Steady solutions of the Kuramoto-Sivashinsky equation for small wave speed. (English) Zbl 0745.34037
The existence of periodic and monotone bounded solutions of the equation \(w'''+w'=c^ 2-w^ 2/2\), \(-\infty<\xi<+\infty\), \(c>0\), referred to as the Kuramoto-Sivashinski equation, is investigated. The main results are summarized in the following two theorems:
Theorem 1. For each \(c>0\) there exists no solution of the considered equation which satisfies \(w'>0\) for all \(\xi\in(-\infty,\infty)\) with \(\lim_{\xi\to\pm\infty}w(\xi)=c\sqrt 2\).
Theorem 2. There exists \(\bar c>0\) such that for each \(c\in(0,\bar c)\) there is an odd periodic solution of the considered equation.
Theorem 1. For each \(c>0\) there exists no solution of the considered equation which satisfies \(w'>0\) for all \(\xi\in(-\infty,\infty)\) with \(\lim_{\xi\to\pm\infty}w(\xi)=c\sqrt 2\).
Theorem 2. There exists \(\bar c>0\) such that for each \(c\in(0,\bar c)\) there is an odd periodic solution of the considered equation.
Reviewer: I.Ginchev (Varna)
MSC:
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
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