a=1$, we recover the Fisher equation. Equation (20), with $a=1$ but $n\ne 1/2$, we may call the generalized Fisher equation [24,25,26]. Starting from (20), we obtain Equation (10) in the form

$$\begin{array}{c}\hfill p{p}_u+cp=u\left(u^n-1\right)\left(u^n+a\right).\end{array}$$

(21)

We can take

$$\begin{array}{c}\hfill \Phi \left(u\right)=u\left(u^n-1\right),\Psi \left(u\right)=\left(u^n+a\right),\end{array}$$

(22)

after which (15) takes the form

$$\begin{array}{c}\hfill P^2\left[(n+1)u^n-1\right]+Pc=u^n+a.\end{array}$$

(23)

We immediately realize the solution of (21), satisfying the boundary conditions (11a,b)

$$\begin{array}{c}\hfill p\left(u\right)=p_0\left(u\right)\equiv Pu\left(u^n-1\right),\end{array}$$

(24)

where

$$\begin{array}{c}\hfill P={\displaystyle \frac{1}{\sqrt{n+1}}},\end{array}$$

(25)

exists if the speed of the wave is

$$\begin{array}{c}\hfill c=c_0\equiv P+{\displaystyle \frac{a}{P}}.\end{array}$$

(26)

Substituting the obtained $p\left(u\right)$ into (9) and integrating, we obtain the solution of (20)

$$\begin{array}{c}\hfill u\left(\xi \right)={\displaystyle \frac{1}{{\left[exp\left(nP\xi \right)+1\right]}^{1/n}}}.\end{array}$$

(27)

The Fisher equation from (27) follows the well-known result [27]

$$\begin{array}{c}\hfill u\left(\xi \right)={\displaystyle \frac{1}{{\left[exp\left(\xi /\sqrt{6}\right)+1\right]}^2}},\end{array}$$

(28)

and the condition (26) takes the form $c=5/\sqrt{6}$.

To additionally illustrate the method of integration used in this Section, in the Appendix A we obtain in the framework of the method the exact solution of the generalized Duffing–Van der Pol equation.

In Appendix B, we can see how the results (24)–(27) can be generalized in the case when advection is taken into account in addition to diffusion.

The connection between the existence of the elementary solution of (10) and the possible symmetry of the equation will be analysed in our next publication.

3. The Correct Boundary Conditions for Equation (10)

In the previous Section, we postulated two boundary conditions (11a,b) for (10). However, the equation is of the first order. The first impression is that the problem is overdetermined, and the existence of the solution of (10) demands the finetuning of the l.h.s. of the equation. Being more specific, one may think that the solution of (21) will not exist for a general value of c. However, the last statement immediately comes into contradiction with the results of the Appendix C, where the solution of (21) for arbitrary $c\gg 1$ is presented. It looks like we have a problem.

To solve this problem, we have to return to Equation (6) with the boundary conditions (7a,b). The boundary conditions are exceptional [28]—$u_{\xi \xi }$ is not defined at $\xi =\pm \infty$. However, for $\xi \to +\infty$ the equation asymptotically becomes

$$\begin{array}{c}\hfill u_{\xi \xi }+c{u}_{\xi \xi }+F^{\prime }\left(0\right)u=0,\end{array}$$

(29)

and for $\xi \to -\infty$

$$\begin{array}{c}\hfill v_{\xi \xi }+c{v}_{\xi }-\left|F^{\prime }\left(1\right)\right|v=0,\end{array}$$

(30)

where $v=1-u$.

The solution of (30) is

$$\begin{array}{c}\hfill v=A{e}^{s_1\xi }+B{e}^{s_2\xi },\end{array}$$

(31)

where A and B are arbitrary constants and $s_{1,2}$ are the roots of the characteristic equation

$$\begin{array}{c}\hfill s^2-cs-\left|F^{\prime }\left(1\right)\right|=0,\end{array}$$

(32)

that is

$$\begin{array}{cc}\hfill s_1& ={\displaystyle \frac{c+\sqrt{c^2+4\left|F^{\prime }\left(1\right)\right|}}{2}},\hfill \end{array}$$

(33a)

$$\begin{array}{cc}\hfill s_2& ={\displaystyle \frac{c-\sqrt{c^2+4\left|F^{\prime }\left(1\right)\right|}}{2}}.\hfill \end{array}$$

(33b)

because of the boundary condition (7a), $B=0$. So, this boundary condition should be specified as follows: for $\xi \to -\infty$

$$\begin{array}{c}\hfill 1-u\sim e^{s_1\xi }.\end{array}$$

(34)

Now, let us come to (29). The characteristic equation is

$$\begin{array}{c}\hfill s^2+cs+F^{\prime }\left(0\right)=0,\end{array}$$

(35)

that is

$$\begin{array}{c}\hfill s_{3,4}={\displaystyle \frac{-c\pm \sqrt{c^2-4F^{\prime }\left(0\right)}}{2}}.\end{array}$$

(36)

We have either the real roots (both being negative), the double (negative) root, or the complex roots (both with the negative real part). In the first case, the solution of (29) is

$$\begin{array}{c}\hfill u=A{e}^{s_3\xi }+B{e}^{s_4\xi },\end{array}$$

(37)

for the case of the double root the solution is

$$\begin{array}{c}\hfill u=e^{-c\xi /2}(A+B\xi ),\end{array}$$

(38)

and in the complex case the solution is

$$\begin{array}{c}\hfill u=e^{-c\xi /2}\left[Acos\left(\alpha \xi \right)+Bsin\left(\alpha \xi \right)\right],\end{array}$$

(39)

where $\alpha =[4F^{\prime }\left(0\right)-c^2]/2$. In any case, the solution of (29) satisfies the boundary condition (7b) for any A and B. So the boundary condition (7b) can be discarded.

Because we have lost one boundary condition, the solution of (6) contains an arbitrary constant. And here, our mechanical analogy helps us to understand that this (irrelevant) constant describes the translation of the solution in “time” $\xi$.

After we understand the true nature of the boundary conditions (7a,b) for (6), the situation with the boundary conditions (11a,b) for Equation (10)— which are also exceptional—becomes obvious.

In the vicinity of $u=1$, Equation (10) becomes asymptotically

$$\begin{array}{c}\hfill q{q}_v-cq=\left|F^{\prime }\left(1\right)\right|v,\end{array}$$

(40)

where again $v=1-u$ and $q\equiv dv/d\xi$. Equation (40) has two solutions

$$\begin{array}{cc}\hfill q& =s_{1}v,\hfill \end{array}$$

(41a)

$$\begin{array}{cc}\hfill q& =s_{2}v,\hfill \end{array}$$

(41b)

where $s_{1,2}$ are given by (33a,b). Only the solution (41a) satisfies the condition (7a). Hence, we realize that the boundary condition (11a) should be specified as follows: for $u\to 1$

$$\begin{array}{c}\hfill p=s_1(u-1).\end{array}$$

(42)

(Note the difference between the proportionality sign in (34) and the equality sign in (42).) One can easily check that our solution (24) satisfies this boundary condition.

In the vicinity of $u=0$, Equation (10) becomes asymptotically

$$\begin{array}{c}\hfill p{p}_u+cp=-F^{\prime }\left(0\right)u.\end{array}$$

(43)

Equation (43) has two solutions (we will consider only the case $c^2>4F^{\prime }\left(0\right)$)

$$\begin{array}{cc}\hfill p& =s_{3}u,\hfill \end{array}$$

(44a)

$$\begin{array}{cc}\hfill p& =s_{4}u,\hfill \end{array}$$

(44b)

where $s_{3,4}$ are given by (36). Because both $s_3$ and $s_4$ are negative, both solutions will satisfy the boundary condition (7b). So, we understand that the second boundary condition (11b) can be discarded—it is satisfied automatically.

Note that for the case considered explicitly in the previous Section, ($F\left(u\right)$ being given by (19) and c being given by (26), obviously fulfilling the inequality $c^2>4a$ for any $n>0$ and $a\ge 0$), we obtain

$$\begin{array}{cc}\hfill s_3& =-{\displaystyle \frac{1}{\sqrt{n+1}}},\hfill \end{array}$$

(45a)

$$\begin{array}{cc}\hfill s_4& =-a\sqrt{n+1}.\hfill \end{array}$$

(45b)

Looking at Equation (24), we realize that the solution (44a) is relevant in this case.

So finally, in the previous Section we were solving the first-order ODE with a single boundary condition. The solution should exist for any given value of the speed c and should not contain an undefined constant. We did finetune the speed c in the previous Section for the solution of (21) to be expressed in terms of elementary functions, and not for the solution to exist.

Our analysis of (29) allows us to understand (partially) the analytic properties of $p\left(u\right)$. If $c^2\ge 4F^{\prime }\left(0\right)$, from (37) and (38) we can expect that $p\left(u\right)$ would be either a single- or a double-valued function. (Further analysis is necessary to decide which option is realized.)

Looking at (39) (which corresponds to $c^2<4F^{\prime }\left(0\right)$), on the other hand, we realize that $p\left(u\right)$ would be an infinitely valued function. Note that for physical reasons, u should be strictly non-negative; such values of speed are unacceptable, because from (39) it follows that u would acquire negative values in the vicinity of $\xi =+\infty$. (Apart from recovering this well-known property, in this paper we will say nothing about the front propagation speed selection, an issue which is both important and complicated [29,30].)

4. Conclusions

We have obtained exact traveling wave solutions of the KPP equation for a wide class of reaction terms and similar for the generalized Burgers–Huxley equation. Our method of integration of the second-order differential equations is based on the choice of the original dependent variable as the new independent variable, and the derivative of the original dependent variable as the new dependent variable. The boundary conditions for the obtained first-order differential equations were thoroughly studied. All this gave us the opportunity to obtain the exact solutions mentioned above.