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Lie symmetries and multiple solutions in \(\lambda-\omega\) reaction-diffusion systems. (English) Zbl 0924.35061

The \(\lambda - \omega\) reaction-diffusion system of the form \[ u_t=D\nabla^2u+\lambda (z)u-\omega (z)v,\qquad v_t=D\nabla^2v+\omega (z)u+\lambda (z)v \] is considered, where \(u=u(x,y,t)\), \(v=v(x,y,t)\), \(D\) is a constant, \(z=(u^2+v^2)^{1/2}\), \(\lambda (z)\) is a positive function of \(z\) for \(0\leq z< z_0\) and negative for \(z>z_0\), \(\omega (z)\) is a positive function. The symmetry algebra generated by translations with respect to \(x, y, t\), rotations in the plane \((x,y)\) and rotations in the plane \((u,v)\) is considered. Invariant solutions with respect to different subalgebras of this algebra are obtained.

MSC:

35K57 Reaction-diffusion equations
58J70 Invariance and symmetry properties for PDEs on manifolds
35K40 Second-order parabolic systems
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