Analysis on the M-rogue wave solutions of a generalized (3+1)-dimensional KP equation

https://doi.org/10.1016/j.aml.2019.106145Get rights and content

Abstract

In this paper, multiple rogue wave solutions of a generalized (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation are studied. Based on the bilinear form of this considered equation and a generalized ansatz form, the N-order rogue waves can be constructed. Here we only show in the first-order rogue waves, the second-order rogue waves and the third-order rogue waves. Then, dynamic characteristics of multiple rogue waves are analyzed by drawing the 3D-and 2D-dimensional plots. We see that the M-rogue wave consists of M-independent single first order rogue wave. This interesting phenomenon can help us better reveal KP equation evolution mechanism.

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 uxxxy+3(uxuy)x+utx+utyuzz=0,u=u(x,y,z,t),which describes three-dimensional solitons in weakly dispersive media, particularly in fluid dynamics and plasma physics [35]. If setting y=x, Eq. (1.1) can be reduced to the classical KP equation [36]; If setting z=y=x, 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 tan(ϕ(ξ)2)-expansion method [39]. Wu and Geng utilized the Wronskian technique to construct the N-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 n=1 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 ς=x+lyht into Eq. (1.1), then Eq. (1.1) can be reduced into the following (1+1)-dimensional equation luςςςς+3(luςuς)ς(h+hl)uςςuzz=0,where l and h are all real parameters. There exist an equilibrium solution u0 and variable transformation u(ς,z)=2(lnϑ(ς,z))ς+u0.Then substituting Eq. (2.2) into Eq. (2.1), we obtain the new type bilinear form of Eq. (1.1) 2l(ϑϑςςςς4ϑςϑςςς+3ϑςς2

The second-order rogue waves

In this section, so as to obtain the second-order rogue waves of Eq. (1.1), we just take n=1 in Eq. (2.5). That is ϑ=ϑ2(ς,z;τ,φ)=H2(ς,z)+2τzP1(ς,z)+2φςT1(ς,z)+(τ2+φ2)H0(ς,z)=(a0,0+a0,2z2+a0,4z4+a0,6z6)+(a2,0+a2,2z2+a2,4z4)ς2+(a4,0+a4,2z2)ς4+ς6+2τz(b0,0+b0,2z2+b2,0ς2)+2φς(c0,0+c0,2z2+c2,0ς2)+(τ2+φ2).Substituting Eq. (3.1) into Eq. (2.3), collecting coefficients of each terms and letting them to zero, we get the following parameters values a0,0=τ29hl+9h+1875l3h3(l+1)3+φ2h2(l+1)2l2(τ2+φ2),a0,2=475

The third-order rogue waves

In order to get the third-order rogue waves solution for Eq. (1.1), we take n=2, have ϑ=ϑ3(ς,z;τ,φ)=H3(ς,z)+2τzP2(ς,z)+2φςT2(ς,z)+(τ2+φ2)H1(ς,z)=(a0,0+a0,2z2+a0,4z4+a0,6z6+a0,8z8+a0,10z10+a0,12z12)+(a2,0+a2,2z2+a2,4z4+a2,6z6+a2,8z8+a2,10z10)ς2+(a4,0+a4,2z2+a4,4z4+a4,6z6+a4,8z8)ς4+(a6,0+a6,2z2+a6,4z4+a6,6z6)ς6+(a8,0+a8,2z2+a8,4z4)ς8+(a10,0+a10,2z2)ς10+ς12+2τz[(b0,0+b0,2ς2+b0,4ς4+b0,6ς6)+(b2,0+b2,2ς2+b2,4ς4)z2+(b4,0+b4,2ς2)z4+z6]+2φς[(c0,0+c0,2z2+c0,4z4+c0,6z6)+(c2,0+c2,2z2+c2,4z4)ς2+(c4,0+c4,2z2)

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 Hn,Pn,Tn (2.6). It is noteworthy that these rogue waves all have the property limx±u(x,y,z,t)=u0, limy±u(x,y,z,t)=u0, and limz±u(x,y,z,t)=u0. 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)

  • LiuW. et al.

    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

    (2018)
  • KohlR.W. et al.

    Cubic-quartic optical soliton perturbation by semi-inverse variational principle

    Optik

    (2019)
  • ZhaoZ. et al.

    Multiple lump solutions of the (3+ 1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation

    Appl. Math. Lett.

    (2019)
  • MaW.X. et al.

    Wronskian and Grammian solutions to a (3+ 1)-dimensional generalized KP equation

    Appl. Math. Comput.

    (2011)
  • Mohyud-DinS.T. et al.

    Exact solutions of (3+ 1)-dimensional generalized KP equation arising in physics

    Res. Phys.

    (2017)
  • WuJ.P. et al.

    Novel Wronskian condition and new exact solutions to a (3+ 1)-dimensional generalized KP equation

    Commun. Theor. Phys.

    (2013)
  • WangX.B. et al.

    On the solitary waves, breather waves and rogue waves to a generalized (3+ 1)-dimensional Kadomtsev-Petviashvili equation

    Comput. Math. Appl.

    (2017)
  • DystheK. et al.

    Oceanic rogue waves

    Annu. Rev. Fluid Mech.

    (2008)
  • ShatsM. et al.

    Capillary rogue waves

    Phys. Rev. Lett.

    (2010)
  • StenfloL. et al.

    Rogue waves in the atmosphere

    J. Plasma Phys.

    (2010)
  • NovikovS. et al.

    Theory of Solitons: The Inverse Scattering Method

    (1984)
  • Cited by (22)

    • Multiple soliton solutions of the generalized Hirota-Satsuma-Ito equation arising in shallow water wave

      2021, Journal of Geometry and Physics
      Citation Excerpt :

      In this paper, we will study the multiple rogue waves for determining the multiple soliton solutions. The multiple rogue waves method used by some of powerful authors for various nonlinear equations including: constructing rogue waves with a controllable center in the nonlinear systems [45], a (3+1)-dimensional Hirota bilinear equation [18], the generalized (3+1)-dimensional KP equation [43], the Boussinesq equation [7]. Structures of this paper as follows, the multiple Exp-function scheme has been introduced in Section 2.

    • Multiple-order line rogue wave, lump and its interaction, periodic, and cross-kink solutions for the generalized CHKP equation

      2021, Propulsion and Power Research
      Citation Excerpt :

      In this paper, we will study the multiple rogue waves for determining the multiple soliton solutions. The multiple rogue waves method used by some of powerful authors for various nonlinear equations including: constructing rogue waves with a controllable center in the nonlinear systems [46], a (3 + 1)-dimensional Hirota bilinear equation [47], the generalized (3 + 1)-dimensional KP equation [48], the Boussinesq equation [49]. We plainly confirm that other published papers do not cover ours, and constructed work is really novel.

    • N-lump and interaction solutions of localized waves to the (2 + 1)-dimensional generalized KP equation

      2021, Results in Physics
      Citation Excerpt :

      Subsequently, a general discussion on the lumps solutions of NPDEs can be found in Ref. [31]. More recently, the lump and interaction solutions were also constructed to linear PDEs and combined equation with nonlinear terms by using symbolic computation [32–38]. The lump solutions and lump-type solutions sometimes were called the rogue waves-type solutions.

    View all citing articles on Scopus
    View full text