Positive and necklace solitary waves on bounded domains

• We present new solitary wave solutions of the two-dimensional NLS on bounded domains.
• These necklace solitary waves become unstable well below the critical power for collapse.
• On the annulus they have a second stability regime well above the critical power.
• We introduce a non-spectral variant of Petviashvili’s renormalization method.

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  Pcr. This is in contrast with the ground-state (“single-pearl”) solitary waves on bounded domains, which are stable at any power below  Pcr.On annular domains, the ground state solitary waves are radial at low powers, but undergo a symmetry breaking at a threshold power well below  Pcr. 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  Pcr. Unlike on convex bounded domains, however, necklace solitary waves on the annulus have a second stability regime at powers well above  Pcr. For example, when the ratio of the inner to outer radii is 1:2, four-pearl necklaces are stable when their power is between 3.1Pcr and 3.7Pcr. 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.