Analysis on the M-rogue wave solutions of a generalized (3+1)-dimensional KP equation
Introduction
The solutions of nonlinear partial differential equations (NLPDEs) play an important role in nonlinear science fields, such as optical fiber communications [1], oceans [2], capillary waves [3], and so on [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. In the last decades, studies of rogue wave solutions have attracted great concern and many good results have been acquired. The rogue wave solution is a kind of special rational solution. The average height of rogue waves is at least twice the height of the surrounding waves, is very unpredictable and so it can be quite unexpected and mysterious. In fact, similar phenomena can be well explained by fractal calculus, for example an extremely fast diffusion or an extremely slow diffusion in a fractal space [14]. Due to the character of coming from nowhere and disappearing with no trace [15], the analysis of rogue waves is rather significant. Recently, the Hirota bilinear method [16], the Bäcklund transformation method [17], the exp-function method [18], [19], [20], [21], Darboux transformation method [22] and the inverse scattering method [23] others [24], [25], [26], [27], [28], [29], [30] are used to construct rogue waves. More recently, Akhmediev and Ankiewicz constructed a hierarchy of rogue wave solutions by using the Darboux transformation method, and that the relations between the rational solutions and rogue wave solutions of the nonlinear Schrödinger equation is presented in [22]. The rogue wave has special wave morphology, so the variational approach has been caught much attention [31], [32]. It is worth noting that the above methods play an essential role in constructing the rogue wave solutions of the NLPDEs. Nextly, the rogue wave solutions of the Boussinesq equation [33] and the potential Yu–Toda–Sasa–Fukuyama equation [34] are constructed by extending the above method.
In this paper, we mainly analyze a generalized (3+1)-dimensional KP equation. That is which describes three-dimensional solitons in weakly dispersive media, particularly in fluid dynamics and plasma physics [35]. If setting , Eq. (1.1) can be reduced to the classical KP equation [36]; If setting , Eq. (1.1) can be reduced to the potential KdV equation [37]. Ma and Abdeljabbar established Wronskian and Grammian formulation of Eq. (1.1) based on Hirota bilinear form [38]. Mohyud-Din and Irshad constructed the hyperbolic function solutions, the exponential solutions, the trigonometric function solutions and the rational solutions of Eq. (1.1) by using the improved -expansion method [39]. Wu and Geng utilized the Wronskian technique to construct the -soliton solutions, periodic solutions and rational solutions of Eq. (1.1) [40]. Wang and Tian derived the solitary waves, breather waves, rogue waves of Eq. (1.1) with the help of Bell’s polynomials [41]. To our best knowledge, multiple rogue waves of Eq. (1.1) have not been proposed yet.
The structure of this paper is as follows. In Section 2, a new type of bilinear form of Eq. (1.1) was established. Then the first-order rogue waves of Eq. (1.1) via a novel ansatz, were constructed. In Section 3, the second-order rogue waves are generated by taking in Eq. (2.5). Section 4 given the third-order rogue waves of the generalized (3+1)-dimensional KP equation. Section 5 contains a short conclusion and further discussions.
Section snippets
The first-order rogue waves
In this section, we focus on the first-order rogue waves for a generalized (3+1)-dimensional KP equation. First taking into Eq. (1.1), then Eq. (1.1) can be reduced into the following (1+1)-dimensional equation where and are all real parameters. There exist an equilibrium solution and variable transformation Then substituting Eq. (2.2) into Eq. (2.1), we obtain the new type bilinear form of Eq. (1.1)
The second-order rogue waves
In this section, so as to obtain the second-order rogue waves of Eq. (1.1), we just take in Eq. (2.5). That is Substituting Eq. (3.1) into Eq. (2.3), collecting coefficients of each terms and letting them to zero, we get the following parameters values
The third-order rogue waves
In order to get the third-order rogue waves solution for Eq. (1.1), we take , have
Conclusions and discussions
In this work, first we reduced the generalized (3+1)-dimensional KP equation into (1+1)-dimensional equation by using a variable transformation. Then we constructed multiple rogue wave solutions with the help of the bilinear equation of the reduced equation and three ansatz polynomial functions (2.6). It is noteworthy that these rogue waves all have the property , , and . As far as we know, if every term in a bilinear
CRediT authorship contribution statement
Hong-Yi Zhang: Methodology. Yu-Feng Zhang: Methodology.
Acknowledgment
This work is supported by the Fundamental Research Funds for the Central University, People’s Republic of China (No. 2017XKZD11).
References (42)
- et al.
Nonlinear dynamics associated with rotating magnetized electron-positron-ion plasmas
Phys. Lett. A
(2010) - et al.
Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions
Appl. Math. Model.
(2015) - et al.
Lump waves, solitary waves and interaction phenomena to the (2+ 1)-dimensional Konopelchenko-Dubrovsky equation
Phys. Lett. A
(2019) - et al.
Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior
Appl. Math. Lett.
(2020) - et al.
Interaction behavior associated with a generalized (2+ 1)-dimensional Hirota bilinear equation for nonlinear waves
Appl. Math. Model.
(2019) - et al.
Diversity of exact solutions to a (3+ 1)-dimensional nonlinear evolution equation and its reduction
Comput. Math. Appl.
(2018) - et al.
Resonant behavior of multiple wave solutions to a Hirota bilinear equation
Comput. Math. Appl.
(2016) - et al.
Waves that appear from nowhere and disappear without a trace
Phys. Lett. A
(2009) - et al.
Solitons, Bäcklund transformation and Lax pair for a (2+1)-dimensional Broer-Kaup-Kupershmidt system in the shallow water of uniform depth
Commun. Nonlinear Sci. Numer. Simul.
(2017) - et al.
Multi-soliton solutions and Breathers for the generalized coupled nonlinear Hirota equations via the Hirota method
Superlattices Microstruct.
(2017)
Resonant multiple wave solutions, complexiton solutions and rogue waves of a generalized (3+ 1)-dimensional nonlinear wave in liquid with gas bubbles
Waves Random Complex Media
Cubic-quartic optical soliton perturbation by semi-inverse variational principle
Optik
Multiple lump solutions of the (3+ 1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation
Appl. Math. Lett.
Wronskian and Grammian solutions to a (3+ 1)-dimensional generalized KP equation
Appl. Math. Comput.
Exact solutions of (3+ 1)-dimensional generalized KP equation arising in physics
Res. Phys.
Novel Wronskian condition and new exact solutions to a (3+ 1)-dimensional generalized KP equation
Commun. Theor. Phys.
On the solitary waves, breather waves and rogue waves to a generalized (3+ 1)-dimensional Kadomtsev-Petviashvili equation
Comput. Math. Appl.
Oceanic rogue waves
Annu. Rev. Fluid Mech.
Capillary rogue waves
Phys. Rev. Lett.
Rogue waves in the atmosphere
J. Plasma Phys.
Theory of Solitons: The Inverse Scattering Method
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