## Stability of patterns with arbitrary period for a Ginzburg-Landau equation with a mean field.(English)Zbl 1146.35012

The authors consider the following real elliptic Ginzburg-Landau system with mean field $$\langle B\rangle=0$$, on the interval $$I:=(0,L/2)$$:
\left\{ \begin{alignedat}{2} -A_{xx}&=A-A^3-AB,&&\qquad t>0,\;x\in I,\\ -B_{xx}&=\mu(A^2)_{xx},&&\qquad t>0,\;x\in I,\\ A_x(0)&=A_x(L/2)=0,\\ B_x(0)&=B_x(L/2)=0.\end{alignedat}\right.\tag{1} It is assumed that $$\mu>1$$ and $$L>0$$. Solutions to (1) are in one-to-one correspondence with $$L$$-periodic solutions of the equation posed on the real line. Of interest is the existence and linear parabolic stability of solutions to (1) in dependence on the parameter $$\mu$$. It is shown that any solution of (1) for which $$A$$ or $$A_x$$ changes sign is linearly unstable. Therefore the article focuses on positive decreasing solutions.
The main result follows: There are constants $$\mu_1>\mu_2>1$$ that only depend on $$L$$, and that are given explicitly in terms of Jacobi Elliptic Integrals, with the following properties: (a) If $$\mu>\mu_1$$ then all positive decreasing solutions are constant. (b) If $$\mu=\mu_1$$ then there is exactly one positive decreasing solution, and it is linearly neutrally stable. (c) If $$\mu_1>\mu>\mu_2$$ then there are exactly two positive decreasing solutions. That with smaller value of $$A(0)$$ is linearly stable, while the other one is linearly unstable. (d) If $$\mu_2\geq\mu>1$$ then there is exactly one positive decreasing solution, and it is linearly unstable. These results give a characterization of the positive decreasing solutions of (1) and a complete bifurcation diagram for this subset of solutions.

### MSC:

 35B35 Stability in context of PDEs 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35B10 Periodic solutions to PDEs 35K57 Reaction-diffusion equations 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 35B32 Bifurcations in context of PDEs
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