A dynamical study of certain nonlinear diffusion–reaction equations with a nonlinear convective flux term

Abstract

We explore the dynamics of quadratic and quartic nonlinear diffusion–reaction equations with nonlinear convective flux term, which arise in well-known physical and biological problems such as population dynamics of the species. Three integration techniques, namely the $({G^\prime }/{G})$-expansion method, its generalised version and Kudryashov method, are adopted to solve these equations. We attain new travelling and solitary wave solutions in the form of Jacobi elliptic functions, hyperbolic functions, trigonometric functions and rational solutions with some constraint relations that naturally appear from the structure of these solutions. The travelling population fronts, which are the general solutions of nonlinear diffusion–reaction equations, describe the species invasion if higher population density corresponds to the species invasion. This effort highlights the significant features of the employed algebraic approaches and shows the diversity in the constructed solutions.

Appendices

Appendix A

The general solutions to Jacobi elliptic equation and their derivative (see e.g. [53, 55]) are listed in table 1.

In table 1, the elliptic modulus m of the Jacobi elliptic functions varies between $0<m<1$ and $\mathrm {i}=\sqrt{-1}$.

Appendix B

When $m \rightarrow 1$, the Jacobi elliptic functions sn($\xi$), cn($\xi$), dn($\xi$), ns($\xi$), cs($\xi$), ds($\xi$), sc($\xi$) and sd($\xi$) degenerate into hyperbolic functions as follows:

$$\begin{aligned} \begin{array}{ll} \mathrm{sn}(\xi )\rightarrow \tanh (\xi ), &{}\quad \mathrm{cn}(\xi )\rightarrow {{\mathrm{sech}}}(\xi ), \\ \mathrm{dn}(\xi )\rightarrow {{\mathrm{sech}}}(\xi ), &{}\quad \mathrm{cs}(\xi )\rightarrow \mathrm {cosech}(\xi ),\\ \mathrm{ds}(\xi )\rightarrow \mathrm {cosech}(\xi ), &{}\quad \mathrm{ns}(\xi )\rightarrow \coth (\xi ),\\ \mathrm{sc}(\xi )\rightarrow \sinh (\xi ), &{}\quad \mathrm{sd}(\xi )\rightarrow \sinh (\xi ) \end{array} \end{aligned}$$

and into trigonometric functions if $m \rightarrow 0$ as follows:

$$\begin{aligned} \begin{array}{ll} \mathrm{sn}(\xi )\rightarrow \sin (\xi ), &{}\quad \mathrm{cn}(\xi )\rightarrow \cos (\xi ), \\ \mathrm{dn}(\xi )\rightarrow 1,&{}\quad \mathrm{ns}(\xi )\rightarrow \mathrm {cosec}(\xi ),\\ \mathrm{cs}(\xi )\rightarrow \cot (\xi ), &{}\quad \mathrm{ds}(\xi )\rightarrow \mathrm {cosec}(\xi ),\\ \mathrm{sc}(\xi )\rightarrow \tan (\xi ), &{}\quad \mathrm{sd}(\xi )\rightarrow \sin (\xi ). \end{array} \end{aligned}$$

Appendix C

$$\begin{aligned} \mathrm{cd}(\xi )=\dfrac{\mathrm{cn}(\xi )}{\mathrm{dn}(\xi )},&\quad \mathrm{dc}(\xi )=\dfrac{\mathrm{dn}(\xi )}{\mathrm{cn}(\xi )},\\ \mathrm{nc}(\xi )=\dfrac{1}{\mathrm{cn}(\xi )},&\quad \mathrm{nd}(\xi )=\dfrac{1}{\mathrm{dn}(\xi )},\\ \mathrm{cs}(\xi )=\dfrac{\mathrm{cn}(\xi )}{\mathrm{sn}(\xi )},&\quad \mathrm{sc}(\xi )=\dfrac{\mathrm{sn}(\xi )}{\mathrm{cn}(\xi )}, \\ \mathrm{sd}(\xi )=\dfrac{\mathrm{sn}(\xi )}{\mathrm{dn}(\xi )},&\quad \mathrm{ds}(\xi )=\dfrac{\mathrm{dn}(\xi )}{\mathrm{sn}(\xi )}. \end{aligned}$$