Novel y -type and hybrid solutions for the ( 2 + 1 ) -dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

In this paper, the research object is ( 2 + 1 ) -dimensional Korteweg–de Vries–Sawada–Kotera– Ramani (KdVSKR) equation. After adding new constraint, new solutions which contain y -type molecules are obtained. The process of lump molecules and y - type molecules before and after the collision is studied by long-wave limit method, and the kinetic behavior analysis is given. The interactions between y - type molecules and resonant soliton molecules, y -type molecules and breather molecules are obtained by combining velocity resonance method and mode resonance method, respectively. Finally, a novel hybrid solution containing lump molecule, breather molecule and y - type molecule is given, and the kinetic behavior is shown.

In recent years, fission solution and fusion solution have also become hot targets. Wang et al [32] first proposed the concepts of fission solution and fusion solution and applied them to [33,34]. The expression is In the process of calculation, the form of (1) is relatively simple, so it can not obtain all of fission solutions and fusion solutions when applying (1) to nonlinear equations. In order to obtain more fission solutions and fusion solutions, Chen [35] extended this method and gave a more convenient expression: With the help of (2), the fission solution and fusion solution of many nonlinear equations have been studied [36][37][38][39]. The fission solution and fusion solution also call y-type solution. In this paper, we will explore the y-type solutions and hybrid solutions of (2 + 1)dimensional KdVSKR equation: where α and β are arbitraries. Zhu et al. obtained a bilinear form of this equation and derived multi-order lumps, interaction between lump and solitons by using the long-wave limit method [40]. Wei et al. obtained soliton molecules by using velocity resonance mechanism and derived multi-breather solutions by means of complex conjugation relations [41]. The (2 + 1)-dimensional KdVSKR equation is obtained by extending the y dimension from the (1+1)dimensional KdVSKR equation: where a and b are any real numbers. Osman constructed the rational double-soliton rational solutions of the KdVSKR equation and discussed the dynamical behavior of traveling wave solutions [42]. Xiong et al. obtained the asymmetric soliton molecule of KdVSKR equation with the help of N -soliton solution and velocity resonance mechanism. A general group invariant solution is derived by using the direct method of symmetric groups [43]. Zhang et al. introduced a new solution method to explore the new exact solitary wave solutions and quasi-periodic traveling wave solutions of the KdVSKR equation [44]. By using the Lie symmetry analysis method, the vector field and optimal system of the equation are derived, respectively, and the complete integrability of the equation is systematically presented by using binary Bell's polynomials in [45].
The (3) can be regarded as a linear combination of the KdV equation and (2 + 1)-dimensional Sawada-Kotera equation.
Therefore, when β = 0 (3) reduces to the standard KdV equation: Kota proved the existence, uniqueness and continuity of local solutions to the KdV equation depending on the initial values in [46]. Ma obtained the Wronskian determinant of the KdV equation by using the characteristic function of Schrödinger spectrum problem and further obtained the complex-valued solution in [47]. While when α = 0 (3) reduces to the (2 + 1)dimensional Sawada-Kotera(SK) equation: Li et al. [48] obtained soliton molecules of the SK equation and obtained asymmetric solitons using the velocity resonance mechanism. Lai et al. [49] obtained the approximate and exact solutions of bidirectional Sawada-Kotera equation with the help of Adomini polynomials. Wang et al. [50] constructed a large number of traveling wave solutions for the (2 + 1)dimensional SK equation using the traveling wave method.
In the following content, the (2 + 1)-dimensional KdVSKR equation is the research object. Our aim is to explore new y-type solutions of the (2 + 1)dimensional KdVSKR equation. The long-wave limit method, velocity resonance method and mode resonance method are used to study a variety of hybrid solutions and explore their dynamic behaviors. This article will be arranged as follows: In Sect. 2, the y-type solutions of (2+1)-dimensional KdVSKR equation are obtained by adding new constraint.
In Sect. 3, the interactions between y-type molecules and lump molecules are studied by combining the longwave limit method, and their kinetic behavior is also demonstrated by data.
In Sect. 4, the first hybrid solution involving three molecules is introduced.
Finally, concluding remarks for the study on the ytype solutions are given in Sect. 5.
2 The y-type solutions of (2 + 1)-dimensional KdVSKR equation applying Hirota bilinear method to (3), we will get where the Hirota's bilinear operator is defined and multi-soliton solution is with In order to obtain the y-type solutions, a restriction condition is given here: then we acquire Here, is a necessary addition: exp(X ) = 0 if and only if X = ln(0). The symbol in (13) determines two diametrically opposed results. Here, it can be considered that "+ determines the fission solution and "− determines the fusion solution. Easy to understand, fission solution refers to the separation of solitary waves over time. In y-type waves, it can be understood as the angle between two or more branches becomes larger. The fusion solution is the opposite.
Next, the fission solutions and fusion solutions of the (2+1)-dimensional KdVSKR equation under three conditions are given. These are N = 2, N = 3, and N = 4. Case 1 When N = 2, the simplest y-type molecule formed by two solitary waves will be obtained. Then, (10) is here, the parameters are α = 2, β = −1, 125 , φ 1 = 0, φ 2 = −10. As time goes by, fission occurs between the two branches of the solution. Then, we obtain 1-fission molecule. The density maps are shown in Fig. 1a. When t takes different values, the motion process of 1-fission molecule will be obtained. Of course, taking the negative sign in (13), 1-fusion molecule will be obtained, its result is just like 1-fission molecule.

Case 2 When N = 3, the expression is
At this moment, the solution is 2-fission molecule with three branches. In other words, one branch and the other two branches undergo fission and fusion, respectively. The result is shown in Fig. 1b. With the passage of time, the positions of the three branches and the relationship between fission and fusion do not change, but the overall positions will move. A particularly interesting phenomenon is that when changing p 2 = − 28 25 125 , we get a solution whose shape like the letter X . Then, fission and fusion occur between one branch of the solution and the other two branches, respectively, and the solution is 1-fissionfusion molecule. Case 3 When N = 4, under the conditions of (12) and (13), the form of the solution determined by (10) is In theory, we can get two y-type molecules. And there are three cases: two fission molecules, two fusion molecules, one fission molecule and one fusion molecule. Their density maps are shown in Fig. 2.
In each case, two y-type molecules collide with each other, and the collisions are elastic, meaning they do not change their energy or shape. But no fission or fusion occurs between the two y-type molecules. Of course, for N = 4, we can also get a solution with four branches, in which case the results are more complicated.

Interaction of y-type molecules and lump molecules
Zhang [51] first proposed the concept of lump molecule. So far, the interactions between lump molecules and other molecules have also been studied in Refs. [52][53][54]. In this section, the interaction between lump molecules and novel y-type molecules will be explored, the processand results are shown as follows. Here, we will use the long-wave limit method in combination with the new constraint: where k i and k * j are conjugated. The mixture solutions of Ξ -order lump molecules and Θ-order y-type molecules can be obtained. And the trajectory of lump molecule is limited by {K 2m−1 , P 2m−1 , K 2m , P 2m }. In this case, the trajectory meets where x −in f and y −in f correspond to the state of the lump before the collision, x in f and y in f correspond to the state of the lump after the collision. However, the peak value of lump molecules does not change before and after the collision, and there is 3.1 Interaction of one-order y-type molecule and one-order lump molecule In this subsection, it can be seen from (17) that when Ξ = 1, Θ = 1, the interaction between one-order y-type molecule and one-order lump molecule can be obtained. At this point, the expression is and Figure 3 shows the interaction between one-order lump molecule and one-order fission molecule. The black line is the trajectory before the collision, and the red line is the trajectory after the collision. The peak value of lump molecule was always 242 195 before and after the collision.
Similarly, Fig. 4 shows the interaction between oneorder lump molecule and one-order fusion molecule. The pre-collision trajectory is the black line and the post-collision trajectory is the red line with the peak value 242 195 .

Interaction of two-order y-type molecules and one-order lump molecule
Under the condition (17), when Ξ = 1, Θ = 2, the solution consisting of two-order y-type molecules and one-order lump molecule is obtained. When the lump molecule collides with either of the two y-type molecules, its trajectory will change. However, it will not be shown here. Figure 5 shows the interaction results of two-order y-type molecules and one-order lump molecule in the three cases, respectively. (a) shows the solution consisting of two 1-fission molecules and one-order lump molecule, (b) shows the solution consisting of 1-fission molecule, 1-fusion molecule and one-order lump molecule, (c) shows the solution consisting of two 1-fusion molecules and one-order lump molecule. In the above three cases, the shape of the molecules does not change before and after the collision, which means that the collision process is elastic.

Hybrid solutions with y-type molecules
In this section, two main methods are used: velocity resonance method and mode resonance method. By combining (12), the interaction process and results between y-type molecules and different forms of molecules were studied.
Their action process and results are shown in Fig. 6. Here, (a) shows the result of interaction between one-order resonant soliton molecule and one 1-fission molecule. (b) shows the result of interaction between one-order resonant soliton molecule and one 1-fusion molecule.

Interaction of y-type molecules and breather molecules
Many equations have been studied by using the mode resonance method. For (3), this method has not been studied. In this section, this method will be used to study the interaction between y-type molecules and breather molecules in combination with new constraint (12). 225 , φ 1 = φ 2 = I π, φ 3 = φ 4 = φ 5 = φ 6 = 0; c interaction of two 1-fusion molecules and one-order lump molecule at t = 0, with α = 1, β = −1, When where the k i and k * j are conjugated, the solution of Θ-order y-type molecules and Γ -order breather molecules can be obtained.
Under the expression of (29), when Θ = Γ = 1, the solution of the composition of one-order ytype molecule and one-order breather molecule is shown in Fig. 7.
Here, (a) shows the result of interaction between one 1-fission molecule and one-order breather molecule. (b) shows the result of interaction between one 1-fusion molecule and one-order breather molecule. In this process, the y-type molecule and the breather molecule collide elastic, and their shapes do not change before and after the collision.

New hybrid solution
In this subsection, we introduce a novel hybrid solution of (3), which contains lump molecules, y-type molecules and breather molecules. In order to get the solution with Ξ -order lump molecules, Γ -order breather molecules and Θ-order y-type molecules, we take where the k i and k * j are conjugated. When Ξ = Γ = Θ = 1, we obtain mixed solution which contains one lump molecule, one breather molecule and one ytype molecule. The y-type molecule in Fig. 8 is fission molecule, and the y-type molecule in Fig. 9 is fusion molecule. It can be seen from the images that before and after the three collide, their shapes do not change, so the process is elastic collision.
In addition, any combination of (17), (28) and (30) can get a variety of different forms of mixed solutions.

Conclusions
The object of this paper is the (2 + 1)-dimensional KdVSKR equation. By giving new constraint, novel ytype solutions are obtained. The y-type solution can be divided into two types, fission solution and fusion solution. Under the new conditions, combined with velocity resonance method, mode resonance method and longwave limit method, a variety of mixed solutions are studied, and the main conclusions are as follows: 1. N = 2, 1-fission molecule is obtained; N = 3, 2-fission molecule and 1-fission-fusion molecule are obtained; N = 4, three cases with two y-type molecules are analyzed. 2. The collision process between y-type molecules and resonant solitons is studied under the condition of velocity resonance method. 3. The collision process between y-type molecules and breather molecules is investigated under the mode resonance method. 4. Furthermore, a new mixture solution containing one lump molecule, one y-type molecule and one breather molecule is also studied.
The above conclusions are simulated and analyzed. The results show that (2 + 1)-KdVSKR equation has fission and fusion solutions. When the solutions are mixed, their collisions are elastic, that is, no energy is expended or changed.
As for the (2 + 1)-dimensional KdVSKR equation, few studies have been done on it, so many of its properties remain to be explored. Zhang et al. [55][56][57] studied the process of nonlinear superposition of Kadomtsev-Petviashvili I equation, (2 + 1)-dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation and (2+1)-dimensional Sawada-Kotera equation. In the following, we will continue to study the nonlinear superposition of the (2 + 1)-dimensional KdVSKR equation. In addition, we will also try to explore the new y-type solutions of models in [58][59][60] in the future. It is hoped that the research results can explain some natural behaviors and further enrich nonlinear dynamical systems and improve the dynamic behaviors of nonlinear equations.
Funding The authors have not disclosed any funding.
Data availability All data generated or analyzed during this study are included in this published article.