Wavepacket Preservation Under Nonlinear Evolution

We study nonlinear systems of hyperbolic PDE’s in Rd , the hyperbolicity is understood in a wider sense, namely multiple roots of the characteristic equation are allowed and dispersive equations are permitted. They describe wave propagation in dispersive nonlinear media such as, for example, electromagnetic waves in nonlinear photonic crystals. The initial data is assumed to be a finite sum of wavepackets referred to as a multi-wavepacket. The wavepackets and the medium nonlinearity are characterized by two principal small parameters β and where: (i) 1 β is a factor describing spatial extension of involved wavepackets; (ii) 1 is a factor describing the relative magnitude of the linear part of the evolution equation compared to its nonlinearity. A key element in our approach is a proper definition of a wavepacket. Remarkably, the introduced definition has a flexibility sufficient for a wavepacket to preserve its defining properties under a general nonlinear evolution for long times. In particular, the corresponding wave vectors and the band numbers of involved wavepackets are “conserved quantities”. We also prove that the evolution of a multi-wavepacket is described with high accuracy by a properly constructed system of envelope equations with a universal nonlinearity. The universal nonlinearity is obtained by a time averaging applied to the original nonlinearity, in simpler cases the averaged system turns into a system of Nonlinear Schrodinger equations.