Rogue wave solutions of the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation

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Abstract

The main attention of this work focus on finding rogue wave solutions. Here, we analytically derived multi-order rogue wave solutions for the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation, through its bilinear form and symbolic computation. First, second and third order rogue waves have been systematically analyzed and the numerical simulations are exhibited to look into their dynamical features. Additionally, we have explored the “circular structure” among the obtained rogue waves.

Introduction

“Rogue waves” and other similar names, such as “killer waves”, “freak waves” have been the topics of recent publications. However, they were considered mysterious due to the risky observational condition, until direct witnesses confirm their appearance. Those who undergo such thing would be busy saving their lives rather than observing it [1]. So, we do not have an incisive comments on this appearance. Theoretical speaking, original linear theory is less convincing in explaining such waves, while its counterpart is more promising. According to nonlinear theory, a giant single wave may stems from the instability [2], [3] of certain initial condition and increases to a blindingly high amplitude, if modulation instability stands. On the basis of modulation theory addressed by Peregrine, the frequencies inside the frequency band or not decides whether small amplitude waves can grow into a giant wave or remain stable. However, the presence of multiple frequencies would results in the collision between solitons, that makes the whole picture rather complicated.

Recently, rogue waves have been found in optical systems [4], [5], which gives the opportunity for generating high energetic optical pulses [6], [7]. The normalized NLSE [8], [9] readsiψx+122ψt2±|ψ|2ψ=0,where ψ=ψ(x,t) is the slowly varying envelope of the electric field, x denotes the normalized propagation distance and t is the delayed time. When it takes “+”, Eq. (1) stands for focusing NLSE and “” indicates defocusing NLSE, which applicable to shallow water regime and deep ocean, separately. Nevertheless, probabilistic nature indicates that there is only one highly energetic pulse among thousands of them and this may serve as an alternative solution in generating short pulses of intense coherent light.

“Rogue waves” [10] is kind of native phenomena in earth, interacting with various fields, such as water wave tank [11], option pricing [12], Bose–Einstein condensates [13], optical fibers [14]. In present work, we review the generalized (3+1)-dimensional Kadomtsev–Petviashvili (KP) equationuxxxy+3(uxuy)x+utx+uty+uxx+uyy+uttuzz=0,where u is the analytical function rely on temporal coordinate t, propagation distance x,y,z, and the subscript stands for partial differential. In hydrodynamics, KP equation profiles weakly dispersive and small amplitude water waves in (2 + 1)-dimensional regime. This kind of equation appears in every major sector.

As an example, Tian [15] has derived the Grammian N-soliton solutions for the cylindrical KP equation by a novel gauge transformation under certain parameter constrain and the rogue wave solutions have also been constructed through the long wave limit method. Wazwaz [16] has proved that the extension terms do not damage the integrability of the extended KP equation in terms of Hereman’s simplified method. In addition, Riemann–Hilbert approach [17], [18] is applied to an integrable three-component coupled nonlinear Schrödinger equation and coupled fourth-order nonlinear Schrödinger equation to get the multi-soliton solutions. The Riemann–Hilbert problem is established at first, through analyzing the corresponding Las pair. Then, multi-soliton can be recovered by solving the Riemann–Hilbert problem under reflectionless cases. Practically, the collision of one- and two-soliton is displayed. Recently, with the aid of the Hirota’s bilinear method and the long wave limit method, rational as well as semi-rational solutions have been obtained for the fully parity-time symmetric (2 + 1)-dimensional nonlinear Schrödinger equation [19]. Except above aspects, KP-like equaitons is also a hot spot. A fourth-order generalized Boussinesq equation [20] has been investigated by the Lie symmetry method. Concretely speaking, hyperbolic-, rational-, Weierstrass elliptic- as well as Jacobi elliptic-type solutions have been found by employing its optimal system. See [21], [22], [23], [24] for references. Speaking of solitary standing waves, such as stripe soliton, lump soliton, Akhmediev breather, Peregrine soliton, etc, have all been reported in nonlinear systems. However, the rogue waves with multiple peaks are rarely considered. It is natural to wonder, can we derive higher order rogue waves for Eq. (2)? What properties do they have? How about their mechanism?

In present work, attention is directed at searching for rogue wave solutions for Eq. (2). In the following work, the reduced form of Eq. (2) is established in terms of suitable variable transformation at first. Then, its bilinear form has been obtained by potential transformation and the first order rogue wave with controllable center is derived through symbolic computation. Except for the specific form of the first order rogue wave solution, extreme values along with their coordinates have also been derived. Sequently, the second order and the third order rogue waves have been derived. In Section 4, we discuss the circular properties of the rogue wave solutions.

Section snippets

First order rogue wave

In this section, we shall construct the first order rogue waves and analyze their dynamical features. Setting ξ=x+λyγt, Eq. (2) is equal toλuξξξξ+6λuξuξξ+(γ2γλ+λ2γ+1)uξξuzz=0,where λ,γ are real parameters. Under the potential transformation u=u0+ln(f)ξ, Eq. (2) will be translated to(λDξ4+(γ2γλ+λ2γ+1)Dξ2Dz2)f·f=0,which is correspond tofz2ffzz+(1γ+γ2γλ+λ2)(fξ2+ffξξ)+λ(3fξξ24fξfξξξ+ffξξξξ)=0,where f is the dependent function in terms of ξ, z. Here, D-operator is specified asDϕmDφnf(ϕ,φ)

Multiple order rogue waves

Assuming f can be expressed asf=Fl+1(ξ,z;μ,ν)=Λl+1(ξ,z)+2μzΦl(ξ,z)+2νξΨl(ξ,z)+(μ2+ν2)Λl1(ξ,z),whereΛl(ξ,z)=p=0l(l+1)/2k=0pal(l+1)2p,2kξl(l+1)2pz2k,Φl(ξ,z)=p=0l(l+1)/2k=0pbl(l+1)2p,2kξl(l+1)2pz2k,Ψl(ξ,z)=p=0l(l+1)/2k=0pcl(l+1)2p,2kξl(l+1)2pz2k,Φ0=Ψ0=Λ1=0, Λ0=1, ai,j,bi,j,ci,j(i,j{0,2,,l(l+1)}) are parameters, and arbitrary constants μ,ν are employed to control the wave center.

Circularity character

In this section, we will explore the circularity character for the rogue wave solutions. Let us start with the second order rogue wave solutionu2=u0+(lnF˜2(ξ,z;μ,ν))ξ.Taking the same parameters in Fig. 4, solve the equations of u2/z=0, u2/ξ=0, we can derive a series of coordinates{9.83402,6.1416},{9.82496,8.17516},{1.34563,13.0744},{1.34689,11.2468},{11.2692,3.72413},{11.2607,5.88244},where the highest and lowest peaks of each breather locate. Here, the first line stands for the

Concluding remarks

In summary, we have derived specific forms of first, second and third order rogue wave, containing two free parameters μ,ν which govern the center of the rogue wave, for a generalized (3+1)-dimensional KP equation, by means of symbolic computation and its bilinear form. These solutions are systematically discussed and the numerical simulations are exhibited to look into their dynamic behaviors. According to Section 4, the breathers show a “circular structure” that the highest and lowest peaks

CRediT authorship contribution statement

Lingfei Li: Writing - original draft, Software. Yingying Xie: Writing - review & editing, Conceptualization, Methodology.

Declaration of Competing Interest

Authors declare that they have no conflict of interest.

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