Positive and necklace solitary waves on bounded domains

We present new solitary wave solutions of the two-dimensional nonlinear Schrödinger equation on bounded domains (such as rectangles, circles, and annuli). These multi-peak "necklace" solitary waves consist of several identical positive profiles ("pearls"), such that adjacent "pearls" have opposite signs. They are stable at low powers, but become unstable at powers well below the critical power for collapse P_cr. This is in contrast with the ground-state ("single-pearl") solitary waves on bounded domains, which are stable at any power below P_cr. On annular domains, the ground state solitary waves are radial at low powers, but undergo a symmetry breaking at a threshold power well below P_cr. As in the case of convex bounded domains, necklace solitary waves on the annulus are stable at low powers and become unstable at powers well below P_cr. Unlike on convex bounded domains, however, necklace solitary waves on the annulus have a second stability regime at powers well above P_cr. For example, when the ratio of the inner to outer radii is 1:2, four-pearl necklaces are stable when their power is between 3.1P_cr and 3.7P_cr. This finding opens the possibility to propagate localized laser beams with substantially more power than was possible until now. The instability of necklace solitary waves is excited by perturbations that break the antisymmetry between adjacent pearls, and is manifested by power transfer between pearls. In particular, necklace instability is unrelated to collapse. In order to compute numerically the profile of necklace solitary waves on bounded domains, we introduce a non-spectral variant of Petviashvili's renormalization method.