Multi-wave, breather and interaction solutions to (3+1) dimensional Vakhnenko–Parkes equation arising at propagation of high-frequency waves in a relaxing medium

In this study, based on the Hirota bilinear form, the exact analytic solutions of the (3 + 1) dimensional Vakhnenko–Parkes equation with various physical properties were constructed with the help of the Maple package program and symbolic computation. These solutions are the type of multi-waves, breather wave, lump–kink, lump–periodic solutions and interaction solutions (between lump and hyperbolic wave solutions). The constructed solutions have expanded and enriched the solution forms of this new model existing in the literature. By means of Maple package program, 3D and 2D graphs were drawn for the special choices of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed. The solutions obtained can be used in the explanation of physical phenomena occurring in the propagation of high-frequency waves in a relaxing medium.


Introduction
The nonlinear evolution equations (NLEEs) model physical phenomena that occur in many areas of science include plasma physics, solid-state physics, materials science, fluid mechanics, oceanology, signal processing, system identification, mechanics, optical fibres, geochemistry, biology, data mining, artificial intelligence and telecommunications.
In order to better understand the physical phenomena modelled by such equations, or in other words, to look at the physical characteristics of the studied problem from a more accurate point and to reveal its possible applications, it is very important to obtain exact analytic solutions.
Exact or numerical solutions of NLEEs can be obtained using developed methods, computers and various computer programs that can perform long and tedious operations faster. Some of the methods developed in the past years are the modified direct algebraic method [1], Lie symmetry method [2][3][4][5][6][7][8], the Hirota method [9], Painleve method [10], the variational iteration algorithm-II [11], the extended direct algebraic method [12][13][14], the integral equation method [15], the extended (G /G) -expansion method [16,17], the Sinh-Gordon function method [18], the extended auxiliary equation method [19,20], the F-expansion method [21] and so on [22][23][24][25][26]. The Vakhnenko equation was described in 1992 [27] as ∂ ∂x occurs in modelling the propagation of high-frequency waves in a relaxing medium [28]. Here u is the dimensionless pressure which is the function of the spatial variable x and temporal variable t. In 1998, Equation (1) has been converted to Vakhnenko-Parkes (VP) equation which is given by by Vakhnenko and Parkes [29]. In 2017, the n-loop soliton solutions for(2 + 1)-dimensional Vakhnenko equation were calculated in [30]. In 2018, the modified form of Equation (2) was introduced by Wazwaz [31] using the meaning of the modified KdV equation. This form can be given as and is called as modified Vakhnenko-Parkes (mVP) equation. It has been shown by Wazwaz that the equation is completely integrable and multiple soliton solutions are obtained. The studies on the VP equation in recent years are quite remarkable [32][33][34].
In this work, we focus our attention on the (3+1) dimensional integrable Vakhnenko-Parkes (VP) equation and present multi-wave, breather-wave solutions and some interaction solutions using symbolic computation. This equation has a Hirota bilinear form [35], and so, we will do a search for some function solutions containing a set of free parameters using this form. Studies conducted in recent years show that the study on lump solutions, breather wave, multi-wave and interaction solutions are quite remarkable [36][37][38][39][40][41]. If we refer to the meaning of some interaction solutions obtained in this study, as known, solitons characteristically contain a localized wave form that is retained upon interaction with other waves. While soliton solutions are local in a certain direction, lump solutions are a kind of rational function solutions that are local in almost all directions in space [42]. The travelling waves that rise or descend from one asymptotic state to another are called kink waves. The kink solution goes to a constant at infinity [43]. Lump-kink solution is interaction of lump and one kink soliton, lump-periodic solution is interaction of lump with periodic waves and lump-kink-periodic solution is interaction of lump with one kink soliton and periodic wave [40]. Multi-wave solutions are solutions that can sometimes be converted into a single very high-energy soliton that propagates without dispersion over large areas of space. They can produce a highly destructive wave (like tsunami). This increases the importance of these type of solutions [44]. Breathers are the partially localized breathing waves with a periodic structure in a certain direction [45].
The paper is established as follows: We discussed the mathematical structure of the higher dimensional integrable Vakhnenko-Parkes equation in Section 2. In Section 3, we obtain multi-wave solutions including hyperbolic and trigonometric functions. In Section 4, we obtain breather wave solution using homoclinic test function. In Sections 5, 6, 7 and 8, we obtain some interaction solutions. In Sections 9 and 10, we present discussion and conclusion of our obtained results.

The governing equation
The complexity of explaining and interpreting phenomena such as the propagation of waves, optical fibres and biological systems with(1 + 1) dimensional systems revealed that higher dimensional systems should be defined. A (2 + 1)-dimensional VP equation which is given by is formally derived by Victor et al. [46] following the demand for higher dimensional integrable systems. Equation (4) models high-frequency wave perturbations in relaxing high-rate active barotropic media and involves x, y (spatial variables) and t (temporal variable). The(3 + 1)-dimensional VP equation that emerges in the work of Wazwaz [47,48] is given as follows: whereu(x, y, z, t) = U(T 1 , T 2 , T 3 , X), and W(T 1 , T 2 , T 3 , From (6) it follows that From Equations (5) and (6), we obtain Substituting (7) and (9) into (8) yields Equation (10) can be written in bilinear form [35]: where W = 6(ln f ) X . (12) Here D j , j = x, y, z, t, are the bilinear differential operators and f = f (x, y, z, t) are real functions [49]: where x , y , z and t are the formal variables, m, n, s and l are the non-negative integers, h 1 depends on x, y, z, t and h 2 depends on x , y , z , t . Taking into account the Equation (11), the following equation is obtained: In [35], the Painleve analysis to prove the complete integrability to Equation (5) is applied and multiple soliton solutions via using the simplified Hirota's method are derived.

Multi-waves solution
Suppose that the solution of Equation (13) is given as [50][51][52][53] where are parameters to be obtained with calculations. Imposing Equation (14) into Equation (13), a set of algebraic equations for w i , p i , r i , s i , q i , b 0 , b 1 , b 2 are obtained. The obtained system of algebraic equations can be solved by using an auxiliary computer program (Maple). As a result, we have obtained many distinct variants of the constraint equations leading to a reduction in the number of parameters encountered in the equation system.

Breather wave solutions
Depending on the homoclinic breather approach [50,51,54,55], suppose that where η i = w i T 1 + p i T 2 + r i T 3 + s i X + q i , i = 1, 2 and v 1 , v 2 are parameters to be determined. Plugging (38) into bilinear form given in (13) and equating the coefficients of exp(−η 1 ), exp(η 1 ), cos(η 2 ), and sin(η 2 ) to zero, we have numerous equations for the parameters. If the algebraic system is solved using Maple programming, the sets of coefficients are yielded as follows: Set 1.

Lump-periodic solution
The following transformation for lump-periodic solution given as [39,40,56,59] is used: where η i = w i T 1 + p i T 2 + r i T 3 + s i X + q i , i = 1, 2, 3, and b 0 , b 1 are real parameters to be set up. We obtained the following set of parameters by removing coefficients of independent variables and trigonometric functions after substituting (57) into Equation (13).

Discussion part of results
The phenomena of interaction between a lump and kink soliton, interaction of lump with periodic waves, and interaction among a lump, periodic waves and one kink soliton for the (3 + 1) dimensional integrable VP equation were generated as illustrative examples in Figures 1-6. In Section 3, the lump-periodic solutions are reported and different structures of periodic-lump waves are demonstrated in Figures 1 and 3. In Section 4, the homoclinic test function (38) is assumed as a solution to the bilinear equation. The solution obtained in Section 5 consists of positive quadratic function and exponential function. The function f is taken as a combination of positive quadratic function and cosine function in Section 6 and hyperbolic cosine function in Section 7. It is assumed that the auxiliary function f includes positive quadratic, exponential and trigonometric functions in Section 8. The interaction among a lump, triangular periodic waves and one-kink soliton of (77) with the settings (78) is presented in Figure 6.

Conclusion
The (3 + 1) dimensional integrable VP equation is studied by employing a direct method based on Hirota bilinear formulation. Some multi-wave, breather wave and lump-interaction solutions via the symbolic computation are obtained. To the best of our knowledge, the solutions obtained are all new. We believe that the results will benefit future research.

Disclosure statement
No potential conflict of interest was reported by the author(s).