1 Introduction

The Fisher–Kolmogorov equation, proposed for population dynamics in 1930, shows the spread of an advantageous gene into a population. As described by Britton [2], generalizations of this equation are needed to more accurately model complex diffusion and reaction effects found in many biological systems. The equation analyzed in this chapter is a generalized Fisher equation in which g(u) is the diffusion coefficient depending on the variable u, x and t being the independent variables, and f(u) an arbitrary function

$$\begin{aligned}u_t=f(u)+\left(g(u)u_x\right)_x\end{aligned}$$
(1)

Equation (1) is also known as the density-dependent diffusion-reaction equation which is mentioned by J. D. Murray in [7]. Reaction-diffusion equations arise from modeling densities of particles such as substances and organisms which disperse through space as a result of the irregular movement of every particle.

Due to the interest of these equations, a lot of attention has been paid to the use of Lie point symmetry methods to exploit the invariance of the generalized equation [4] and references therein. In [5], we determined the subclasses of these equations which are nonlinear self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov and the symmetry generators we found conservation laws for these partial differential equations.There is no existing general theory for solving nonlinear partial differential equations (PDEs) and the machinery of the Lie group theory provides the systematic method to search for the special group-invariant solutions. The knowledge of the optimal system of subalgebras gives the possibility to construct the optimal system of solutions and permits the generation of new solutions starting from invariant or noninvariant solutions.

Due to the great advance in computation in the last few years, a great progress has been made in the development of methods and their applications for finding solitary travelling-wave solutions of nonlinear evolution equations [3,9,6]. For (1) the list of nontrivial Lie generators were derived in [4] by combining the standard method of group classification and the form-preserving transformation.

The aim of this chapter is to study the density-dependent diffusion-reaction Eq. (1) from the point of view of the theory of symmetry reductions in partial differential equations. We construct the reductions from the optimal system of subalgebras.

Then, due to the fact that Eq. (1) admits groups of space and time translations, we search for travelling wave solutions of the density-dependent diffusion-reaction Eq. (1), with physical interest, when the diffusion coefficient g(u) follows a power law. In order to do that we apply the well known \(\frac{G'}{G}\)-expansion method [3,9], to the reduced equation.

2 Symmetry Reductions

The Lie classical method applied to (1) yields (see [4]):

For f(u) and g(u) arbitrary, the only symmetry that is admitted by (1) is

$$\bf{v}_2=\frac{\partial}{\partial t}.$$

For some special choices of the functions f(u) and g(u) it can also be extended to the cases listed below:

1.For \(f(u)=u^m\) and \(g(u)=u^n\) (\(m \neq n+1\)) we obtain the following generators:

$$\begin{array}{ll}{\bf v}_2,& {\bf v}_1= \displaystyle x\frac{\partial}{\partial x}+\frac{2(m-1)}{n-m+1}\frac{\partial}{\partial t}+\frac{2u}{n-m+1}\frac{\partial}{\partial u}\end{array}$$

2.For \(f(u)=\frac{u^{n+1}}{n+1}\) and \(g(u)=u^n\) (\(n\neq 0\) and \(n\neq -1\)) we obtain the following generators:

$$\begin{array}{ll}{\bf v}_2,& {\bf v}_3 =\displaystyle n t\frac{\partial}{\partial t}-u\frac{\partial}{\partial u}\end{array}$$

3.For \(f(u)=\frac{3\,c_1\,u}{4}+\frac{c_2}{{u}^{\frac{1}{3}}}\) and \(g(u)=u^{-\frac{4}{3}}\) we obtain the following generators:

$$\begin{array}{ll}{\bf v}_2,& {\bf v}_4 =\displaystyle \displaystyle{4\,{e}^{c_1\,t}}\frac{\partial}{\partial t}+\displaystyle{3\,c_1\,{e}^{c_1\,t}\,u}\frac{\partial}{\partial u}\end{array}$$

4.For \(f(u)=-\frac{c_1\,u}{n}\) and \(g(u)= u^n\) (\(n\neq 0\)) we obtain the following generators:

$$\begin{array}{lll} {\bf v}_2,&{\bf v}_5 =\displaystyle n x\frac{\partial}{\partial x}+2u\frac{\partial}{\partial u},&\bf{v}_6 =\displaystyle{\it n}\,e^{{\it c_1}\,t}\frac{\partial}{\partial t}-{{\it c_1}\,e^{{\it c_1}\,t}\,u}\frac{\partial}{\partial u}\end{array}$$

5.For \(f(u)=c_1 u\) and \(g(u)= u^{-1}\) we obtain the following generators:

$$\begin{array}{llll}{\bf v}_2,&{\bf v}_5,&{\bf v}_6,& {\bf v}_7 =\left( x\,\log\left( x\right) -x\right) \frac{\partial}{\partial x}-2\,u\,\log( x) \frac{\partial}{\partial u}\end{array}$$

6.For \(f(u)={\it c_2}\,e^{n\,u}-{{\it c_1}\over{n}}\) and \(g(u)= de^{nu}\) (\(n\neq 0\)) we obtain the following generators:

$$\begin{array}{ll}{\bf v}_2, & {\bf v}_8 =\displaystyle{\it n}\,e^{{\it c_1}\,t}\frac{\partial}{\partial t}-{{\it c_1}\,e^{{\it c_1}\,t}}\frac{\partial}{\partial u}\end{array}$$

7.For \(f(u)=-{{\it c_1}\over{n}}\) and \(g(u)= de^{nu}\) (\(n\neq 0\)) we obtain the following generators:

$$\begin{array}{lll}{\bf v}_2,& {\bf v}_8, &{\bf v}_9 =\displaystyle n x\frac{\partial}{\partial x}+2\frac{\partial}{\partial u}\end{array}$$

8.For \(f(u)={\it c_2}\,e^{n\,u}\) and \(g(u)= d\,e^{n\,u}\) (\(n\neq 0\)) we obtain the following generators:

$$\begin{array}{lll}{\bf v}_2, &{\bf v}_{10} =\displaystyle nt\frac{\partial}{\partial t}-\frac{\partial}{\partial u}\end{array}$$

2.1 Optimal Systems and Reductions

The corresponding generators of the optimal system of subalgebras, [8] are:

1.For \(f(u)=u^{n}\) and \(g(u)=u^m\)

$$\begin{array}{ll}a{\bf v}_1+{\bf v}_2, & {\bf v}_3.\end{array}$$

2.For \(f(u)=e^{nu}\) and \(g(u)=e^{mu}\)

$$\begin{array}{ll}a{\bf v}_1+{\bf v}_2, & {\bf v}_4.\end{array}$$

3.For \(f(u)=c_2u^{n+1}-\frac{c_1u}{n}\) and \(g(u)=u^{n}\)

$$\begin{array}{ll}a{\bf v}_1+{\bf v}_2, & {\bf v}_5.\end{array}$$

4.For \(f(u)=u^{-\frac{1}{3}}\) and \(g(u)=u^{-\frac{4}{3}}\)

$$\begin{array}{llll}a{\bf v}_1+{\bf v}_2, & b{\bf v}_1+{\bf v}_3,& c{\bf v}_3+{\bf v}_6, & d{\bf v}_3+{\bf v}_7,\end{array}$$

where \(a,b,c,d \in R\) are arbitrary.

In the following, reductions of Eq. (1) to ordinary differential equations (ODEs) are obtained by using the generators of the optimal system.

Reduction 1

Generator \({\bf v}_1+{\bf v}_2\). Substituting the similarity variable and similarity solution: {

$$\begin{array}{ll}z=x-c\,t, & u=h\left( z\right)\end{array}$$

}into (1) we obtain:

$$\begin{aligned}g\,h_{zz}+g_{h}\,\left(h_{z}\right)^2+c\,h_{z}+f=0\end{aligned}$$
(2)

Reduction 2

Generator \({\bf v}_3.\) Substituting the similarity variable and similarity solution: {

$$\begin{array}{ll}z={t}^{\frac{n-m+1}{2\,m-2}}\,x,& u=h\,{x}^{\frac{2}{n-m+1}}\end{array}$$

} into (1) we obtain:

$$\begin{array}{l}{h}^{n}\,{\left( n-m+1\right)}^{2}\,\left( { h_{z} }^{2}\,n+h\,{h}_{zz} \right) \,{z}^{2}+4\,{h}^{n+1}\,{h}_{z} \,\left( n+1\right) \,\left( n-m+1\right) \,z\\ +2\,{h}^{n+2}\,\left( n+m+1\right) +{h}^{m+1}\,{\left( n-m+1\right)}^{2}=0\end{array}$$

Reduction 3

Generator \({\bf v}_4.\) Substituting the similarity variable and similarity solution with \(n\neq m\): {

$$\begin{array}{ll}z={t}^{\frac{n-m}{2\,m}}\,x, & u=\frac{2\,\log\left( x\right)}{n-m}+h\end{array}$$

} into (1) we obtain:

\(\begin{array}{l} h_{zz}\,{\left( n-m\right)}^{2}\,{e}^{\frac{h\,{n}^{2}+h\,{m}^{2}}{n-m}}\,{z}^{\frac{2\,n+m}{n-m}+2}+{ h_{z} }^{2}\,n\,{\left( n-m\right)}^{2}\,{e}^{\frac{h\,{n}^{2}+h\,{m}^{2}}{n-m}}\,{z}^{\frac{4\,n-m}{n-m}}\\ +\left( \left( 2\,n+2\,m\right) \,{e}^{\frac{h\,{n}^{2}+h\,{m}^{2}}{n-m}}+\left({n}^{2}-2\,m\,n+{m}^{2}\right) \,{e}^{\frac{2\,h\,m\,n}{n-m}}\right) \,{z}^{\frac{2\,n+m}{n-m}}\\ +4\, h_{z} \,n\,\left( n-m\right) \,{e}^{\frac{h\,{n}^{2}+h\,{m}^{2}}{n-m}}\,{z}^{\frac{3\,n}{n-m}}=0\end{array}\)

Reduction 4

Generator \({\bf v}_5.\) Substituting the similarity variable and similarity solution with \(n\neq 0\): {

$$\begin{array}{ll}z=x,& u=h\,e^ {- {{{\it c_1}\,t}\over{n}} }\end{array}$$

} into (1) we obtain:

\(\begin{array}{l}h^{n}\,\left(h_{z}\right)^2\,n+{\it c_2}\,h^{n+2}+h^{n+1}\,h_{zz}=0\end{array}\)

Reduction 5

Generator \(b{\bf v}_1+{\bf v}_5.\) Substituting the similarity variable and similarity solution: {

$$\begin{array}{ll}z={{8\,x-3\,b\,\log t}\over{8}},& u=h\,t^{{3}\over{4}}\end{array}$$

} into (1) we obtain:

\(\begin{array}{l}-12\,h\,h_{zz}+16\,\left(h_{z}\right)^2-{{9\,b\,h^{{7}\over{3}} \,h_{z}}\over{2}}+9\,h^{{10}\over{3}}-12\,h^2=0\end{array}\)

Reduction 6

Generator \(c{\bf v}_3+{\bf v}_6.\) Substituting the similarity variable and similarity solution: {

$$\begin{array}{ll}z=-{{4\,c\,e^ {- {{2\,x}\over{\sqrt{3}}} }}\over{\sqrt{3}}}-\log t,& u=h\,e^{-\sqrt{3}\,c\,e^ {- {{2\,x}\over{\sqrt{3}}} }-\sqrt{3}\,x}\end{array}$$

} into (1) we obtain:

\(\begin{array}{l}27\,h^{{7}\over{3}}\,h_{z}\,e^{z}+192\,c^2\,h\,h_{zz}-256\,c^2 \,\left(h_{z}\right)^2-96\,c^2\,h\,h_{z}-36\,c^2\,h^2=0\end{array}\)

3 Travelling Wave Solutions

We are interested in finding exact travelling wave solutions for Eq. (1) when the diffusion coefficient follows a power law \(g(u)=u^m\). From generators \({\bf v}_1\) and \({\bf v}_2\) we can obtain travelling wave solutions for Eq (1).

To apply the \(\frac{G'}{G}\)-expansion method to Eq. (2) we suppose that the solutions can be expressed by a polynomial in \(\frac{G'}{G}\) in the form

$$\begin{aligned}h=\displaystyle\sum_{i=0}^n\, a_i\left(\frac{G'}{G}\right)^i,\end{aligned}$$
(3)

where \(G=G(z)\) satisfies the linear second order ODE

$$\begin{aligned}G''(z)+\omega G'(z)+\zeta G(z)=0,\end{aligned}$$
(4)

with a i , \(i=0,\ldots,n\), ω and ζ constants to be determined later and \(a_n\neq 0\). The general solutions are well known.

The homogeneous balance between the leading terms provides us with the value of n [6]. Considering the homogeneous balance between \(h''\) and h 2 in (2), we require that \(nm+n+2=n(m-1)+(n+1)^2\Rightarrow n=1\), we can write (3) as

$$\begin{aligned}\begin{array}{ll}\displaystyle h=a_0+a_1\left(\frac{G'}{G}\right),& a_1\neq 0.\end{array}\end{aligned}$$
(5)

From the general solutions of (4), setting without loss of generality \(a_0=a_1=1\), we obtain for

$$\begin{array}{l}g(h)=h^m,\\ f(h)=\left(h^2-2\,h+2\right)\,\left(\lambda-2\,h^{m+1}+2\,h^{m}\right)-2 \,h^{m-1}\,\left(h^2-2\,h+2\right)^2\,m\end{array}$$

the solution

$$h_1(z)={{{\it c_1}\,\cos z-{\it c_2}\,\sin z}\over{{\it c_1}\,\sin z+ {\it c_2}\,\cos z}}+1.$$

The corresponding solution for the generalized Fisher equation is

$$u_1={{{\it c_2}\,\sin \left(t\,\lambda-x\right)+{\it c_1}\,\cos \left(t \,\lambda-x\right)}\over{{\it c_2}\,\cos \left(t\,\lambda-x\right)- {\it c_1}\,\sin \left(t\,\lambda-x\right)}}+1.$$

For

$$\begin{array}{l}g(u)=u^m\\ f(u)=\left(u-2\right)\,u\,\left(\lambda-u^{m+1}\,m+2\,u^{m}\,m-2\,u^{m+1}+2\,u^{m}\right),\end{array}$$

the corresponding solution for the generalized Fisher equation is

$$\begin{aligned}u_2={{{\it c_2}\,\sinh (x-\lambda t)+{\it c_1}\,\cosh (x-\lambda t)}\over{{\it c_1}\,\sinh (x-\lambda t)+ {\it c_2}\,\cosh (x-\lambda t)}}+1.\end{aligned}$$
(6)

Considering the case \(\zeta=0\), \(\omega \neq 0\), for

$$\begin{array}{l}g(u)=u^m\\ f(u)=-u^{m+1}\,m\,\omega^2-u^{m+1}\,\omega^2-2\,u^{m+2}\,m\,\omega-3\,u^{m+2}\,\omega+c\,h\, \omega-u^{m+3}\,m-2\,u^{m+3}+c\,u^2\end{array}$$
$$\begin{aligned}u(x-\lambda t)=-{{{\it c_2}\,\omega\,e^ {- \omega\,(x-\lambda t)}}\over{{\it c_2}\,e^ {- \omega\,(x-\lambda t)}+ {\it c_1}}},\end{aligned}$$
(7)

which is a kink solution (Fig. 1).

Fig. 1
figure 1

Kink solution (7), \(c_1=c_2=\omega=1\), \(\lambda=-1\)

Full size image

4 Acknowledgments

The authors acknowledge the financial support from Junta de Andalucía group FQM–201, University of Cadiz.