Eur. Phys. J. Plus (2016) 131: 416
https://doi.org/10.1140/epjp/i2016-16416-8

Regular Article

Higher-order rogue waves with new spatial distributions for the (2 + 1) -dimensional two-component long-wave-short-wave resonance interaction system

¹ Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, 100191, Beijing, China
² Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, 100190, Beijing, China

Received: 12 August 2016
Accepted: 9 November 2016
Published online: 30 November 2016

Abstract

In this paper, a two-component (2 + 1) -dimensional long-wave-short-wave (LWSW) system with nonlinearity coefficients, which describes the nonlinear resonance interaction between two short waves and a long wave, is studied. Via the Hirota's bilinear method and Pfaffian, N -order rogue waves for the LWSW system are constructed. Furthermore, correction of the N -order rogue waves is proved directly via the Pfaffian, which is cumbersome or inaccessible in other methods. Results of the first- and second-order rogue waves are presented: 1) For the first-order rogue waves, the two short-wave components are bright, while the long-wave component is dark. The position of maximum amplitude of the rogue wave is analyzed. Evolution process for the first-order rogue wave is also presented and discussed. 2) Choosing different forms of the elements defined in the Pfaffian, we obtain some kinds of the second-order rogue waves with new spatial distributions: when the elements defined in Pfaffian are the same as the first-order rogue waves, we find that the second-order rogue waves for the two short-wave components are split into two first-order rogue waves and the two bumps coexist and interact with each other; when we change the combination of the elements in Pfaffian, we find that the second-order rogue waves for the two short-wave components are split into three and four first-order rogue waves. 3) N -order rogue waves for a general M -component LWSW system are constructed.