School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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In this paper, we study the general rogue wave solutions and their patterns in the vector (or M-component) nonlinear Schrödinger (NLS) equation. By applying the Kadomtsev-Petviashvili hierarchy reduction method, we derived an explicit solution for the rogue wave expressed by τ functions that are determinants of K×K block matrices (K=1,2,⋯,M) with an index jump of M+1. Patterns of the rogue waves for M=3,4 and K=1 are thoroughly investigated. We find that when a specific internal parameter is large enough, the wave patterns are linked to the root structures of generalized Wronskian-Hermite polynomial hierarchy in contrast with rogue wave patterns of the scalar NLS equation, the Manakov system and many others. Moreover, the generalized Wronskian-Hermite polynomial hierarchy includes the Yablonskii-Vorob'ev polynomial hierarchy 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 {\it 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.

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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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