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New conditional symmetries and exact solutions of nonlinear reaction-diffusion-convection equations. (English) Zbl 1128.35358

$Q$-conditional symmetries [see I. W. Fushchich and W. M. Shtelen, Lett. Nuovo Cimento (2) 34, No. 16, 498–502 (1982); erratum ibid. 36, No. 4, 96 (1983; Zbl 0952.22002)] for two nonlinear reaction-diffusion-convection (RDC) equations with power diffusivities of the form

$\left(1\right)\phantom{\rule{1.em}{0ex}}{U}_{t}={\left[{U}^{m}{U}_{x}\right]}_{x}+\lambda {U}^{m}{U}_{x}+C\left(U\right);\phantom{\rule{2.em}{0ex}}\left(2\right)\phantom{\rule{1.em}{0ex}}{U}_{t}={\left[{U}^{m}{U}_{x}\right]}_{x}+\lambda {U}^{m+1}{U}_{x}+C\left(U\right),$

where $\lambda$ and $m$ are arbitrary constants and $C\left(U\right)$ is an arbitrary function are considered. The main results are presented in the form of two theorems.

Theorem 1: Equation (1) is $Q$-conditional invariant under the operator $Q={\partial }_{t}+\xi \left(t,x,U\right){\partial }_{x}+\eta \left(t,x,U\right){\partial }_{U}$ if and only if it and relevant operator have the following forms:

(i) $C\left(U\right)=\left({\lambda }_{1}{U}^{m+1}+{\lambda }_{2}\right)\left({U}^{-m}-{\lambda }_{3}\right)$, $m\ne -1$, ${\lambda }_{2}\ne 0$; $Q={\partial }_{t}+\left({\lambda }_{1}U+{\lambda }_{2}{U}^{-m}\right){\partial }_{U}$;

(ii) $C\left(U\right)=\left({\lambda }_{1}lnU+{\lambda }_{2}\right)\left(U-{\lambda }_{3}\right)$, $m=-1,{\lambda }_{1}\ne 0$; $Q={\partial }_{t}+\left({\lambda }_{1}lnU+{\lambda }_{2}\right)U{\partial }_{U}$;

(iii) $C\left(U\right)=\left({\lambda }_{1}U+{\lambda }_{2}{U}^{1/2}+{\lambda }_{3}\right)$, $m=-1/2$; $Q={\partial }_{t}+f\left(t,x\right){\partial }_{x}+2\left(g\left(t,x\right)U+h\left(t,x\right){U}^{1/2}\right){\partial }_{U}$,

where $2f{f}_{x}+{f}_{t}+fg=0$, ${f}_{xx}-\lambda {f}_{x}-2{g}_{x}-fh=0$, $\left(g-{\lambda }_{1}/2\right)\left(g+2{f}_{x}\right)+{g}_{t}=0$, $2gh-{\lambda }_{1}h+2{f}_{x}h-{\lambda }_{2}{f}_{x}+{h}_{t}-\lambda {g}_{x}-{g}_{xx}=0$, ${h}^{2}-\frac{{\lambda }_{2}}{2}h-{\lambda }_{3}{f}_{x}+\frac{{\lambda }_{3}}{2}g-\lambda {h}_{x}-{h}_{x}x=0$, where ${\lambda }_{1}$, ${\lambda }_{2}$ and ${\lambda }_{3}$ are arbitrary constants. The obtained $Q$-conditional symmetries are applied for constructing a wide range of exact solutions of the nonlinear RDC equations.

##### MSC:
 35K57 Reaction-diffusion equations 58J70 Invariance and symmetry properties 35C05 Solutions of PDE in closed form