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

https://doi.org/10.1016/j.geomphys.2021.104312Get rights and content

Abstract

Under investigation in this paper is the generalized (2+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. Based on the bilinear Hirota method, the M-lump solution and N-soliton solution are constructed by giving some special activation functions in the considered model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are obtained with the aid of Maple software. Then, by employing the long wave limit method to the N-soliton solutions, M-lump solutions including 1-lump, 2-lump and 3-lump and the hybrid solutions between lump and solitons and between M-lump and soliton were obtained. Finally, via symbolic computation, their dynamic structures and physical properties were vividly shown by plotting different three-dimensional designs, two-dimensional designs, density designs. These solutions have greatly enriched the exact solutions of (2+1)-dimensional generalized (2+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation on the existing literature.

Introduction

It is known that the integrability of mathematics physics area has been better investigated in recent years. There are different definitions of integrability of nonlinear evolution differential equations. Among them, there are existing some indicators such as Bäcklund transforms, Lax pairs, infinite conservation laws, N-soliton solutions and infinite symmetry, etc. Therefore, in order to study the integrability of nonlinear evolution differential equations with applications in diverse subjects of sciences similar fluid flow, mechanics, biology, nonlinear optics, substance energy, system identification, and geo-optical filaments, etc. are explained in fractional features [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].

In the past few decades, the plenty of concern has been enforced to detect the novel and further exact traveling wave solution of PDEs by multitude research. With the collaboration of potential symbolic computer programming software they have been interested for researching appropriate solution to the nonlinear PDEs by executing powerful techniques, for examples, the tan-(ϕ/2) expansion method, the sin-Gordon expansion method, the G/G expansion method, the bilinear method, the advanced exponential expansion method [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], and so on.

This work mainly investigate the generalized (2+1)-dimensional Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation by utilizing the Hirota bilinear method [26], [27], [28], [29], [30] to finding M-lump solution and N-soliton solution. Recently Feng and co-authors [31] have studied bilinear form, solitons, breathers and lumps of a (3+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. Authors of [32] employed velocity resonance, module resonance, and long wave limits methods for this equation and provided some simple general soliton molecules and some novel hybrid solutions. Liu et al. [33] obtained lump, lumpoff and rogue waves with predictability for this equation by help of the properties of logarithmic function method. In [34], the Nth-order Pfaffian and Wronskian solutions were derived via the Pfaffian and Wronskian techniques by Deng and co-workers. The periodic-wave solutions and asymptotic behaviors of the aforementioned equation were studied based on Bell's polynomials, the bilinear formalism and Nsoliton solution [35].

In this paper, we mainly consider the following dynamical model, which can be used to describe some interesting (2+1)-dimensional waves of physics, namely, generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation [32]. That isut+λ1uxxx+λ2uux+λ3uxxxxx+λ4x1uyy+λ5uxxy+λ6(uxx1uy+uuy)+λ7(uxuxx+uuxxx)+λ8u2ux=0. The Hirota derivatives is considered asi=13Dσiωif.ζ=i=13(σiσi)ωif(σ)ζ(σ)|σ=σ, where the vectors σ=(σ1,σ2,σ3)=(x,y,t), σ=(σ1,σ2,σ3)=(x,y,t) and ω1,ω2,ω3 are the free amounts. The bilinear form of the generalized KDKK equation is as:BgKDKK(f):=(DxDt+λ1Dx4+λ3Dx6+λ4Dy2+λ5Dx3Dy)f.f=2[(ffxtfxft)+λ1(ffxxxx4fxfxxx+3(fxx)2)+λ3(ffxxxxxx6fxfxxxxx+15fxxfxxxx10(fxxx)2)+λ4(ffyy(fy)2)+λ5(ffyxxxfyfxxx+3fxxfxy3fxfxxy))]=0. Utilize the below bilinear frameΨ=12λ1λ21(lnf)xx. The Bell polynomial will be asPgKDKK(Ψ)=[BgKDKK(f)f]xx.

Theorem 1.1

f solves (1.4) if and only if u=12λ1λ21(lnf)xx illustrates a solution to the gKDKK Eq. (1.1)

(DxDt+λ1Dx4+λ3Dx6+λ4Dy2+λ5Dx3Dy)f.f=2[(ffxtfxft)+λ1(ffxxxx4fxfxxx+3(fxx)2)+λ3(ffxxxxxx6fxfxxxxx+15fxxfxxxx10(fxxx)2)+λ4(ffyy(fy)2)+λ5(ffyxxxfyfxxx+3fxxfxy3fxfxxy))]=0. The M-lump solution and N-soliton solution, which have been studied to find the accurate solutions of nonlinear partial differential equations (NLPDEs) by utilizing Hirota bilinear method (HBM). Most classical test functions for solving NLPDEs by using the Bells polynomial function method can be constructed via multiple soliton waves method by using HBM. The M-lump solution and N-soliton solution employed by several powerful authors for solving various nonlinear equations such as (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation [36], the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation [37], the third-order evolution equation arising in the shallow water [38], and also for investigating other valuable studies can see herein references ([39], [40], [41], [42], [43], [44]). In two prominent works, Ma computed N-soliton solutions and analyzed the Hirota N-soliton conditions according to the Hirota bilinear formulation for (1+1)-dimensional equations [45] and for (2+1)-dimensional equations [46] which could lead to lumps as long wave limits. Also, Ma and co-authors computed lump solutions to a combined fourth-order equation involving three types of nonlinear terms [47] and a combined soliton equation involving three fourth-order nonlinear terms in dispersive waves [48] in (2+1)-dimensions via symbolic computations.

The pattern of this article is summarized as: In section 2, the properties of and the detail of M-lump solution are given, which is to be utilized for taking the exact solutions of the gKDKK equation. Also, in section 3, along with numerical simulation and detail of in section 2, point to be noted finding N-soliton solution with the plenty of cases with abundant illustrations. Finally, some conclusions are given in the end. To the best of our knowledge, the results have not been reported in other places.

Section snippets

Multi-soliton of the generalized KDKK equation

Hither, the N-soliton of Eq. (1.1) will be detected via engaging Hirota operator, in which Eq. (1.4) can be indicated as belowf=fN=q=0,1exp(i<jNqiqjij+i=1Nqiϖi), whereϖi=si(x+piy(si4λ3+si2λ5pi+si2λ1+λ4pi2)t)+ϖi(0),expij=(s1s2)2(5λ3s125λ3s1s2+5λ3s22+3λ1)+λ5(s1s2)(2p1s1p1s2+p2s12p2s2)λ4(p1p2)2(s2+s1)2(5λ3s12+5λ3s1s2+5λ3s22+3λ1)+λ5(s2+s1)(2p1s1+p1s2+p2s1+2p2s2)λ4(p1p2)2. Let q1=0,1,q2=0,1,...,qN=0,1. In the following, (2.2) can be written as:f1=1+eϖ1,f2=1+eϖ1+eϖ2+12eϖ1+ϖ2,f3=1+eϖ1

N-Soliton solutions of the (2+1)-D gKDKK equation

To solve the (2+1)-dimensional gKDKK equation, the below function getf=1+ν1=1NeΓl+ν1<ν2Nν1ν2eΓν1+Γν2+ν1<ν2<ν3Nν1ν2ν1ν3ν2ν3eΓν1+Γν2+Γν3+...+(ν1<ν2Nν1ν2)eν1=1NΓν1, withΓν1=aν1x+bν1y+ων1t+ιν1s,ων1=sν14λ3+sν12λ5pν1+sν12λ1+λ4pν12,ν1ν2=(s1s2)2(5λ3s125λ3s1s2+5λ3s22+3λ1)+λ5(s1s2)(2p1s1p1s2+p2s12p2s2)λ4(p1p2)2(s2+s1)2(5λ3s12+5λ3s1s2+5λ3s22+3λ1)+λ5(s2+s1)(2p1s1+p1s2+p2s1+2p2s2)λ4(p1p2)2, in which the constants as, aν1,bν1,cν1 and ν1ν2 are values of the N-th soliton. In below

Conclusion

In this work, we have constructed the M-lump solution and N-soliton solution and the novel analytical solution of generalized KDKK equation by choosing some generalized activation functions in the considered model. By means of symbolic computation, these analytical solutions and corresponding rogue waves are acquired. Via diverse two-dimensional plots, density plot, and three-dimensional plots, dynamical characteristics of these rouge waves are exhibited. Next, by carrying out the long wave

Funding

There is no fund for this manuscript.

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant No. 62061048).

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