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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

(1)U t =[U m U x ] x +λU m U x +C(U);(2)U t =[U m U x ] x +λU m+1 U x +C(U),

where λ and m are arbitrary constants and C(U) 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= t +ξ(t,x,U) x +η(t,x,U) U if and only if it and relevant operator have the following forms:

(i) C(U)=(λ 1 U m+1 +λ 2 )(U -m -λ 3 ), m-1, λ 2 0; Q= t +(λ 1 U+λ 2 U -m ) U ;

(ii) C(U)=(λ 1 lnU+λ 2 )(U-λ 3 ), m=-1,λ 1 0; Q= t +(λ 1 lnU+λ 2 )U U ;

(iii) C(U)=(λ 1 U+λ 2 U 1/2 +λ 3 ), m=-1/2; Q= t +f(t,x) x +2(g(t,x)U+h(t,x)U 1/2 ) U ,

where 2ff x +f t +fg=0, f xx -λf x -2g x -fh=0, (g-λ 1 /2)(g+2f x )+g t =0, 2gh-λ 1 h+2f x h-λ 2 f x +h t -λg x -g xx =0, h 2 -λ 2 2h-λ 3 f x +λ 3 2g-λh x -h x x=0, where λ 1 , λ 2 and λ 3 are arbitrary constants. The obtained Q-conditional symmetries are applied for constructing a wide range of exact solutions of the nonlinear RDC equations.

35K57Reaction-diffusion equations
58J70Invariance and symmetry properties
35C05Solutions of PDE in closed form