Resonance Soliton, Breather and Interaction Solutions of the Modified Kadomtsev–Petviashvili-II Equation

In this paper, we investigate the modified Kadomtsev–Petviashvili-II (mKP-II) equation, which has important applications in fluid dynamics, plasma physics and electrodynamics. By utilizing the Hirota bilinear method, the N-soliton solutions of the mKP-II equation are obtained. The resonance Y-type soliton, and the interaction between M-resonance Y-type solitons and P-resonance Y-type solitons are constructed by imposing some constraints to the parameters of the N-soliton solutions. Moreover, the novel type of double opening resonance Y-type soliton solutions are obtained by selecting some appropriate parameters in 3-soliton solutions. By making some conjugate assumptions in the parameters, the multiple breathers are presented. Furthermore, the hybrid solutions consisting of multiple breathers and resonance Y-type solitons are investigated. The dynamics of these hybrid solutions are analyzed using both numerical simulations and graphical methods.


Introduction
With the development of nonlinear science, the nonlinear evolution equations (NLEEs) have received extensive attention.Subsequently, many effective methods for calculating nonlinear evolution equations have been formed, including the Hirota bilinear method [1], the inverse scattering transform [2], the Bäcklund transformation [3], the Darboux transformation [4][5][6] and Riemann-Hilbert method [7].Among these methods, the Hirota bilinear method plays an important role in the study of soliton.Based on the method, researchers have obtained a variety of solutions to the NLEEs, such as the 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1272-1281 solitons, lumps, breathers and rogues waves and so on [8][9][10][11][12].Combining the KP hierarchy reduction method with the Hirota's bilinear method, the general N-bright, N-dark soliton solutions and higher-order rogue waves can be constructed [13,14].Resonant solutions have been proven to exist in many nonlinear systems, and they can be constructed using the linear superposition principle [15,16].Resonance phenomena, such as the fission and fusion of solitary waves, are also commonly observed.Breathers, which are special cases of solitons, have attracted significant attention [17][18][19][20].The breathers of the NLEEs can be obtained by making some conjugate assumptions to the parameters in the N-soliton solutions.
The modified Kadomtsev-Petviashvili-II (mKP-II) equation is widely recognized as a critical equation in the field of nonlinear evolution equations (NLEES) and has attracted the attention of numerous scholars [21][22][23][24].The mKP-II equation is of the form [25]: Huang et al. obtain the general rational and semi-rational solutions of this equation [26].Seadawy et al. investigate exact solitary waves solutions of mKP equation [27].Chang investigates some types of kink solitons by using the totally non-negative Grassmannian approach [28].Interaction solutions for the mKP equation are constructed by the nonlocal symmetry reductions and consistent tanh expansion method [29].In this paper, we focus on studying the resonance Y-type solitons, multiple breathers and hybrid solutions of the mKP-II equation by adding some constraints in N-soliton solutions.
The paper is structured as follows.In Sect.2, the N-soliton solutions of the mKP-II equation are obtained by the Hirota bilinear method.The resonance Y-type soliton solutions be derived by adding some constraints in N-soliton solutions.Furthermore, by selecting N = 3 , a novel type of double opening resonance Y-type soliton is constructed.Based on these findings, we investigate the interaction of M-resonance Y-type soliton and P-resonance Y-type soliton.In Sect.3, breather solutions are investigated by making conjugate assumptions to the parameters in the N-soliton solutions.The hybrid solutions including breathers and resonance Y-type solitons are obtained.The dynamics of these phenomena are analyzed.The conclusions are given in the last section.

N-Soliton Solutions and Resonance Y-Type Soliton of the mKP-II Equation
Via the transformation the bilinear form of Eq. ( 1) is obtained as where g = g(x, y, t) and f = f (x, y, t) are real functions, D x , D y and D t are the bilinear derivative operators defined by [1] where a(x, y, t) and b(x, y, t) are differentiable functions, and x ′ , y ′ and t ′ are the inde- pendent variables, l, n and m are nonnegative integers.It is apparent while f and g are solved by ( 3), then u = u(x, y, t) is the solution of (1) by the transformation (2).
To obtain the N-soliton solutions of (1), we employ the Hirota bilinear method.The auxiliary functions f and g are expressed in terms of an expansion parameter by the Hirota bilinear method [30] Substituting (5) into the bilinear form (3) and eliminating the coefficients of all powers of , the series of equations are obtained.By solving these equations, the N-soliton solutions are with i , j = 0, 1 and where the constants k i , l i , i and s i are arbitrary.The notation ∑ =0,1 indicates a summation over all possible combinations of i = 0, 1, Journal of Nonlinear Mathematical Physics (2023) 30:1272-1281 summation ∑ 1≤i≤j is over all possible combinations of the N elements with the specific condition 1 ≤ i ≤ j.
To derive the resonance Y-type soliton solutions, certain terms in (6) need to be removed.It is well-known that exp(x) = 0 is true if and only if x = ln(0) .Therefore, if setting all B ij = 0 , the N-soliton solutions can be reduced to the res- onance Y-type soliton solutions with the form The simplicity of the aforementioned structure renders the derivation of hybrid solutions betweem resonance Y-type solitons and other types of solitons challenging.To overcome this limitation, Li et al. proposed a universal method for establishing connections between the parameters of the N-soliton, which allows for the generation of resonance Y-type solitons in various integrable systems [31].Motivated by their approach, this study aims to investigate the general constraints on the parameters in (6).
Theorem.The nonlinear superposition of M-resonance Y-type solitons and P-resonance Y-type solitons can be obtained from the N-soliton solutions by letting the parameters as [32,33].

and acquiring
The symbol in (10) determines two diametrically opposed results, which shows that the resonance Y-type soliton not only refers to fission solitons, but also to fusion soliton produced by stripe solitons.The formation of fission or fusion solutions are formed by the separation of solitary waves over time.To differentiate between these two distinct physical phenomena, the resonance Y-type soliton, in which the angle between two branches increases as time progresses, is referred to as a fission soliton, while the opposite is true for a fusion soliton.For above formula, the symbol " + " can be interpreted as representing a fusion Y-type soliton, whereas the symbol "−" is interpreted as representing a fission Y-type soliton.
Based on ( 8) and (10), two types of resonance Y-type solitons are plotted in Fig. 1 with the parameters as The density plots of the fusion and fission Y-type solitons are shown in Fig. 1a, b, respectively.
On the other hand, based on the theorem, one takes the double opening resonance Y-type soliton and interaction between M-resonance Y-type soliton and P-resonance Y-type soliton of the mKP-II equation with N = 3 and N = 4 , respectively.
Case 1.For N = 3 , the expression is where the parameters l 2 and l 3 satisfy the condition (10), and A 3 and B 23 is deter- mined by (7).By choosing the appropriate parameters, we obtain a double opening resonance Y-type soliton, which is called a resonance X-type soliton.The intersection position is changed with wave propagation from the Fig. 2.
(  where the relevant parameters j , A j and B ij are determined by (7), ( 9) and (10).By selecting the appropriate parameters, this study has identified three distinct types of ( 13)

Multiple Breathers and Interaction Solutions of the mKP-II Equation
Breather has been widely studied by the Hirota bilinear method and the Darboux transformation [34,35].The multiple breathers can be constructed by selecting the complex conjugate relations of the parameters k i , l i in (6).One takes the fol- lowing parameters constraints in ( 6) "*" denotes the complex conjugate, the multi-order breathers can be obtained from the N-soliton solutions and the conditions (14).
The interaction between M-resonance Y-type soliton and P-order breather can be constructed by taking the parameters as For instance, the interaction between a resonance Y-type soliton and one-order breather are constructed by letting M = 2 , P = 1 .In this case, the expression is By choosing the appropriate parameters, the interaction between a fission Y-type soliton and one-order breather are obtained.This type of solution is shown in Fig. 6 by selecting different time.

Conclusions
This paper employs the Hirota bilinear method to derive the N-soliton solutions of the equation.The resonance Y-type soliton solutions and the interaction between M-resonance Y-type soliton and P-resonance Y-type soliton are constructed by imposing additional constraints on the parameters.In addition, a novel type of double opening resonance Y-type soliton is discovered by choosing N = 3 .The study also investigates multiple breather solutions by making conjugate assumptions on the parameters, and presents hybrid solutions between resonance Y-type solitons and breathers.Through extensive simulation, the dynamical characteristics of various types of waves are analyzed in detail, which may contribute to explaining natural phenomena and expanding the applications of the nonlinear dynamics system.