We investigate rogue-wave solutions in a three-component coupled nonlinear Schrödinger equation. With certain requirements on the backgrounds of components, we construct a multi-rogue-wave solution that exhibits a structure like a four-petaled flower in temporal-spatial distribution, in contrast to the eye-shaped structure in one-component or two-component systems. The results could be of interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids.

In this paper, we obtain a uniform Darboux transformation for multi-component coupled NLS equations, which can be reduced to all previous presented Darboux transformation. As a direct application, we derive the single dark soliton and multi-dark soliton solutions for multi-component coupled NLS with defocusing case and mixed focusing and defocusing case. Some exact single and two-dark solitons of three-component NLS equation are shown by plotting the picture.

We find that diverse nonlinear waves such as soliton, Akhmediev breather, and rogue waves, can emerge and interplay with each other in a two-mode coupled system. It provides a good platform to study interaction between different kinds of nonlinear waves. In particular, we obtain dark rogue waves analytically for the first time in the coupled system, and find that two rogue waves can appear in the temporal-spatial distribution. Possible ways to observe these nonlinear waves are discussed.

We study on dynamics of high-order rogue wave in two-component coupled nonlinear Schrödinger equations. We find four fundamental rogue waves can emerge for second-order vector RW in the coupled system, in contrast to the high-order ones in single component systems. The distribution shape can be quadrilateral, triangle, and line structures through varying the proper initial excitations given by the exact analytical solutions. Moreover, six fundamental rogue wave can emerge on the distribution for second-order vector rogue wave, which is similar to the scalar third-order ones. The distribution patten for vector ones are much abundant than the ones for scalar rogue waves. The results could be of interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids. Introduction-Rogue wave (RW) is the name given by oceanographers to isolated large amplitude wave, which occurs more frequently then expected for normal, Gaussian distributed, statistical events [1][2][3]. It depicts a unique event that seems to appear from nowhere and disappear without a trace, and can appear in a variety of different contexts [4][5][6][7]. RW has been observed experimentally in nonlinear optics by Solli group and Kibler group [8,9], water wave tank by Chabchoub group [10], and even in plasma system by Bailung group [11]. These experimental studies suggest that the rational solutions of related dynamics equations can be used to describe these RW phenomena [12,13]. Moreover, there are many different pattern structures for high-order RW [14][15][16], which can be understood as a nonlinear superposition of fundamental RW (the first order RW). Recently, many efforts are devoted to classify the hierarchy for each order RW [17,18], since these superpositions are nontrivial and admit only a fixed number of elementary RWs in each high order solution (3 or 6 fundamental ones for the second order or third order one).