Chapter 6 - Two-Dimensional Solitons

Abstract

The fundamental physical mechanism behind the transverse instability induced by the self-focusing of soliton stripes is similar to the mechanism governing the self-focusing and modulation instabilities of small-amplitude, quasi-harmonic, wave packets.The instability occurs when transverse modulations on the wave front of a planar solitonstripe decrease the local value of the soliton energy. Two analytical methods can be used for studying the transverse instability of solitons, namely, ray-optics approach and linear stability analysis. The ray -optics approach is based on the assumption that a transversely modulatedplane wave remains locally close to its steady-state profile, so each individual segment of the wave evolves along an individual ray used for analyzing the self-focusinginstabilities of a soliton stripe. It makes use of a linear eigenvalue problem that is obtained by linearizing the (2 + 1)-dimensional NLS equation near the exactone-dimensional soliton solution. Because two-dimensional spatial solitons are unstable in a Kerr medium, the soliton interaction and spiraling can be observed only in non-Kerr media. The interaction can bemodeled theoretically either by using an NLS equation with saturable nonlinearity or by employing the cubic-quintic NLS equation. If the two-dimensional spatial solitons are treated as “effective particles” their interaction potential can be calculated by using the collective-coordinate approach.