Abundant ﬁssion and fusion solutions in the ( 2 + 1 )-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

Fission and fusion are important phenomena, which have been observed experimentally in many physical areas. In this paper, we study the above two phenomena in the (2 + 1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation. By introducing some new constraint conditions to its N - solitons, the ﬁssion and fusion are obtained. Numerical ﬁgures show that the two types of solutions look like the capital letter Y in spatial structures. Then, by taking a long wave limit approach and complex conjugation restrictions, some hybrid resonance solutions are generated, such as the interaction solutions between the L - order lumps and Q -ﬁssion (fusion) solitons, as well as hybrid solutions mixed by the T -order breathers and Q - ﬁssion (fusion) solitons. Dynamical behaviors of these solutions are analyzed theoretically and numerically. The results obtained can be helpful for understand-ing the fusion and ﬁssion phenomena in many physical models, such as the organic membrane, macromolecule material and even-clump DNA, plasmas physics, and so on.

Usually, the interactions between soliton solutions derived by the above methods for nonlinear soliton equations are regarded to be completely elastic because the velocity, shape and amplitude of solitons keep unchanged after the interactions. However, for some nonlinear soliton models, completely inelastic interactions may occur when the wave vectors and velocities of the solitons satisfy some special conditions. For example, at certain time, one soliton may fission to two or more solitons. Contrarily, at some time, two or more solitons may fusion to one. In the terminology of soliton theory, the above two phenomena are termed soliton fission and soliton fusion, respectively. Interestingly, investigations show that the fission and fusion phenomena have found their applications in many models, such as in organic membrane and macromolecule material [32], in even-clump DNA [33], in Sr-Ba-Ni oxidation crystal and waveguide [34], in nuclear physics, and so on [35]. Therefore, it motivates many scholars to seek fission and fusion solutions in nonlinear differential equations. However, in most work, the authors obtained the fission and fusion solutions by taking a logarithmic transformation with f in a form of [36][37][38]: e k j x+ p j y+w j t+φ j .
(1.1) What needs to be pointed out is that because of the specialty of the approach, it loses the power to construct the interaction solutions between fission/fusion and other types of waves. Subsequently, Chen et al. considered a general transformation via: (1.2) and thereby the interaction solution between a lump and (N − 2)-fissionable wave of the Sawada-Kotera equation was obtained [39]. It is a pity that the particular form of the expression (1.2) makes many terms lost compared with the classical N -soliton solution, which lead to the interaction solutions between fusion and fission waves unobtainable. Recently, Li et al. [40] introduced a typical relation to the parameters involved in the N -solitons, and they obtained the fusion, fission and some hybrid solutions of the fifth-order KdV system. Inspired by the work mentioned in [36][37][38][39][40], here we would like to seek fission and fusion solutions in the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff (gCBS) equation: (1.3) which was introduced by Bogoyavlenskii [41] and Schiff [42] in two different ways and constitutes a generalization of the (2 + 1)-dimensional CBS equation derived from the Korteweg-de Vires equation [43]. Due to the importance and applications of the CBS and gCBS equations, many experts have paid attention to the constructions of their exact solutions [44][45][46][47][48].
Unlike the above work, our plan here is to search the fission, fusion solutions as well as their interactions with some localized waves via an alternative method. We begin our work with a new constraint associated with the N -solitons. It is shown that under the new constraint, the fission and fusion solutions may be readily constructed. Interestingly, numerical results reveal that the spatial structure of the solutions looks like the capital letter "Y". After that, we introduce the complex conjugation restrictions and long wave limit to the constraint, and then some interaction solutions are generated, including the hybrid solutions consisting of the fission solitons and fusion solitons, of the fission/fusion solitons and T -order breathers, as well as L-order lumps. In order to exhibit the dynamical behaviors that the interaction solutions may possess, we make some theoretical analysis and numerical simulations.

Fission and fusion solutions of the gCBS equation
With the aid of the Hirota's bilinear method, the Nsoliton solution of the (2 + 1)-dimensional gCBS Eq. (1.3) may be readily obtained via where the function f (x, y, t) is given by i, j, s = 1, 2, · · · , N .
In the above, the parameters k i , p i , w i and φ i are arbitrary constants related to the amplitude and phase of the i-soliton, while μ=0,1 means the summation of all possible combinations of μ i , μ j = 0, 1.
In order to construct the fission and fusion solutions of Eq. (1.3), we first give a necessary supplement to the range of ln x: the relation exp(ln x) = 0 holds if and only if x = 0. According to the supplement, one can see that: if all A js = 0, all the terms of exp(ln A js ) are removal and so that the expression (2.2) degenerates to a form of (1.1) given in [36], and if A js = 0 with 3 ≤ j < s ≤ N , then the expression (2.2) reduces to a form of (1.2) established in [39]. In this sense, we conclude that the supplement associated with Eq. (2.2) is more general, which can be used to construct the fission and fusion solutions of the (2 + 1)-dimensional gCBS equation.

Proposition 1
Assume that the parameters A js described in (2.3) satisfy the following constraint:

4)
which is equivalent to

then a kind of fission solutions combined by P-fission and Q-fission solitons can be constructed.
It is necessary to point out that for convenience, we record the solution obtained by Proposition 1 as the fission solution. In fact, the fusion and interaction solution between the fission and fusion are also contained due to the symbols "±" involved in (2.5). Generally, if the symbol "+" corresponds to the fission, then the symbol "-" corresponds to the fusion. In order to distinguish the two physical phenomena, we call the phenomenon fission wherein a soliton is divided into two or more solitons with y changing from minus infinity to positive for a fixed time. In the following, Proposition 1 will be applied to generate the fission and fusion solutions as well as their interaction for the (2 + 1)-dimensional gCBS equation.
Here, we take P = 2 and Q = 2 in Proposition 1; then the resonance solution is described by where the parameters η i and A js are described by (2.3).
Just as we point out in the above, the formula (2.6) represents three different types of solutions when appropriate parameters are chosen. The first type is an interaction solution between two different 2-fission waves, seen in Fig. 1. The second type is an interaction between two different 2-fusion waves, seen in Fig. 2. And the third is a solution mixed by a 2-fission and 2-fusion waves, seen in Fig. 3. Interestingly, it is seen from the  figures that the spatial structures of these solutions look like the capital letter Y . Therefore, the fission and fusion solutions obtained can also be called as resonance Ytype solutions. In order to examine the properties of these interaction solutions to be elastic or inelastic, we take the interaction solution between the fission and fusion waves in Fig. 3 as an example. The illustrative numerical simulations are undertaken in Fig. 4. From this figure, one can see that with time evolution, the amplitudes of each branch of the mixed solution consisting of the fission and fusion keep unchanged for a fixed y value. Therefore, it is concluded that interactions between the fission and fusion waves are elastic. It is noted that if we take one of P and Q to be zero, then we can obtain a pure fission or fusion solution when N = 2, whose dynamical behaviors are shown in Fig. 5a-b. Except that, when N = 3 and appropriate parameters are selected, we can also derive a new kind of solution which fuses first and then splits rapidly, seen in Fig. 5c. To the best of our knowledge, the solution fusing first and then splitting rapidly fast together with the interaction solutions in Figs. 1, 2 and 3 constructed by Proposition 1 has not been discussed before.

Hybrid solutions between the L-order lumps and Q-fission (fusion) solitons
In this section, we shall show that when the constraint condition introduced in Proposition 1 is incorporated with the long wave limit approach, a kind of hybrid solution consisting of the L-order lumps and Q-fission (or fusion) solitons can be obtained.
While the trajectories of the L-order lump waves are governed by the parameters {K 2i−1 , K 2i , P 2i−1 , P 2i−1 }, the central coordinates of the lumps before and after the interactions with the Q-fission (or fusion) waves are described by (3.5) The height of the lump waves of u is given by , (3.6) which keeps unchanged before and after the interactions.
According to Proposition 2, when L = 1, Q = 2 and N = 4, the hybrid solution mixed by a lump and 2-fission (fusion) waves can be written as with (3.9) Here we choose K 1 = K * 2 = 2 5 − 6i 5 , P 1 = P 2 = 1, k 3 = 1 2 , k 4 = 3 4 , p 3 = 1, p 4 = 9 8 , δ 1 = −1 and δ 2 = 1, a special interaction solution between a lump and fission waves is obtained, whose behaviors are depicted in Fig. 6. From the figure, one can see that the lump wave propagates along the trajectory y = 4 5 x − 25033248 2688085 (marked by red in Fig. 6) before it collides with the fission waves at t = 0. After the collision, it changes the trajectory to y = 4 5 x (marked by black in Fig. 6). When we select p 3 = 1 3 , p 4 = 7 Fig. 6 The time evolution behaviors of the interaction solution between a lump and 2-fission solitons of u determined by Eq. interactions of the lump with the fusion wave, its trajectory changes from y = 4 5 x − 406635209568 50586174665 (marked by red in Fig. 7) to y = 4 5 x (marked by black in Fig. 7). By observing Figs. 6 and 7, one can find that, before and after the interactions, the shape, velocity and amplitude of the lump keep unchanged; however, only the trajectory of the lump is shifted, which coincides with the theoretical analysis made in Proposition 2. It is noted that the shift of the trajectory of the lump is a very important phenomenon, which may be used to explain some related phenomena in the ocean.
Making analogous analysis to Proposition 2. On taking L = Q = 2 while N = 6 and setting the parameters to satisfy the following conditions Moreover, if setting L = 1, N = 6, and requiring the parameters to satisfy then we can derive three different types of interaction solutions: (1) a hybrid solution mixed by two different fission waves and a lump, seen in Fig. 10; (2) a hybrid solution combined by two different fusion waves and a lump, seen in Fig. 11; (3) a hybrid solution mixed by a fission, a fusion and a lump, seen in Fig. 12.

Hybrid solutions between the T -order breathers and Q-fission (fusion) solitons
In this section, we shall show that when we combine the complex conjugation restrictions with Proposition 1, the hybrid solutions between the T -order breathers and Q-fission (fusion) solitons can be obtained.
Proposition 3 Assume that the parameters satisfy Proposition 1 and the following complex conjugation conditions:
Similarly, when we take N = 6, T = 2 and set two pairs of parameters to satisfy the complex conjugation conditions via based on the 6-solitons expression given in (2.1), we can derive the interaction solutions between the 2-order breathers and 2-fission (fusion) waves, whose dynamical behaviors are shown in Figs. 14 and 15. Moreover, if we set N = 6, T = 1 and only require a pair of parameters to satisfy the complex conjugation conditions via then the hybrid solution between a 1-order breather, a fission and a fusion can be constructed, whose behaviors are displayed in Fig. 16. It is known that the breather, as a kind of localized periodic wave, has been found applications in optics, biophysics and condensed matter physics [49]. Here, we have shown that the localized periodic wave can interact with the fission (fusion) to form an interaction solution. Based on the important applications of the breather and fission (fusion) in many physical fields, we hope that the interaction solutions obtained can provide experts some new insights for investigations of some related problems.

Remark 1
We would like to point out that the interaction solutions with fission/fusion phenomena we derive in this paper is different to what were given by Hossen et al. in Ref. [50]. Following their method [50], we can also obtain similar results which are shown in Appendix I.

Conclusions
Fission and fusion are very important physical phenomena, which have been observed in many fields, such as organic membrane, biophysics, plasma physics, nuclear physics and life science. In this paper, we investigate the fission and fusion phenomena in the (2 + 1)dimensional generalized CBS equation by introducing a more general constraint to the N -solitons (seen in Proposition 1). Numerical simulations show that such solutions look like the capital letter Y from the spatial structure. In addition, we obtain some interaction solutions, such as the mixed solutions by a lump and fission (fusion) waves, by a lump, a fission and fusion, by 2order lumps and fission (fusion) waves, by a breather and fission (fusion) waves, by 2-order breathers and fission (fusion) waves. Dynamical behaviors of these solutions are discussed numerically and theoretically, which indicate that the selections of the parameters have great impact on the solutions. The method given in the paper is effective, which can be applied to investigate the fission and fusion phenomena in other physical models. However, there still exist many interesting problems that need further considerations. For example, how to use the fusion and fission solutions to explain any physical phenomena in the related areas? Can such type solutions be obtained by other methods, such as the bilinear neural network method? Except the solutions given here, does other kind of interesting solutions exist? Based on the importance and applications of fission and fusion phenomena, all these solutions are deserved deep investigations. 3) and collecting all the coefficients of x, y, t, cosh(ξ 1 ), sinh(ξ 1 ), cos(ξ 2 ) and sin(ξ 2 ), then we get a set of algebraic equations in a j , k j , m j , p, q, δ 1 and δ 2 . After solving these algebraic equations, we get the following relations of the parameters:  −x+δ 1 t))) a 4 1 x 2 +a 2 1 (a 2 4 x 2 +a 2 5 (y−δ 1 t) 2 +a 7 )+a 2 4 a 2 5 (y−δ 1 t) 2 + pa 2 1 cosh(m 1 (−x+δ 1 t))+qa 2 1 cos(k 1 (−x+δ 1 t)) .

(A.4)
It is found that when different conditions are imposed onto the parameters p and q, different interaction solutions can be obtained: 1) On setting p = 0 and q = 0, the solution v in (A.4) exhibits a single lump, which has one valley and one peak (seen in Fig. 17a). However, on setting p = 0 and q = 0, the solution v represents an interaction solution between a lump and a periodic wave, whose behaviors are displayed in Fig. 17b-c. Comparison shows that there are one peak and one valley in Fig.  17b and with q gradually increasing it splits into two peaks and two valleys by fission in Fig. 17c. That's to say, the fission phenomenon occurs in the lump wave.
2) On setting p = 0 and q = 0, the solution v in (A.4) displays an interaction solution wherein the lump get into a double kink waves (seen in Fig. 18a), while when p = 0 and q = 0, the solution v is shown to be an interaction solution among the lump, double kinks and periodic waves. Inspection reveals that one valley and one peak of the lump in Fig. 18b split into two valleys and two peaks in Fig. 18c with q gradually increases.