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Rogue Waves and Their Patterns in the Vector Nonlinear Schrödinger Equation
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Journal Article
Rogue Waves and Their Patterns in the Vector Nonlinear Schrödinger Equation
2023
Overview
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 reduction method, we derive an explicit solution for the rogue wave expressed by
τ
functions that are determinants of
K
×
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, the Manakov 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.
Publisher
Springer US,Springer Nature B.V
