Research paperPredictability, fast calculation and simulation for the interaction solutions to the cylindrical Kadomtsev-Petviashvili equation
Introduction
Nonlinear evolution equations (NLEEs) have been used to describe the phenomena in many fields involving plasma, ocean dynamics, nonlinear optics, Bose-Einstein condensates, physiology and biology [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Numerous researchers have contributed themselves to finding the exact solutions to NLEEs [11], [12], [13], [14]. Many effective ways have been discovered to derive exact solutions, such as the Hirota bilinear method [15], the Bäcklund transformation [15], the inverse scattering method [16] and the Darboux transformation [17]. Soliton, which is exponentially localized in certain direction, has many special characteristics and is widely used in the nonlinear phenomena simulation [18], [19], [20], [21]. Rational solution is another type of exact solution, and lump solution is a special kind of rational solution attracting much attention recently. In 2006, lump solution was derived by using the variable separation method [22]. With symbolic computation, lump solutions to the Kadomtsev-Petviashvili (KP) equation were obtained through a direct method [23]. Since the development of symbolic computation [24], more and more lump solutions were successfully obtained [25], [26], [27], [28]. In nature, some nonlinear phenomena can be described by the interaction between soliton and rational waves. Interaction solutions are useful in analyzing how two or more waves interact with each other, such as interaction between a lump wave with solitons. Some research implies that the appearance of rogue waves has a strong relationship with such solutions [4], [29], [30]. Moreover, people have recently derived lumps and the interaction solutions to both linear and nonlinear partial differential equations in (2+1)-dimensions [31], [32], [33], and even solitonless solutions [34]. However, there are few studies on interaction solutions to the NLEEs with variable coefficients, for example, the cylindrical Kadomtsev-Petviashvili (cKP) equation (which will be discussed in this paper).
In 1895, Korteweg and de Vries firstly derived the KdV equation, by balancing weak nonlinearity against weak (linear) dispersion [35]. Based on the research in the KdV equation, scientists started to pay attention to equations in two spacial dimensions. The KP equation, as the two-dimensional KdV equation, was one of the important models [36]. Another research direction was to explore equations related to different geometries. The equation for purely concentric waves related to cylindrical geometry was deduced by Maxon Viecelli at the first time, and it was named after the cylindrical KdV (cKdV) euqation [37]. The KdV equation can be extended to higher dimensions (e.g., two dimensions), and the cKdV equation can also be generalized to the following cKP equation [38], [39], [40], [41]This equation was firstly introduced by Johnson in 1978 [38]. Lipowskii later derived the same equation for internal waves in a stratified medium [39]. The cKP equation was also deduced for internal waves in a stratified medium and for magnetized plasmas with pressure effects and transverse perturbations [40]. The solutions to the cKP equation describe some waves in reality, for instance, picture 19 on page 38 in Ref. [41] is an example of horseshoe-like-front soliton.
Johnson’s derivation process started from the classical water-wave problem (the derivation can be seen in [42]), where there is an incompressible and irrotational fluid bounded above by a free surface and below by a rigid horizontal surface. The fluid extends to infinity in all horizontal directions. In the absence of any disturbance, the fluid will be stationary with a constant depth. He interpreted incompressibility as in any given position in fluid (ρ represents the density of the fluid which is a function of x, y and t). In reality, the flow is hardly irrotational anywhere, while the vorticity of some flows is sometimes very small. It is reasonable to assume irrotationality in a simplified model. The surface is named the free surface when the atmosphere exerts only pressure on the surface, without including a viscous component, which is often taken as a constant (the atmospheric pressure). At the bottom of the fluid, he assumed that the bed is impermeable. With these assumptions, the cKP equation was derived in cylindrical co-ordinate by balancing weak nonlinearity and weak dispersion.
Depending on the real constant α2, we call Eq. (1) the cKPI equation when α2 is negative, and the cKPII equation when α2 is positive. When the cKP equation obviously reduces to the cKdV equation.
Through the dependent variable transformationthe Hirota bilinear form of the cKP equation can be written aswhere DxDt, and are bilinear operators [15] defined by
Based on the research in Refs. [17] and [43], the Lax pair corresponding to the KP and cKP equations were shown to be equivalent to some extent. By using Darboux transformation method, a class of solutions (include solitons and lumps) were derived, and the difference between the solutions to the KP equation and the cKP equation was briefly compared. For instance, unlike the line-front soliton and the lump solution to the KP equation, the cKP equation has the horseshoe-like-front soliton and dissipative lump solutions. One-decay mode and two-decay mode solutions to the cKP equation were discovered with the simplified homogeneous balance method [44]. Recently, a type of variable-coefficient KP (vcKP) equation has been studied [45], but it does not involve the cKP equation studied in this paper.
The outline of this paper is as follows. In Section 2, the one-soliton, two-soliton and three-soliton solutions will be derived and discussed. The formula of the N-soliton solution will be given as well. In Section 3, supposing the solution f as a polynomial function, we will derive several families of rational solutions to the cKP equation. In Section 4, the lump solutions and some properties will be given, and we will briefly compare the lump solutions to the KP equation and the cKP equation. The propagation behaviors will be shown with some figures. In Section 5, we will discuss the interaction solutions between a lump wave and a stripe wave by considering a combination of two positive quadratic functions and an exponential function. The interaction solutions between a lump and a two-soliton wave will be considered as a combination of two positive quadratic functions and a hyperbolic cosine function. The motion process of interaction solutions will be indicated by sets of graphs at different time t. Section 6 will be our conclusions.
Section snippets
One-soliton solution
We define the form of f(x, y, t) as followswhere P, Q, Ω and η0 are all real constants to be determined.
With symbolic computation, we can solvewhere P ≠ 0 and the equation is known as the nonlinear dispersion relationship. The one-soliton solution to Eq. (1) can be written as
According to the properties of the function sech2(x), the solitary peak value is
Rational solutions
In this section, we will derive the rational solutions with symbolic computation. Rational solutions are the basis of research for rogue waves, thus solving the rational solutions is of importance.
Suppose f is a polynomial function of variables x, y and t aswhere ci,j,k () are all real constants. In this paper, we consider the situation of . With symbolic computation, we obtain five families of solutions as below.
The first class of
Lump solutions
We find the cKPII equation has no lump solution, and we will investigate the cKPI equation of which the Hirota bilinear form iswhere α2 is a positive real constant. We suppose the form of f iswherewhile ci (1 ≤ i ≤ 9) are real constants to be determined and c9 > 0.
For any fixed value of t, the extreme point of the lump locates at
Interaction solutions
In this section, we will focus on the interaction solutions between lump wave and soliton. With symbolic computation, we find that cKPII has no interaction solution. Hence we will analyze the cKPI equation, that is, Eq. (13).
Concluding remarks
We have discussed various exact analytical solutions to the cKP equation (especially the cKPI equation) based on its bilinear representation and four types of testing function (see Eqs. (12), (14), (18) and (24)). Firstly, one-, two-, and three-soliton solutions have been provided. Supposing the solution f to the bilinear equation as a polynomial, rational solutions have been obtained with symbolic computation. As a special type of rational solutions, lump solutions have been derived via
CRediT authorship contribution statement
Jun-Wen Xia: Writing - review & editing. Yi-Wei Zhao: Writing - original draft. Xing Lü: Supervision, Methodology, Software.
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.
Acknowledgements
This work is supported by the Fundamental Research Funds for the Central Universities of China (2018RC031), and the National Natural Science Foundation of China under Grant No. 71971015. J.W. Xia and Y.W. Zhao are supported by the Project of National Training Program of Innovation and Entrepreneurship for Undergraduates under Grant No. 201910004096.
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