New interaction solutions of the KdV-Sawada-Kotera-Ramani equation in various dimensions

In this paper, the KdV-Sawada-Kotera-Ramani(KdVSKR) equation in various dimensions are studied. The bilinear form of the (1+1)-dimensional and (2+1)-dimensional KdVSKR equation are obtained by the independent transformation. Based on the Hirota bilinear method, we constructed new interaction solutions by studying the unknown nonlinear differential equations for the corresponding parameters. Three dimensional plots, density plots and contour plots provide us with a better understanding of visualizing the dynamic behavior of solutions.


Introduction
Recently, two large tides in the Qiantang River intersected to form a Y-shape, and such a cross-tide spectacle is also a soliton wave in shallow water. In nonlinear physics, soliton wave, lump wave and rouge wave have received a lot of attention due to the localized nature of their structure. It has been found that the interaction between various waves can produce some interesting phenomena. For example, when lump wave interact with soliton pair, the soliton pair can excite unknown rouge wave, which may cause incalculable damage to the ship. As we all know, many phenomena in physics and other fields are described by nonlinear differential equations. The study of interaction solutions, a mixture of various functions reflecting nonlinear wave interactions, can contribute to a deeper and more comprehensive understanding of the nonlinear world, so as to reduce the damage caused by mysterious interaction waves as much as possible. Therefore, when we want to understand the physical mechanism of phenomena in nature, interaction solutions for the nonlinear differential equations have to be explored.
In the current course of analysis, we have examined the integrable KdV-Sawada-Kotera-Ramani(KdVSKR) equation in (1+1) and (2+1) dimensions, given as where α and β are arbitrary constants. They can be expressed as under the common logarithmic transformation In the above equations, D x , D y and D t are the Hirota bilinear operators [23][24][25] defined by Through the generalized auxiliary equation method, many new exact solutions to equation (1) are obtained including Jacobi elliptic function solutions, the Weierstrass elliptic function solutions and rational solutions [26]. Solitary wave solutions, periodic and quasi-periodic traveling wave solutions are constructed by reducing the KdVSKR equation into two systems of ordinary differential equation [27]. The equation (2) can be obtained by extending the y dimension from equation (1). Dynamics of multi-order lumps and interaction between lump and solitons are derived by the long wave limit method [28]. The soliton molecules are obtained by velocity resonance mechanism and multi-breather solutions are constructed through complex conjugation relations [29]. New y-type molecules are studied by giving new constraint and some hybrid solutions with y-type molecules is obtained by using velocity resonance method and mode resonance method [30].
The primary purpose of the work is to explore the various interaction solutions of the KdVSKR equation in various dimensions. The paper is arranged in the following manner: in section 2, we will study the (1+1)dimensional KdVSKR equation for the solution of hyperbolic cosine and cosine functions and the solution of hyperbolic sine, sine and exponential functions. The dynamics behaviors of the new interaction solutions are shown by selecting the particular parameters. In section 3, we will attain the (2+1)-dimensional KdVSKR equation for the interaction solution of lump solution with hyperbolic cosine function and the interaction solution of lump solution with one exponential function. Interaction phenomena is also analyzed by observing the three dimensional plots at different times. In section 4, we will give concluding remarks.

The solution of hyperbolic cosine and cosine function
Consider the first test function as a solution to equation (7) where a i , b i (i = 0, 1, 2) and k j ( j = 1, 2) are arbitary constants to be determined. Substituting expression (8) into (7), then equating all coefficients of ( ) ( ) ( ) ( ) cos , sin , cosh , sinh q q q q have coefficients of different powers to zero. We get the system of bilinear equation for a i and b i as follows: Case-1: where a 2 , b 0 , b 2 , k 1 , k 2 are arbitary real constants. Through the transformation (5), we get the following solution where a 2 , b 2 , k 1 , k 2 are arbitary real constants. Through the transformation (5), we get the following solution A large number of arbitrary constants exist in the above equation, and exact solutions with a rich spatial structure are obtained by a special selection of the relevant parameters. Figure 1 represent the three dimensional plot, density plot, contour plot and the wave along x-axis of (12). Obviously, multiwaves appear after interaction of different types of waves and the shape of the periodic wave does not change over time.

The solution of hyperbolic sine, sine function and exponential functions
Then, we consider the another test function as a solution to equation (7) 1,2) and k j ( j = 1, 2, 3, 4) are arbitary constants to be determined.
Substituting (13) into (7), yields a polynomial in the powers of trigonometric, hyperbolic and exponential functions. Collecting the coefficients of the same power, and equating each summations to zero, produces an algebraic system of equations. We solve the system of equations to obtained the values of the parameters involved. Substituting the values of the parameters into (13) and then into (5), yields the interaction solutions to equation (1): Through the transformation (5), we get the following solution It can be found that the value of c 0 affects the coefficients of x and t in the exponential function and hyperbolic function at the same time. In order to observe the motion trajectory of the solutions more clearly, we change the value of c 0 to draw the three dimensional plots, density plots and contour plots of (15) respectively. Figure 2 shows that the splitting phenomenon of the interaction solution becomes more vivid and the amplitude is larger with the increase of c 0 . They correspond to the parameter sets a 2 = 0.

The (2+1)-dimensional KdVSKR equation
The (2+1)-dimensional KdVSKR equation has the following bilinear form      Case-4: e  e  e  e  e k  k  a  b  k  k   ,  ,  ,  ,  In order to better observe the dynamics of the interaction solution of lump solution with hyperbolic cosine function, a set of parameters of (25) is selected. Figure 3 shows the three dimensional plots and density plots of (25) at different times when b 0 = 0.6. Obviously, the hump is the lowest at t = 0. With the increase of t, the lump wave and double stripe soliton moved in the specific direction, and the intersection of two stripe solitons was due to the appearance of lump wave. Figure 4 shows the three dimensional plots and density plots of (25) at different times when b 0 = 1. Without losing generality, the lump wave still appears at the intersection of two stripe solitons. But the double stripe soliton is more clear and the hump is lower under this set of parameters.

The interaction solution of lump solution with one exponential function
To obtain the interaction solution of lump solution with one exponential function, we assume that where a i , b i , e i (i = 0, 1, 2, 3) and k j ( j = 0, 1, 2) are unknown constants to be determined. Substituting expression (26) into (16), and equating all coefficients of ( ) t x y , , , exp 3 q to zero, produces an algebraic system of equations. Solving the system of equations to get the values of the parameters involved, yields the following interaction solutions:    . 32   In order to better observe the dynamics of the interaction solution of lump solution with one exponential function, a set of parameters of (34) is selected. Figure 5 shows the three dimensional plots and density plots of (34), which shows that peaks and troughs persist and the density distribution remains constant at different times, but the peaks decrease gradually under the interaction of different waves.

Conclusions
In this paper, the KdVSKR equation in (1+1)-dimensional and (2+1)-dimensional are discussed. This paper constructs new interaction solutions based on the Hirota bilinear method. Some of the solutions are hyperbolic sine, sine with exponential function, lump solution with hyperbolic cosine function, and lump solution with exponential function. Three dimensional plots, density plots and contour plots of the interaction solutions were made, and we can observe the physical structure and characteristics of the interaction phenomenon. The amplitudes and shapes of the interaction waves remain unchanged, which is a completely elastic collision in a sense. These interaction solutions to the KdVSKR equation enrich the field of research on the resonances of solitons. It is hoped that the obtained results can find the potential applications in nonlinear physics and improve our understanding on the dynamical behavior of relevant fields in physical and engineering fields. However, the interaction solutions of the KdVSKR equation in higher dimensions is not further explored in this paper, and it is hoped that it will be explored in the future research.