Stability Properties of Space Periodic Standing Waves

Abstract

Equilibrium solutions of parabolic systems of the form $u=\Delta u+F(\alpha ,u)$ are considered, where $D$ designates as $2\times 2$ diagonal matrix, $\alpha$ a bifurcation parameter, $u=(u_1,u_2)$ a state vector and $F$ a polynomial nonlinearity. A trivial solution branch $u(\delta )\in {\mathbb{R}}^2,\delta \in I=(–ϵ,ϵ)$, is supposed to be given, i.e. $F({\alpha }_0+\delta ,u(\delta ))=0$ for some ${\alpha }_0$ and every $\delta \in I$. Then a period $L$ is fixed and under suitable assumptions space-$L$-periodic bifurcating standing waves are constructed. It is shown that these bifurcating branches become generically unstable as $L\uparrow \infty$. Under the condition of $d_{uu}F({\alpha }_0,u(0))=0$ however, they will remain stable against $nL$-periodic perturbations $(1<n\in \mathbb{N})$, provided that the trivial solution-branch $u(\delta )$ behaves alike for small $\delta <0$. The so-called Landau-Ginzburg equations arising in Landau’s theory of phase transitions constitute a special example in physics.