Conditional Lie-Bäcklund symmetry of evolution system and application for reaction-diffusion system (Q2925253)

The authors develop the conditional Lie-Bäcklund symmetry (CLBS) method with the aim of applications to evolution equations and reaction-diffusion systems. It is proved that the reducibility of a system of evolution equations to an ODE system can be completely characterized by its CLB-symmetry. The main attention in this article is paid to the basic reduction theorem for the CLB-symmetries of the evolution system NEWLINE\[NEWLINE u^{(i)}_t=F^{(i)}(t,x,u^{(i)},u^{(i)}_1,\dots,u^{(i)}_{k_1},\dots,u^{(m)}_1,\dots,u^{(m)}_{k_m}),\quad i=1,\dots,m. NEWLINE\]NEWLINE For the illustration of the reduction theorem, the authors also consider the two-component nonlinear diffusion equation of the form NEWLINE\[NEWLINE U_t=[P(U)U_x]_x+G(U,V)V_x+R(U,V),\quad V_t=[Q(U)V_x]_x+H(U,V)U_x+S(U,V). NEWLINE\]NEWLINE For the last system, the classification on admitted CLB-symmetries is fulfilled. As a corollary, the exact solutions in the invariant subspaces determined by the admitted CLB-symmetries are constructed in the last section.