Unstable simple modes of the nonlinear string

Thierry Cazenave, Fred B. Weissler
1996 Quarterly of Applied Mathematics  
We prove instability of high-energy simple modes for the nonlinear vibrating string equation J utt -(a+ b /* ul(t, x) dx) uxx = 0, I u\dn = 0, in f] = (0,7r), where a > 0, b > 0. Resume. On montre l'instabilite des modes simples d'energie assez grande pour l'equation des cordes vibrantes non-lineaire utt -(a+ b ul(t, x) dx) uxx = 0, u\dn = 0, dans O = (0, ir), ou a > 0, b > 0. 1. Introduction. A classical model of the nonlinear vibrating string is given by the equation J utt -(a+ b fo ul(t,x)
more » ... -(a+ b fo ul(t,x) dx) uxx = 0, I U\gn = 0, in = (0,7i"), where a > 0, b > 0. This model has been extensively studied, for example, by Carrier [2], Bernstein [1], and Narashimha [8]. See also Dickey [4], Medeiros and Milla Miranda [7] and the references therein. Our main interest in this model is the stability of simple modes, i.e., solutions of the form u(t,x) = u>(t) sm(jx), such solutions necessarily being periodic in time. Dickey [4] showed that small amplitude simple modes are stable. Here, we prove that simple modes of sufficiently large energy are unstable, i.e., that they have nontrivial unstable manifolds. We study equation (1.1) using Fourier series. Setting Uj (t) = ^ J u(t, x) sin(jx) dx,
doi:10.1090/qam/1388017 fatcat:2c2icgk7tjegdijaomrffczgom