The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen

Full-Text PDF (1409 KB) | Metadata | Table of Contents | ZAA summary
Volume 6, Issue 6, 1987, pp. 485–503
DOI: 10.4171/ZAA/267

Published online: 1987-12-31

Stability Properties of Space Periodic Standing Waves

B. Scarpellini[1]

(1) Universität Basel, Switzerland

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 $u(\delta) \in \mathbb R^2, \delta \in I = (– \epsilon, \epsilon)$, 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.

No keywords available for this article.

Scarpellini B.: Stability Properties of Space Periodic Standing Waves. Z. Anal. Anwend. 6 (1987), 485-503. doi: 10.4171/ZAA/267