Interaction phenomena between solitons, lumps and breathers for the combined KP3-4 equation

The Hirota’s bilinear method is used to determine the N-soliton solutions of the combined KP3 and KP4 equation, from which the M-lump solutions that decay to zero in all directions in the plane are obtained by using long wave limit when N=2M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=2M$$\end{document}. Then, we discuss some novel hybrid solutions between lumps, breathers and solitons. In order to shade more light on the dynamical characteristics of the acquired solutions, numerical simulations have been performed by means of the 3D figures under careful choice of the values of the parameters involved. These results may be useful for understanding the propagation phenomena of nonlinear localized waves. The hybrid solutions describing molecules between lumps, breathers and solitons are also presented.


Introduction
Our world comprises so many complex phenomena of nature. The shifts, transitions and interactions between solitons, breathers, lumps and other nonlinear waves are common phenomena and have often been visualized in diverse fields of sciences, for instance, fluid mechanics [1,6,16], Bose-Einstein condensates [21] and nonlinear optics [8,14]. The study of both the formation and interaction of these localized wave structures becomes progressively substantial because of their prominent features [4,9,18,19,[22][23][24][25][26].
The nonlinear partial differential equations are fundamental to model these complex phenomena. There are many well-reputed models, one of which is the Kadomtsev-Petviashvili (KP) equation. This equation has been used in several domains among which are fiber optics, hydrodynamics, plasma physics, nonlinear electrical transmission lines, and so on [2,12,13]. In the KP equation, two special bound states of lump waves decaying in all directions were first found in Ref. [15]. However, the lump solution derived from the classical multi-soliton solution cannot constitute the lump molecule. The same situation also occurs in soliton molecule [10]. Therefore, with higher-order effects added, a novel integrable (2+1)-dimensional KdV equation u t = a(6uu x + u x x x − 3w y ) + b(2wu x − z y + u xxy + 4uu y ), namely, a combination of the KP3 equation and KP4 equation u t = 12(2wu x − z y + u xxy + 4uu y ), u y = w x , is considered. The combined KP3-4 equation is quite different from any member of the KP hierarchy because some interesting properties such as soliton molecules are valid only for the combined system but not for the separated system. In the simplest case, the bound state of two-soliton molecule is characterized by velocity resonance mechanism [10]. The key features of multi-soliton complexes are the equal group velocity of elementary solitons and the molecule-like behavior [5,7,11,20,27]. By contrast, the existence of bound states of solitons, lumps and breathers, has remained largely unexplored. Owing to this, we get motivated to construct novel hybrid and molecule solutions between solitons, lumps and breathers of the cKP3-4 equation.

Multi-soliton solutions of the cKP3-4 equation
Following Hirota's method and plugging the transformation (2) into the cKP3-4 Eq. (1) provide the trilinear equation The equivalence of trilinear equation may be given by [10] ( after introducing an auxiliary variable τ such that where D x , D y and D τ are the Hirota's bilinear derivative operators, met by Finding the multi-soliton solution of Eq. (1) is quite straightforward since we may solve the simpler bilinear form (4) and (5) rather than the trilinear equation (3). The N -soliton solution of Eq. (1) is ascertained by Hirota's formula and depends on the parameters is over all conceivable mixtures of the N elements with the special condition i < j.

Multi-lump solutions of the cKP3-4 equation
The long wave limit method to obtain lump solutions which decay to a uniform state in all directions from Nsoliton solution (6) relies on the freedom of choosing the phase constant η 0 i [3]. For example, taking every exp(η 0 i ) = −1 in (6), then f N may be written as . A rational solution obtained as a long wave limit of k i → 0 of N -soliton solution may be expressed as where in the above formula i, j,...,m,n means the summation over all possible combinations of i, j, . . . , m, n, which are taken from 1, 2, . . . , N and all different. The determinant form of (7) can be found in [17]. In general, if choosing p M+i = p * i (i = 1, 2, . . . , M) for N = 2M, B i j > 0, where the asterisk * denotes the complex conjugation, long wave limit method yields formula for an M-lump solution. In continue, we bring one-, two-and three-lump solutions of the cKP3-4 Eq. (1) in the following subsection.

One-lump wave solution
If the parameters of two-soliton solution are adequately chosen, the long wave limit of the solution gives a twodimensional permanent nonsingular lump that decays in all directions. Exerting exp(η 0 i ) = −1 and taking the long wave limit in (6), yield With regard to a solution of the cKP3-4 equation, f 2 is equivalent to by virtue of (2), where (9) generally gives a singular solution (2) at some position, a nonsingular solution is obtained by taking p 2 = p * 1 = p 1r − i p 1i , p 1i = 0, in which case the wave system is positive dispersive. Thus, we have Inserting (10) into the transformation (2), we have a permanent lump solution of Eq. (1) moving with the velocity v x = −( p 2 1r + p 2 1i )(3a + 2bp 1r ) and v y = 6ap 1r + 3bp 2 1r − bp 2 1i . In Fig. 1, one-lump solution is drawn for a particular choice of the constants. The one-lump wave with one peak and two valley is a standard localized wave decaying in all the directions of space.  (7), the f 4 can be reduced to where , with the aid of variable transformation (2), yields the two-lump wave solution which is localized in space and time. In Fig. 2, the envelope of two-lump is drawn for a particular choice of the parameters.

Three-lump wave solution
As before, three-lump solution is deduced by taking N = 6, M = 3 in (7). The expression is tedious, and here we shall omit it. Then, letting the parameters in (7) to be p 4 and substituting this polynomial function (7) into (2), one can see the 3D profile of three-lump wave solution in Fig. 3.

Hybrid solutions of the cKP3-equation
Following our strategy for the multi-lump solutions of Eq. (1) in Sect. 3, we start with the N -soliton solution. (6). When k i → 0 (i = 1, . . . , 2M), the N -soliton solution of cKP3-4 equation (1) deduces to the hybrid solution of M-lump, Pbreather, and Q-soliton, where N = 2M + 2P + Q, in which M, P and Q are nonnegative integers and represent the numbers of lump, breather, and soliton, respectively.

One-lump and one-soliton solution
The three-order hybrid solution, which consists of onelump wave and one line soliton, can be obtained from the three solitary waves (6) as N = 3, M = 1, P = 0 and Q = 1. In this case, by catching exp(η 0 1 ) = −1, exp(η 0 2 ) = −1, as k 1 , k 2 → 0, then f 3 will be as follows Fig. 4 show the typical time development of the hybrid solution. The collision of lump and soliton waves with different velocities does not alter their final shape or velocity after interaction, and their key physical properties remain unchanged.
In the hybrid solution, the lump wave moves uniformly along a straight line parallel to x = −p 1 p 2 (bp 1 + bp 2 + 3a)t + c 1 , Under the constraints of (13), Eq. (12) is transformed into with will not change over time [28]. This means the direction of motion trajectory of the lump wave is parallel to that of the soliton wave. Further, the molecule structure is formed when the velocities of lump and soliton Fig. 4 The interaction between one lump and one soliton propagating in the opposite direction with parameters a = 1, b = 1, In this new bound state, the direction and magnitude of velocity of the lump wave are exactly the same as those of the soliton wave. In Fig. 5, the lump and soliton waves keep their shapes, amplitudes and velocities unchanging during the propagation.

One-lump and two-soliton solution
Applying the same principle as before, for brevity we omit some of the details, setting exp(η 0 1 ) = −1, exp(η 0 2 ) = −1, then taking k 1 , k 2 → 0, the f 4 can be rewritten as The hybrid solution of the cKP3-4 equation between one lump and two solitons is given by substituting f , Eq. (15), into Eq. (2). In Fig. 6, we plot the hybrid When t increases, one-lump wave passes the intersection of the crossed two solitary waves. As time goes by, they start to separate and keep their shapes and velocities unchanging, which reflects the collision is elastic.
In view of the velocity resonance condition As t goes on, the lump moves toward the breather. During the interaction, the lump merges with breather and then separates. After the interaction, the amplitudes of the lump and breather remain unchanged. Therefore, their interaction is elastic.
It is clear that molecule between one lump and one breather is possible provided the resonance condition v lump = v br eather and constraints between parameters In lump-breather molecule shown in Fig. 8, both of lump and breather travel at exactly the same speed and remain relatively stationary. During the propagation, the lump-breather molecule changes its position along the x y-plane while keeps its shape and amplitude unchanged.

Two-lump and one-soliton solution
Indeed, taking exp(η 0 i ) = −1, i = 1..4, and acting long wave limit on five-soliton solution as the previous section, we have where Substituting Eqs. (16)-(17) into the variable transformation (2), the hybrid solution presents two lumps and one soliton. In Fig. 9, the dynamical behavior of the interaction between two lumps and a solitary wave is drawn for a particular choice of the parameters a = 1, b = 1, k 5 = 1, As we can see in the figures, two lumps and one soliton move in different directions and after the collision, no changes have happened to the shapes, velocities and energies of two lumps and one soliton.

One-lump and three-soliton solution
In this subsection, we report the hybrid solution consisting of one-lump and three-soliton waves with exp(η 0 1 ) = −1, exp(η 0 2 ) = −1 in f 5 (6). Then taking k 1 , k 2 → 0, a typical one-lump and three-soliton solution is shown in Fig. 10. The heights of the waves do not change before and after the collision. The lump and solitons move freely without loss of their energy.  We are now in a position to derive a hybrid solution consisting of one lump, one breather and one soliton for the cKP3-4 equation by constraining result of the preceding subsection. Imposing more constraints on solution of one lump and three solitons as k 4 = k * 3 , p 4 = p * 3 , η 0 4 = η 0 * 3 leads to a hybrid solution associated with one lump, one breather and one soliton. In Fig.  11, we plot the interaction between these waves for Once again, we see these waves moving from the mutual interaction with their profiles unchanged.
A lump-breather-soliton molecule is constructed by one-lump, one-breather and one-soliton restricted by the velocity resonant condition v lump = v br eather = v soliton and one constraint α 5 = 0 between arbitrary parameters. When velocities and trajectories of lump, breather and soliton are perfectly synchronized, lump-breather-soliton molecule can be formed for copropagative waves, see Fig. 12.

Conclusions
In application, the phenomena of shifts, transitions and interactions have often been visualized between solitons, lumps, breathers and other nonlinear localized waves. In this study, we utilize bilinear method through defining appropriate transformation to calculate the results of multiple soliton solutions up to lumps, breathers and molecules by a limiting procedure and velocity resonance in the integrable cKP3-4 equation.
The physical characteristics of the solutions were graphically depicted to shed more light on the obtained results under the choice of suitable values of the parameters, that builds this article more effective and aggressive. It is shown that the collision does not destroy the structure of the waves. Because of existence of different random parameters, the attained solutions are more significant and supportive to explain physical nature. The attained solutions also show that the proposed method is very reliable, aggressive and simple, and so, the recommended idea could be extended for further nonlinear models in mathematical physics. Data availability All data generated or analyzed during this study are included in this published article.