N-lump and interaction solutions of localized waves to the (2+1)-dimensional generalized KDKK equation
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- expansion method, the sin-Gordon expansion method, the 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 is The Hirota derivatives is considered as where the vectors , and are the free amounts. The bilinear form of the generalized KDKK equation is as: Utilize the below bilinear frame The Bell polynomial will be as Theorem 1.1 f solves (1.4) if and only if illustrates a solution to the gKDKK Eq. (1.1)
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 below where Let . In the following, (2.2) can be written as:
N-Soliton solutions of the (2+1)-D gKDKK equation
To solve the (2+1)-dimensional gKDKK equation, the below function get with in which the constants as, and 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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