Let us assume that equation (1), with independent variables *t*, *x* and dependent variable *u*, is invariant under one parameter continuous transformations:

$$\begin{aligned}& \breve{t}=t+\epsilon\xi^{1}(t,x,u)+ O\bigl(\epsilon^{2} \bigr), \\& \breve{x}=x+\epsilon\xi^{2}(t,x,u)+ O\bigl(\epsilon^{2} \bigr), \\& \breve{u}=u+\epsilon\phi(t,x,u)+ O\bigl(\epsilon^{2}\bigr), \end{aligned}$$

where *ϵ* is the group parameter. The associated Lie algebra of infinitesimal symmetries is the set of vector fields of the form

$$ V=\xi^{1}(t,x,u)\frac{\partial}{\partial t}+\xi^{2}(t,x,u) \frac{\partial}{\partial{x}} +\phi(t,x,u)\frac{\partial}{\partial{u}}. $$

(18)

If the above vector field generates a symmetry of equation (1), then

$$ \operatorname{Pr}^{(2)}V(F)\vert_{F=0}=0,\quad F:= \frac{\partial u}{\partial t}-\frac{\partial}{\partial x} \biggl(u^{m}\frac{\partial u}{\partial x} \biggr)-u(1-u) (u-\alpha), $$

(19)

where \(\operatorname{Pr}^{(2)}X\) denotes the second prolongation of *V* and it concludes determining the equations containing an overdetermined system of linear PDEs in \(\xi^{1}\), \(\xi^{2}\), and *ϕ*.

These determining equations can easily be integrated to show that the symmetry group of equation (1) is spanned by the vector fields:

$$ V_{1}=\frac{\partial}{\partial t},\qquad V_{2}=\frac{\partial}{\partial x} $$

(20)

for arbitrary *m* and *α*,

$$ V_{1}=\frac{\partial}{\partial t}, \qquad V_{2}=\frac{\partial}{\partial x}, \qquad V_{3}=e^{-2t}\frac{\partial}{\partial t}+e^{-2t}u \frac{\partial}{\partial u} $$

(21)

for \(\alpha=-1\) and \(m=2\). However, it is important that the ambient concentration *α* should be in \((0,1)\). Thus, the last case \(\alpha=-1\) has no physical interpretation and we consider it just from the mathematical viewpoint. Here, we consider two nonzero conjugacy classes of one-dimensional subalgebras as follows:

$$ \mathcal{L}_{1,1}=\langle V_{1}+cV_{2} \rangle, \qquad \mathcal{L}_{2,1}=\langle V_{1}+cV_{3} \rangle. $$

(22)

Below, we list the corresponding similarity variables and similarity solutions as well as the reduced ODEs obtained from the generators of optimal system.

### Reduction 1

Using the subalgebra \(\mathcal{L}_{1,1}\), we obtain the similarity variables and similarity solutions \(u(t,x)=F(\xi)\), \(\xi=x-ct\), and the reduced ODE is

$$ \bigl(\bigl(F(\xi)^{m}F'(\xi)\bigr) \bigr)'+cF'(\xi)+F(\xi) \bigl(1-F(\xi) \bigr) \bigl(F( \xi)-\alpha \bigr)=0, $$

(23)

where \(c\in\mathbb{R}\) is the wave speed. Solutions of equation (23) are indeed traveling wave solutions of (1), which previously were discussed in the literature [17–21]. The solutions of equation (23) are indeed traveling wave solutions of (1) which previously were discussed in the literature [17–21]. Now, if we solve (23) by GPS, then the obtained results are indeed the solutions of (1) achieved by LSGPS. The natural conditions

$$ \lim_{\xi\rightarrow-\infty}F(\xi)=A,\qquad \lim_{\xi\rightarrow \infty}F( \xi)=B, $$

(24)

where \(A,B\in\{0,1,\alpha\}\) are taken into account for dealing with (23). Also, the initial conditions \(F(0)=0.5\) and \(F'(0)=\lambda\) (\(\lambda\in\mathbb{R}\) and specially \(\lambda=0\)), which are determined by the application at hand are applied in our computations. As can be seen from the figures of this paper, it tries to achieve the equilibrium \(\lim_{\xi\rightarrow +\infty}F(\xi)=\alpha\). To illustrate the behavior of traveling wave solutions with different parameters *m*, *c* and *α*, we provide several numerical plots obtained from GPS. Figures 1 and 2 show the solutions with varying initial conditions \(F'(0)\) with respect to \(\alpha=0.6\) and \(\alpha=0.2\), respectively. Because of the singularity for large values of \(\|F'(0)\|\), small values are singled out. Equilibriums \(F(\xi)\rightarrow0.6\) as \(\xi\rightarrow+\infty\) in Figure 1 and \(F(\xi)\rightarrow0.2\) as \(\xi\rightarrow+\infty\) in Figure 2 are depicted. Now, in Figures 3-12, we take the pragmatic convection \(F'(0)=0\).

Traveling waves, by varying wave speeds *c*, are demonstrated in Figure 3. A high smoothness of the solutions by varying *c* from 0.1 to 4 is reported. Similar to the two previous figures, by increasing the *c*, a smooth tending of the solutions to \(\alpha=0.2\) is reported in Figure 4.

As mentioned in [17], the power *m* appearing in the nonlinear density dependence serves to smooth out the solutions of the density-dependent diffusion Nagumo equation. Indeed, the oscillations diminish while *m* is increasing, and Figures 5 and 6 exhibit that there are more oscillations for the standard Nagumo equation (\(m=0\)) than \(m>0\) and monotone solutions for \(m\geq2\) are reported.

Now, we present the obtained traveling wave solutions for variable density parameter \(\alpha\in(0,1)\), with respect to \(c=1\) and \(c=0.3\). Comparison of Figures 7, 9, and 11 with their related Figures 8, 10, and 12 shows that the solutions with respect to \(c=1\) tend directly to *α* in a more streamlined manner than the solutions with respect to \(c=0.3\).

### Reduction 2

Using the subalgebra \(\mathcal{L}_{2,1}\), we obtain the similarity variables and similarity solutions \(u(t,x)=\frac{e^{t}\Psi(\xi)}{\sqrt{e^{2t}+c}}\), \(\xi=x\), and the reduced ODE is

$$ \Psi^{2}(\xi)\Psi''(\xi)+2 \Psi(\xi) \bigl(\Psi'(\xi) \bigr)^{2}+\Psi(\xi )- \Psi^{3}(\xi)=0. $$

(25)

Now, we utilize the reduction method [11, 28–32], instead of using the usual method based on invariants. Obtaining the first integrals of ODEs is often sophisticated work, however, using the mentioned reduction method, the first integral of the reduced ODE (25) is easily obtained. Equation (25) can be written as an autonomous system of two ODEs of first order, *i.e.*:

$$ \left \{ \textstyle\begin{array}{l} w_{1}'= w_{2}, \\ w_{2}'= \frac{w_{1}^{2}-1-2w_{2}^{2}}{w_{1}}, \end{array}\displaystyle \right . $$

(26)

using the obvious change of dependent variables

$$ w_{1}(\xi)=\Psi(\xi),\qquad w_{2}(\xi)=\Psi'( \xi). $$

(27)

It is possible to choose \(w_{1}\) as a new independent variable, because (26) is autonomous. In this way, a first-order nonautonomous ODE can be extracted from system (26) as follows:

$$ \frac{dw_{2}}{dw_{1}}=\frac{w_{1}^{2}-1-2w_{2}^{2}}{w_{1}w_{2}}, $$

(28)

of which its integration leads to

$$ w_{2}=\sqrt{\frac{6a_{1}+2w_{1}^{6}-3w_{1}^{4}}{6w_{1}^{4}}}, $$

(29)

with \(a_{1}\) an arbitrary constant. Clearly, corresponding first integral of equation (25) is

$$ \bigl(\Psi'(\xi) \bigr)^{2}\Psi^{4}(\xi)- \frac{1}{3}\Psi^{6}(\xi)+\frac {1}{2}\Psi^{4}( \xi)=a_{1}. $$

(30)

Lastly we replace (29) from (26) into the first equation of system (26) from which one concludes to a first-order separable as follows:

$$ w_{1}'=\sqrt{\frac{6a_{1}+2w_{1}^{6}-3w_{1}^{4}}{6w_{1}^{4}}}. $$

(31)

The exact solution of equation (31) can be implicitly expressed by

$$ \int\frac{6w_{1}^{2}}{\sqrt{36a_{1}+12w_{1}^{6}-18w_{1}^{4}}}\, dw_{1}=\xi+a_{2}, $$

(32)

and replacing \(w_{1}\) with \(\Psi(\xi)\) yields the exact solution of (25). Assuming \(a_{1}=0\) yields the following explicit solutions:

$$ \Psi_{\pm}(\xi)=\frac{\sqrt{2}}{4} \biggl( \frac{3+e^{\pm\frac{2(\xi +a_{2})}{\sqrt{3}}}}{e^{\pm\frac{\xi+a_{2}}{\sqrt{3}}}} \biggr), $$

(33)

and therefore

$$ u_{\pm}(t,x)=\frac{\sqrt{2}e^{t}}{4\sqrt{e^{2t}+c}} \biggl(\frac{3+e^{\pm\frac {2(x+a_{2})}{\sqrt{3}}}}{e^{\pm\frac{x+a_{2}}{\sqrt{3}}}} \biggr). $$

(34)

The solution of \(u_{+}(t,x)\) in three dimensions is plotted in Figure 13 with respect to \(c=a_{2}=1\).