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Zeitschrift für Analysis und ihre Anwendungen


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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.

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