Research article

New lump interaction complexitons to the (2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential in plasmas

  • Tukur Abdulkadir Sulaiman ,
  • Abdullahi Yusuf ,
  • Alrazi Abdeljabbar d ,
  • Mustafa Bayram a
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  • a Department of Computer Engineering, Biruni University, Istanbul, Turkey
  • b Department of Mathematics, Federal University Dutse, Jigawa, Nigeria
  • c Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
  • d Department of Mathematics, Khalifa University for Science and Technology, Abu Dhabi, UAE

Received date: 2022-02-23

  Revised date: 2022-04-21

  Accepted date: 2022-04-21

  Online published: 2022-04-28

Abstract

A soliton is a packet of self-reinforcing waves that maintains its structure when moving at a constant speed. Solitons are caused by the cancellation of the medium's nonlinear and dispersive effects. In plasmas, the bilinear form of Hirota will be utilized to investigate the (2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential. Solutions for complexiton lump interaction have been developed. To throw further light on the physical qualities of the recorded data, certain 3-dimensional and contour plots are presented to illustrate the interaction elements of these solutions.

Highlights

● The (2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential in plasmas.

● New lump interaction complexitons are successfully constructed.

● Numerical simulation of the presented results are presented.

Cite this article

Tukur Abdulkadir Sulaiman , Abdullahi Yusuf , Alrazi Abdeljabbar , Mustafa Bayram . New lump interaction complexitons to the (2+1)-dimensional Korteweg-de Vries equation with electrostatic wave potential in plasmas[J]. Journal of Ocean Engineering and Science, 2024 , 9(2) : 173 -177 . DOI: 10.1016/j.joes.2022.04.020

1. Introduction

For addressing Cauchy problems for ordinary and partial differential equations, classical methods like as the Laplace method and the Fourier transform approach were developed. In recent soliton theory [1], [2], Cauchy issues for nonlinear ordinary and partial differential equations have been solved using isomonodromic transformation and inverse scattering methods. One of the most fascinating and promising scientific research areas is the study of exact solutions and the associated challenge of providing results for a wide class of nonlinear equations [3].
Exact solutions to partial differential equations have significant mathematical and physical aspects. In both time and space, a soliton solution is a wave solution with exponentially localized functions in all directions. Exact solutions to partial differential equations have significant mathematical and physical aspects. In addition, interaction solutions between lump and soliton solutions are well-known for allowing nonlinear processes to be shown.
Internal waves and ocean topography have been a major focus of research for a long time. The driving mechanism of wave solutions affects the propagation of surface and internal gravity waves. Oceans, lakes, and the sky all have these waves, and the mechanism that creates them could be beneficial in ocean engineering. Shallow-water equations are a collection of hyperbolic partial differential equations that explain fluid flow beneath a pressure surface [4]. Water wave dispersion refers to frequency dispersion in fluid dynamics, which means waves of different wavelengths travel at varying phase speeds. In this context, water waves are waves that propagate on the water's surface, with gravity and surface tension acting as restoring forces. As a result, water with a free surface is commonly thought of as a dispersive medium [5]. Ocean engineering involves the development, design, operation, and maintenance of watercraft propulsion and ocean systems using a variety of engineering sciences such as mechanical engineering, electrical engineering, electronic engineering, and computer science. Power and propulsion plants, machinery, piping, automation and control systems for all types of marine vehicles, as well as coastal and offshore constructions, are all included [6]. Water wave equations describe various nonlinear physical phenomena in the above mentioned areas. Breather wave solutions, for example, are crucial in the study of ocean waves since they are formed from these types of equations. Breathers are pulsating localized structures that have been used to simulate high waves in a variety of nonlinear dispersive environments. They are commonly used in coastal engineering. They use a narrow banded beginning procedure to replicate severe waves in a range of nonlinear dispersive mediums. Breathers, on the other hand, may be able to exist in more complex ecosystems, such as random seas, despite the inherent physical limitations, according to several recent studies [7]. This study is going to present some important two-wave and breather wave solutions to the (2+1)-dimensional Korteweg-de Vries equation.
The study of many complicated physical aspects and other nonlinear engineering difficulties, such as lump solutions, kink solutions, and so on, relies heavily on these solutions and their generalized forms. Lump type solutions have been developed and their dynamics examined for a number of nonlinear equations. The study of lump-kink solutions, or the collisions between lump and kink solutions, has piqued the curiosity of many researchers [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Lump solutions, on the other hand, are very different and haven't been thoroughly investigated. There exist prerequisites for a high number of nonlinear equation solutions in this regard. This prompted us to look for lump interaction complexitons solutions to the (2+1)-dimensional Korteweg-de Vries equation in plasmas with an electrostatic wave potential. The (2+1)-dimensional KdV equation describes the electrostatic wave potential in plasmas [22]
χt+χxxx3(χy1χx)x=0.
Consider the variable transformation
χ(x,y,t)=k2(lng(x,y,t))xy.
Plugging (2) into (1), the following bilinear form is secured
2gy(gt+gxxx)2g(gyt3kgxx+gxxxy)+6gx(gxxykgx)6gxygxx=0.
Equation (3) may equivalently be written in Hirota's operators form as [22]
(DyDt+Dx3Dy3kDx2)g·g=0.

2. Two-waves solutions

In this section, we successfully construct two-wave solutions. We use the following test function to ensure that the two-wave solutions are secured:
g(x,y,t)=δ1ex+γ1y+γ2t+δ2e(x+γ1y+γ2t)+δ3sin(x+γ3y+γ4t)+δ4sinh(x+γ5y+γ6t).
Plugging (5) into (3), provides a polynomial in exponential, trigonometric and hyperbolic functions. The following collection of solutions are obtained by solving a system of algebraic equations collected based on the powers of the exponential, trigonometric and hyperbolic functions:
(I): γ2=3k4γ1γ1,γ3=2γ13kγ14+γ1kk2γ1,γ4=3γ1k6γ13(kγ1)γ122,γ5=γ1,γ6=3k4γ1γ1,δ3=0.
(II): γ1=γ3,γ2=2γ3+3kγ3,γ4=2γ33kγ3,δ1=δ324δ2,δ4=0. .
Considering set-I, we provide
gI(x,y,t)=δ1et(3k4γ1)γ1+x+γ1y+δ2et(3k4γ1)γ1xγ1y+δ4sinh(t(3k4γ1)γ1+x+γ1y).
Thus,
χI(x,y,t)=k(2((δ1eϕ1+δ2eϕ1+δ4sinh(ϕ1))(γ1δ1eϕ1+γ1δ2eϕ1+γ1δ4sinh(ϕ1))(δ1eϕ1δ2eϕ1+δ4cosh(ϕ1))(γ1δ1eϕ1γ1δ2eϕ1+γ1δ4cosh(ϕ1))))/(δ1eϕ1+δ2eϕ1+δ4sinh(ϕ1))2,
where ϕ1=t(3k4γ1)γ1+x+γ1y .
Considering set-II, we provide
gII(x,y,t)=δ32et(2γ3+3k)γ3+xγ3y4δ2+δ2et(2γ3+3k)γ3x+γ3y+δ3sin(t(2γ33k)γ3+x+γ3y).
Thus,
χII(x,y,t)=k2((δ32eϕ24δ2+δ2eϕ2+δ3sin(ϕ3))(γ3δ32eϕ24δ2γ3δ2eϕ2γ3δ3sin(ϕ3))(δ32eϕ24δ2δ2eϕ2+δ3cos(ϕ3))(γ3δ32eϕ24δ2+γ3δ2eϕ2+γ3δ3cos(ϕ3)))/(δ32eϕ24δ2+δ2eϕ2+δ3sin(ϕ3))2,
where ϕ2=t(2γ3+3k)γ3+xγ3y,ϕ3=t(2γ33k)γ3+x+γ3y .

3. Breather wave solutions

We use the following test function to ensure that the breather wave solutions are secured:
g(x,y,t)=η1cos(p0(λ0t+x+y))+η2exp(p1(β0t+x+y))+exp(p1(β0t+x+y)).
Plugging (10) into (3), the following collection of solutions are obtained by conducting some symbolic computations:
(I): β0=3k+3p02p12,λ0=3k+p023p12,η2=η12p024p12.
(II): p1=ip0,β0=6kλ0+8p02,η2=η124.
(III) p1=ip0,β0=3k+4p02,λ0=3k+4p02.
Considering set-I, we provide
gI(x,y,t)=η12p02ep1(t(3k+3p02p12)+x+y)4p12+η1cos(p0(t(3k+p023p12)+x+y))+ep1(t(3k+3p02p12)+x+y).
Thus,
χI(x,y,t)=k(2((14η12p02eϕ4η1p02cos(ϕ5)+p12eϕ4)(η1cos(ϕ5)η12p02eϕ44p12+eϕ4)(η12p02eϕ44p1η1p0sin(ϕ5)p1eϕ4)2))/(η1cos(ϕ5)η12p02eϕ44p12+eϕ4)2,
where ϕ4=p1(t(3k+3p02p12)+x+y),ϕ5=p0(t(3k+p023p12)+x+y).
Considering set-II, we provide
gII(x,y,t)=14η12eip0(t(6kλ0+8p02)+x+y)+eip0(t(6kλ0+8p02)+x+y)+η1cos(p0(λ0t+x+y)).
Thus,
χII(x,y,t)=k(2((14η12eϕ6+η1cos(p0(λ0t+x+y))+eϕ6)(14η12p02eϕ6η1p02cos(p0(λ0t+x+y))p02eϕ6)(14iη12p0eϕ6η1p0sin(p0(λ0t+x+y))+(i)p0eϕ6)2))/(14η12eϕ6+η1cos(p0(λ0t+x+y))+eϕ6)2,
where ϕ6=ip0(t(6kλ0+8p02)+x+y).
Considering set-III, we provide
gIII(x,y,t)=η2eip0(t(3k+4p02)+x+y)+η1cos(p0(t(3k+4p02)+x+y))+eip0(t(3k+4p02)+x+y).
Thus,
χIII(x,y,t)=k(2((η2eϕ7+η1cos(ϕ8)+eϕ7)(η2p02eϕ7η1p02cos(ϕ8)+p02(eϕ7))(iη2p0eϕ7η1p0sin(ϕ8)+ip0eϕ7)2))/(η2eϕ7+η1cos(ϕ8)+eϕ7)2,
where ϕ7=ip0(t(3k+4p02)+x+y),ϕ8=p0(t(3k+4p02)+x+y).

4. Conclusion

Generally, the Korteweg De Vries equation is a mathematical model of waves on shallow water surfaces used in mathematics. It is renowned as the classic example of a perfectly solvable model whose solutions can be described precisely and precisely. Via the properties of Bell's polynomial, Wang et al. [22] gave the general lump solutions, lumpoff solutions, and rogue wave solutions to (1). Rosa et al. [23] presented the exact solutions of (1) via Lie symmetry analysis. Wazwaz [24] provided single and multiple-soliton solutions to (1) via tanh-coth method and the cosh ansatz. The (2+1)-dimensional KdV equation with plasma electrostatic wave potential has been studied in this paper. The bilinear Hirota technique is used to effectively design new lump interaction complexions. The evaluated nonlinear model was fulfilled by all of the recorded solutions. Numerical simulations of the resulting results are also shown using 3-dimensional and contour graphs. The findings given in this study have never been published before, to the best of our knowledge (Fig. 1, Fig. 2, Fig. 3, Fig. 4).
Fig.1 The 3D and density view of Eq. (7) under k=2.4,γ1=4,δ2=5.52,δ4=4.4,δ1=9.94 .
Fig.2 The 3D and density view of Eq. (9) under k=1.4,γ3=2.1,δ2=3.1,δ3=2.2 .
Fig.3 The 3D and density view of Eq. (14) under k=1.8,λ0=2.6,;p0=1,η1=5.52 .
Fig.4 The 3D and density view of Eq. (16) under k=1,p0=1.8,η2=5.02 .

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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