Soliton and lump-soliton solutions in the Grammian form for the Bogoyavlenskii–Kadomtsev–Petviashvili equation

This paper investigates the Bogoyavlenskii–Kadomtsev–Petviashvili (BKP) equation by using Hirota’s direct method and the Kadomtsev–Petviashvili (KP) hierarchy reduction method. Soliton solutions in the Grammian determinant form for the BKP-II equation are obtained and soliton collisions are shown graphically. Lump-soliton solutions for the BKP-I equation are presented in terms of the Grammian determinants. Various evolution processes of the lump-soliton solutions are demonstrated graphically through the study of three kinds of lump-soliton solutions. The fusion of lumps and kink solitons into kink solitons and the fission of kink solitons into lumps and kink solitons are observed in the interactions of lumps and solitons.

It is easy to see that BKP equation (1.1) can be rewritten in the following form by employing the scaling transformation t → 1 4 t: (u xt + u xxxy + 8u x u xy + 4u xx u y ) x + δu yyy = 0, δ = ±1.
where α is an arbitrary constant, s is an auxiliary independent variable, D x , D y , D s , and D t are the Hirota bilinear operators defined by [6] The aim of the paper is to derive Grammian determinant solutions for the BKP equation by using the KP hierarchy reduction method which was first proposed by the Kyoto school [44]. The plan of this paper is as follows. In Sect.

Soliton solutions for the BKP-II equation in the Grammian determinant form
In this section, we derive Grammian determinant solutions for the BKP-II equation (u xt + u xxxy + 8u x u xy + 4u xx u y ) x + u yyy = 0. (2.1) The main idea is to get solutions for Eq. (2.1) from the Grammian determinant solutions of the KP hierarchy under the reduction of the KP hierarchy. For this purpose, we first characterize the following result on the KP hierarchy [45].

Lemma 1 The bilinear equations
in the KP hierarchy admit determinant solutions where the matrix element m (n) ij satisfies

4)
and ϕ (n) i , ψ (n) j are arbitrary functions satisfying In order to derive solutions for BKP-II equation where p i , q j are arbitrary constants, δ ij is the Kronecker delta notation. It is easy to verify that ϕ (n) i , ψ (n) j , and m (n) i,j satisfy (2.4) and (2.5). By taking bilinear equations (2.2a)-(2.2b) are reduced to the bilinear form (1.5a)-(1.5b) of the BKP-II equation with δ = 1. Therefore, we get the following theorem. and According to Theorem 1, the one-soliton solutions for BKP-II equation (2.1) are given by taking N = 1. In this case, and the one-soliton solutions take the form u = e ξ 1 +η 1 1 + 1 p 1 +q 1 e ξ 1 +η 1 .
The two-soliton solutions can be written down from (1.4) and the above f . Similarly, the three-soliton solutions in the Grammian determinant form can be obtained. Figure 1 presents the interaction of the three solitons for BKP-II equation (2.1). As the figure shows, all the solitons are kink-type and the velocity and amplitude remain unchanged before and after the collision, and therefore the interaction of the three solitons is an elastic interaction. Recently, Rao and his collaborators have proposed an effective method to derive semirational solutions for the third-type Davey-Stewartson equation, the multi-component long-wave-short-wave resonance interaction system, and the Fokas system [26][27][28]. In what follows, by using the method proposed by Rao, we present semi-rational solutions for the BKP-I equation in the following theorem.

2)
where the matrix element is given by p i , p j , c ik , c jl are arbitrary complex constants, γ ij are arbitrary real constants, and

4)
Proof Consider the functions ϕ (n) i , ψ (n) j , and m (n) i,j defined by where A i and B j are differential operators given by the matrix element m (n) ij can be rewritten as By applying the variable transformation s, x 4 = -2it, and taking

Discussion on the interactions between lumps and kink solitons
In this section, we investigate the dynamics of lump-soliton solutions for the BKP-I equation. It is shown that the fascinating phenomena of fusion and fission can be observed in the study of lump-soliton solutions for the equation.

Dynamics of one lump and one soliton
Taking N = 1 and n 1 = 1 in (3.2), we can get the fundamental lump-soliton solutions for the BKP-I equation as 11 is an arbitrary complex constant. When γ 11 = 0, the fundamental lump-soliton solutions reduce to lump solutions, therefore we take γ 11 = 1. For the nonsingularity of u, p 1 is not purely imaginary.
We observe the time evolution of the fundamental lump-soliton solutions (4.1) on the xy plane. From Fig. 2(a) and Fig. 2(b), we can see that when t 0 this solution contains only one kink soliton and a lump appears from the kink soliton gradually. In Fig. 2(c), the lump separates from the kink soliton and then spreads along the opposite direction to the kink soliton. When t 0, this solution contains one kink soliton and one lump. In this case, the fundamental lump-soliton solution shows the fission phenomenon of a lump from a kink soliton. From Fig. 3(a) and Fig. 3(b), we can observe that this solution comprises one kink soliton and one lump when t 0. As time goes on, the lump moves towards the kink soliton and then collides with the kink soliton. After the collision, the lump merges with the kink soliton in Fig. 3(c). When t 0, the fundamental lump-soliton solution comprises only one kink soliton. In this case, solution (4.1) shows the fusion phenomenon of a lump into a kink soliton.

Dynamics of multiple lump-soliton solutions
Taking N > 1, n i = 1, γ ii = 1 in (3.2), the multiple lump-soliton solutions for the BKP-I equation can be constructed. Considering the case of N = 2, the function f is given by where ξ i and ξ i are given by (3.4), p 1 , p 2 , c 11 , c 21 are arbitrary complex constants. Now we observe the dynamical behaviors of the multiple lump-soliton solutions. From Fig. 4, we observe that two lumps emerge from two kink solitons gradually and then separate completely. This multiple lump-soliton solution comprises only two kink solitons when t 0 and evolves into two lumps and two kink solitons when t 0. From Fig. 5, we observe that two lumps move towards two kink solitons and then merge with the two kink solitons after the interaction. This solution comprises two lumps and two kink solitons when t 0 and reduces to two kink solitons when t 0. Therefore, Fig. 4 and Fig. 5 show the fission process of two lumps from two kink solitons and the fusion process of two lumps and two kink solitons respectively. However, from Fig. 6, we observe multiple lump-soliton solution which has different dynamics from those in Fig. 4 and Fig. 5. Obviously, the multiple lump-soliton solution is composed of one lump and two kink solitons all the time and there is no fission or fusion of the lump and the two kink solitons, but the amplitude and the propagation direction of the lump change after the collision. Therefore, the collision is an inelastic collision.

Dynamics of higher-order lump-soliton solutions
Taking N = 1 and n 1 > 1 in (3.1), the higher-order lump-soliton solutions for the BKP-I equation can be derived. To demonstrate the interactions of the higher-order lump-soliton solutions, we set n 1 = 3 to derive third-order lump-soliton solutions as where ξ i and ξ i are given by (3.4), γ 11 is nonzero real constant, c 11 , c 12 , c 13 are arbitrary complex constants.
To illustrate the dynamics of the third-order lump-soliton solutions, the time evolution of the solution is plotted in Fig. 7 and Fig. 8. As Fig. 7 shows, three lumps arise from a kink soliton and then separate from the soliton gradually. As shown in Fig. 8, three lumps come to interact with a kink soliton and then fuse into the kink soliton completely. Therefore, the dynamics of these higher-order lump-soliton solutions is broadly analogous to that of fundamental lump-soliton solutions, which is shown in Fig. 2 and Fig. 3, but more lumps interact with a kink soliton. However, the multiple lump-soliton solutions possess more complex structures and dynamical behaviors than the fundamental lump-soliton solutions and higher-order lumpsoliton solutions. When N = 2, the multiple lump-soliton solutions not only exhibit the fission of two lumps from two kink solitons and the fusion of two lumps into two kink solitons, but also the inelastic collision of one lump and two kink solitons. It should be noted that the interaction scenarios of nonlinear waves for NLEEs are diverse [3,4,[15][16][17][26][27][28][29][30][31]. In practice, the fusion and fission phenomena have been explored in some physical fields such as hydrodynamics, nuclear physics, and plasma physics [46,47]. It is hoped that the obtained results will enrich the applications of NLEEs in nonlinear scientific fields.