Nonlinear Localized structures for solving wave equations over long distances

A new method is described in this paper that has the potential to greatly extend the range of application of Eulerian computational methods for many problems. The new method has many of the advantages of Green's Function based integral equation methods for long distance propagation since the propagation distance can be indefinitely long. However, unlike Green's Function schemes, which are useful for uniform index of refraction in simple domains, since the new method is an Eulerian finite difference technique; it allows short pulses to automatically propagate through regions of varying index of refraction and undergo multiple scattering in complex domains. It also can automatically capture produced waves (on sufficiently fine grids) near a source (antennas and scatterers) and transfer them to a sequence of coarser grids for efficient long range propagation. Unlike Ray Tracing schemes, the new method propagates entire smooth surfaces, greatly simplifying the computation of solutions over extended domains.