# Q-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems

## Abstract

**:**

## 1. Introduction

## 2. Main Result

**Theorem 1.**In the cases ${\lambda}_{2}={\lambda}_{1}$ or ${\eta}_{v}^{1}={\eta}_{u}^{2}=0$ with conditions (8), the system of determining equations for finding of the Q-conditional operators of the form (5) for system (6) coincide with the system of determining equations for finding Lie operators.

**Proof.**Substituting (13), with ${\lambda}_{2}={\lambda}_{1}$, into system (7) we find that Equations $1)-11)$ are transformed into identities, and Equations $12)$ and $13)$ take the form

**Theorem 2.**The nonlinear reaction–diffusion system (6) is Q-conditionally invariant under operator (5) with coefficients (17) if and only if the nonlinearities ${C}^{1},\phantom{\rule{4pt}{0ex}}{C}^{2}$ are the solutions of linear system (18).

**Theorem 3.**Reaction–diffusion system (6) is Q-conditionally invariant under operator (5) with conditions (8), and ${\eta}_{u}^{2}=0$, if and only if the system and corresponding operator have one of the seven following forms (moreover ${\lambda}_{2}\ne {\lambda}_{1}$):

**Proof.**To prove this theorem, it is necessary and sufficient to construct the general solution of system (22) for all possible ratios between parameters ${\beta}_{1},\phantom{\rule{4pt}{0ex}}{\beta}_{2},\phantom{\rule{4pt}{0ex}}{\gamma}_{2}.$ To do this we need to investigate the following seven cases:

## 3. Ansätze and Exact Solutions of the Reaction–Diffusion System

No. | Ansätze | Systems of ODEs |
---|---|---|

(23) | $u=\left(t+\psi \right){e}^{{\beta}_{1}t+\phi}$ | ${\phi}^{\prime \prime}+{\left({\phi}^{\prime}\right)}^{2}-{\beta}_{1}\left(g\left({e}^{{\beta}_{1}\psi -\phi}\right)+\frac{{\lambda}_{1}+{\lambda}_{2}}{2}\right)=0$ |

$v={e}^{{\beta}_{1}t+\phi}$ | ${\psi}^{\prime \prime}+2{\phi}^{\prime}{\psi}^{\prime}+g\left({e}^{{\beta}_{1}\psi -\phi}\right)\left({\beta}_{1}\psi -\phi \right)-h\left({e}^{{\beta}_{1}\psi -\phi}\right)-\frac{{\lambda}_{1}+{\lambda}_{2}}{2}=0$ | |

(24) | $u=\psi \left({e}^{{\beta}_{2}t+\phi}\right){}^{\frac{{\beta}_{1}}{{\beta}_{2}}}-\frac{{e}^{{\beta}_{2}t+\phi}}{{\beta}_{1}-{\beta}_{2}}$ | ${\phi}^{\prime \prime}+{\left({\phi}^{\prime}\right)}^{2}+g\left(\left({\beta}_{1}-{\beta}_{2}\right)\psi \right)\left({\beta}_{1}-{\beta}_{2}\right)-{\beta}_{2}\frac{{\lambda}_{1}+{\lambda}_{2}}{2}=0$ |

$v={e}^{{\beta}_{2}t+\phi}$ | ${\psi}^{\prime \prime}+\frac{2{\beta}_{1}}{{\beta}_{2}}{\phi}^{\prime}{\psi}^{\prime}+\frac{{\beta}_{1}}{{\beta}_{2}}\left(\frac{{\beta}_{1}}{{\beta}_{2}}-1\right)\psi \left({\left({\phi}^{\prime}\right)}^{2}-{\beta}_{2}g\left(\left({\beta}_{1}-{\beta}_{2}\right)\psi \right)\right)-h\left(\left({\beta}_{1}-{\beta}_{2}\right)\psi \right)=0$ | |

(25) | $u=\phi t+\psi $ | ${\phi}^{\prime \prime}-\phi g\left(\phi \right)=0$ |

$v=\phi $ | ${\psi}^{\prime \prime}-g\left(\phi \right)\psi -h\left(\phi \right)-{\lambda}_{1}\phi =0$ | |

(26) | $u=\frac{1}{2}{t}^{2}+\phi t+\psi $ | ${\phi}^{\prime \prime}-g\left(2\psi -{\phi}^{2}\right)-\frac{{\lambda}_{1}+{\lambda}_{2}}{2}=0$ |

$v=t+\phi $ | ${\psi}^{\prime \prime}-g\left(2\psi -{\phi}^{2}\right)\phi -h\left(2\psi -{\phi}^{2}\right)-\frac{{\lambda}_{1}+{\lambda}_{2}}{2}\phi =0$ | |

(27) | $u=\frac{1}{{\beta}_{2}}{e}^{{\beta}_{2}t+\phi}-{\tau}_{2}t+\psi $ | ${\phi}^{\prime \prime}+{\left({\phi}^{\prime}\right)}^{2}-{\beta}_{2}\phantom{\rule{4.pt}{0ex}}g\left({\tau}_{2}(\phi +1)+{\beta}_{2}\psi \right)-\frac{{\lambda}_{1}+{\lambda}_{2}}{2}{\beta}_{2}=0$ |

$v={e}^{{\beta}_{2}t+\phi}-{\tau}_{2}$ | ${\psi}^{\prime \prime}-h\left({\tau}_{2}(\phi +1)+{\beta}_{2}\psi \right)+{\tau}_{2}g\left({\tau}_{2}(\phi +1)+{\beta}_{2}\psi \right)+\frac{{\lambda}_{1}+{\lambda}_{2}}{2}{\tau}_{2}=0$ | |

(28) | $u=\psi {e}^{{\beta}_{1}t}-\phi $ | ${\phi}^{\prime \prime}-g\left(\phi \right)=0$ |

$v=\phi $ | ${\psi}^{\prime \prime}-\left(h\left(\phi \right)+{\beta}_{1}{\lambda}_{1}\right)\psi =0$ | |

(29) | $u=\psi {e}^{{\beta}_{1}t}-\frac{{\gamma}_{2}t}{{\beta}_{1}}-\frac{\phi}{{\beta}_{1}}-\frac{{\gamma}_{2}}{{\beta}_{1}^{2}}$ | ${\phi}^{\prime \prime}-g\left({\beta}_{1}^{2}\psi {e}^{-\frac{{\beta}_{1}\phi}{{\gamma}_{2}}}\right){\beta}_{1}-\frac{\left({\lambda}_{1}+{\lambda}_{2}\right)}{2}{\gamma}_{2}=0$ |

$v={\gamma}_{2}t+\phi $ | ${\psi}^{\prime \prime}-{e}^{\frac{{\beta}_{1}\phi}{{\gamma}_{2}}}h\left({\beta}_{1}^{2}\psi {e}^{-\frac{{\beta}_{1}\phi}{{\gamma}_{2}}}\right)-{\beta}_{1}\frac{{\lambda}_{1}+{\lambda}_{2}}{2}\psi =0$ |

## 4. Solutions and Their Properties of Some Generalization of the Lotka–Volterra System

**Figure 1.**Exact solution to (51).

**Figure 2.**Exact solution of (55).

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

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Pliukhin, O.
*Q*-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems. *Symmetry* **2015**, *7*, 1841-1855.
https://doi.org/10.3390/sym7041841

**AMA Style**

Pliukhin O.
*Q*-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems. *Symmetry*. 2015; 7(4):1841-1855.
https://doi.org/10.3390/sym7041841

**Chicago/Turabian Style**

Pliukhin, Oleksii.
2015. "*Q*-Conditional Symmetries and Exact Solutions of Nonlinear Reaction–Diffusion Systems" *Symmetry* 7, no. 4: 1841-1855.
https://doi.org/10.3390/sym7041841