Korteweg-de Vries description of Helmholtz-Kerr dark

A wide variety of different physical systems can be described by a relatively small set of universal equations. For example, small-amplitude nonlinear Schrödinger dark solitons can be described by a Korteweg-de Vries (KdV) equation. Reductive perturbation theory, based on linear boosts and Gallilean transformations, is often employed to establish connections to and between such universal equations. Here, a novel analytical approach reveals that the evolution of small-amplitude Helmholtz-Kerr dark solitons is also governed by a KdV equation. This broadens the class of nonlinear systems that are known to possess KdV soliton solutions, and provides a framework for perturbative analyses when propagation angles are not negligibly small. The derivation of this KdV equation involves an element that appears new to weakly-nonlinear analyses, since transformations are required to preserve the rotational symmetry inherent to Helmholtz-type equations. PACS number(s): 42.65.–k (nonlinear optics), 42.65.Tg (optical solitons), 05.45.Yv (solitons, nonlinear dynamics of) Submitted to Journal of Physics A: Mathematics and General as a Paper Accepted on 21 September 2006 3 Corresponding author KdV description of Helmholtz-Kerr dark solitons 2