Abstract
In this paper, the bilinear method is employed to investigate the rogue wave solutions and the rogue type multiple lump wave solutions of the (2+1)-dimensional Benjamin-Ono equation. Two theorems for constructing rogue wave solutions are proposed with the aid of a variable transformation. Four kinds of rogue wave solutions are obtained by means of Theorem 1. In Theorem 2, three polynomial functions are used to derive multiple lump wave solutions. The 3-lump solutions, 6-lump solutions, and 8-lump solutions are presented, respectively. The 3-lump wave has a “triangular” structure. The centers of the 6-lump wave form a pentagram around a single lump wave. The 8-lump wave consists of a set of seven first order rogue waves and one second order rogue wave as the center. The multiple lump wave develops into low order rogue wave as parameters decline to zero. The method presented in this paper provides a uniform method for investigating high order rational solutions. All the results are useful in explaining high dimensional dynamical phenomena of the (2+1)-dimensional Benjamin-Ono equation.
Highlights
Rogue wave is an isolated huge wave, which plays an important role in analyzing many science problems, such as ocean’s waves [1,2,3], optical fibers [4], Bose-Einstein condensates [5, 6], and financial markets [7, 8]
These show that the number of the higher peaks of the multiple rogue wave solutions is equivalent to the subscript of un
We focus on investigating the rogue wave solutions and the rogue type multiple lump solutions of the (2+1)-dimensional Benjamin-Ono equation by means of the bilinear method
Summary
Rogue wave is an isolated huge wave, which plays an important role in analyzing many science problems, such as ocean’s waves [1,2,3], optical fibers [4], Bose-Einstein condensates [5, 6], and financial markets [7, 8]. Rogue wave solutions are an interesting class of lump-type solutions [11]. In 2015, Ma investigated the lump solutions of the (2+1)-dimensional Kadomtsev-Petviashvili equation by means of the bilinear forms and positive quadratic functions [13]. There are many methods to investigate the lump solutions and the interaction solutions among solitons, such as the inverse scattering transformation [19], the Painleveanalysis [20,21,22], the symmetry analysis [23,24,25,26,27], and the Darboux transformation [28, 29]. Tang et al investigated the nonelastic collision between a lump wave and a stripe soliton of the (2+1)-dimensional Ito equation [31].
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