Rogue Waves and Their Patterns in the Vector Nonlinear Schrodinger Equation

In this paper, we study the general rogue wave solutions and their patterns in the vector (or M-component) nonlinear Schrodinger (NLS) equation. By applying the Kadomtsev-Petviashvili reduction method, we derive an explicit solution for the rogue wave expressed by tau functions that are determinants of K x K block matrices (1 <= K <= M) with an index jump of M + 1. Patterns of the rogue waves for M = 3, 4 and K = 1 are thoroughly investigated. It is found that when one of the internal parameters is large enough, the wave pattern is linked to the root structure of a generalized Wronskian-Hermite polynomial hierarchy in contrast with rogue wave patterns of the scalar NLS equation, theManakov system, and many others. Moreover, the generalized Wronskian-Hermite polynomial hierarchy includes the Yablonskii-Vorob'ev polynomial and Okamoto polynomial hierarchies as special cases, which have been used to describe the rogue wave patterns of the scalar NLS equation and the Manakov system, respectively. As a result, we extend the most recent results by Yang et al. for the scalar NLS equation and the Manakov system. It is noted that the case M = 3 displays a new feature different from the previous results. The predicted rogue wave patterns are compared with the ones of the true solutions for both cases of M = 3, 4. An excellent agreement is achieved.