Stability of simple modes of the Kirchhoff equation (Q2757156)

The authors study the hyperbolic partial differential equation with a non-local nonlinearity of Kirchhoff type \(U_{tt}-m(\int_\Omega |\nabla u|^2 dx)\Delta u=0\), where \(m\) is a smooth function. The equation admits infinitely many simple modes, i.e. time-periodic solutions with only one Fourier component in the space variables. The main result is that these simple modes are stable provided their energy is small enough. Here stable means orbitally stable as solutions of the two-mode system obtained considering initial data with two Fourier components.