VI-9

E.N. Kalaidin and O. Palaguta

Kuban State

We present a spectral theory for the stability of localized roll waves with a jump discontinuity. Due to the localized wave structure, the disturbances are spanned by both the discrete eigenfunctions, which decay to zero away from the jump, and essential eigenfunctions which approach bounded oscillations. Since the substrate is unstable, the essential spectrum contains a band of continuous unstable modes. However, this entire unstable band may be contained in a localized wavepacket disturbance which travels slower than the roll wave. Such disturbances will not destabilize the roll wave. To remedy this short-coming of the classical spectral theory, we develop a convective weighted spectral theory that condenses the continuum of essential modes into a single discrete mode, the resonant pole. This infinite reduction of order allows us to capture both the linear and nonlinear dynamics of the roll wave.