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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
\[ (1)\quad U_t=[U^mU_x]_x+\lambda U^mU_x+C(U);\qquad (2)\quad U_t=[U^mU_x]_x+\lambda U^{m+1}U_x+C(U), \]
where \(\lambda\) 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 \(Q=\partial_t+ \xi(t,x, U)\partial_x+\eta(t,x,U)\partial_U\) if and only if it and relevant operator have the following forms:
(i) \(C(U)=(\lambda_1U^{m+1}+ \lambda_2)(U^{-m}-\lambda_3)\), \(m\neq-1\), \(\lambda_2\neq0\); \(Q= \partial_t+(\lambda_1U+\lambda_2U^{-m})\partial_U\);
(ii) \(C(U)= (\lambda_1\ln U+\lambda_2)(U-\lambda_3)\), \(m=-1,\lambda_1\neq0\); \(Q=\partial_t+(\lambda_1\ln U+\lambda_2)U\partial_U\);
(iii) \(C(U)=(\lambda_1U+\lambda_2U^{1/2}+\lambda_3)\), \(m=-1/2\); \(Q=\partial_t+f(t,x)\partial_x+2(g(t,x)U+h(t,x)U^{1/2}) \partial_U\),
where \(2ff_x+f_t+fg=0\), \(f_{xx}-\lambda f_x-2g_x-fh=0\), \((g-\lambda_1/2)(g+2f_x)+g_t=0\), \(2gh-\lambda_1h+2f_xh-\lambda_2f_x+ h_t- \lambda g_x-g_{xx}=0\), \(h^2-\frac{\lambda_2}2h-\lambda_3f_x+ \frac{\lambda_3}2g-\lambda h_x- h_xx=0\), 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 for PDEs on manifolds
35C05 Solutions to PDEs in closed form

Citations:

Zbl 0952.22002
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