Rogue Wave Patterns

In this chapter, we study rogue wave patterns. Clear rogue patterns will arise when certain internal parameters in the rogue wave solutions get large. These patterns manifest as triangles, pentagons, heptagons, rings, and other shapes, in the spatial-temporal plane or just the spatial plane. They can be asymptotically predicted by root structures of certain polynomials, such as the Yablonskii-Vorob’ev polynomial hierarchy, Adler-Moser polynomials, and Okamoto polynomial hierarchies, in the (1+1)-dimensional case, and by root curves of certain double-real-variable polynomials in the (2+1)-dimensional case. In addition, these patterns are often universal in the sense that they would arise in many different integrable systems. We will develop the asymptotic theory for the prediction of these rogue patterns, and verify our predictions by comparing them to true rogue solutions. Profiles of super rogue waves at high order in the nonlinear Schrödinger equation are also summarized at the end of this chapter.