Rogue Waves and Lump Solitons of the (3+1)-Dimensional Generalized B-type Kadomtsev–Petviashvili Equation for Water Waves

Abstract

In this paper, the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order, higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.

Rogue waves of a (3+1)-dimensional BKP equation

“…Moreover, it is worth noting the following: (I) The above semi-rational solutions will be reduced to rational solutions if the constant δ jk is removed, which are exactly those solutions reported in Ref. [19] of Eq. ( 1).…”

“…[8][9][10] Among these models, studies of the high dimensional NLEEs are concentrated to investigate the rich nonlinear waves interactions. [11][12][13][14][15] In this paper, we will investigate a (3 + 1)-dimensional BKP equation, [16][17][18][19][20] u ty − u xxxy − 3(u x u y ) x + 3u xz = 0, (1) where u is an analytic function of the scaled spatial coordinates x, y, z and temporal coordinate t, with the subscripts denoting the partial differential derivatives. Equation (1) has been proposed as an opposite procedure on applying the linear superposition principle.…”

“…[18] The lump and linerogue wave have been derived via the KP hierarchy reduction method. [19] Certain hybrid solutions and breather solutions have been constructed. [20] Recently, some semi-rational solutions with certain parameter constraints which reveal special resonant interaction [The resonant interaction describing a special interaction between the N-th order rational solutions and N + 1 solitons, which occurs that the phase of the lump (line rogue wave) varies to infinite suddenly when such lump (line rogue wave) interacts with a soliton (based on the semi-rational solutions obtained by the Hirota method)] between the lump (line rogue wave) and soliton have been investigated via the KP hierarchy reduction [21,22] and bilinear method for the KP I equation, [23,24] the Fokas system, [25] the Davey-Stewartson (DS) I equation, [26] and some other models.…”

Rational and semi‐rational solutions of a nonlocal (2 + 1)‐dimensional nonlinear Schrödinger equation

“…Recently, many attentions have been focused on the research of breathers, rogue waves, and lump waves. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] Lump and rogue waves, both are rational solutions, are observed in many physical phenomena, such as nonlinear optic media, superfluids and Bose-Einstein condensates, and so on. [22][23][24][25][26][27][28][29] In addition, searching for semi-rational solutions of NLEEs also has caused concern for scientists.…”

“…For instance, Hirota bilinear method, Darboux transformation (DT), inverse scattering transformation (IST), and so on. Recently, many attentions have been focused on the research of breathers, rogue waves, and lump waves . Lump and rogue waves, both are rational solutions, are observed in many physical phenomena, such as nonlinear optic media, superfluids and Bose‐Einstein condensates, and so on .…”