Breather wave, resonant multi-soliton and M-breather wave solutions for a (3 + 1)-dimensional nonlinear evolution equation

: In this paper, a (3 + 1)-dimensional nonlinear evolution equation is considered. First, its bilinear formalism is derived by introducing dependent variable transformation. Then, its breather wave solutions are obtained by employing the extend homoclinic test method and related ﬁgures are presented to illustrate the dynamical features of these obtained solutions. Next, its resonant multi-soliton solutions are obtained by using the linear superposition principle. Meanwhile, 3D proﬁles and contour plots are presented to exhibit the process of wave motion. Finally, M-breather wave solutions such as one-breather, two-breather, three-breather and hybrid solutions between breathers and solitons are constructed by applying the complex conjugate method to multi-soliton solutions. Furthermore, their evolutions are shown graphically by choosing suitable parameters.


Introduction
It is known that Hirota bilinear method [1] is a direct and effective method to solve a large number of nonlinear evolution equations, which can depict physical phenomenon in nonlinear science. Recently, Professor Ma considered the Hirota conditions of a (2+ 1)-dimensional combined equation and derived its the N-soliton solution in [2]. Furthermore, Ma presented N-soliton solutions of the combined pKP-BKP equation in [3] and soliton solutions to the B-type Kadomtsev-Petviashvili equation under general dispersion relations in [4] by using Hirota bilinear method. In recent literature, many types of exact solutions have been obtained by employing the Hirota bilinear method, such as soliton solutions [5][6][7], breather solutions [8,9], lump wave solutions [10], interaction solutions [11,12], rogue wave solutions [13,14], resonant multi-soliton solutions [15][16][17], bifurction solitons [18,19], bright and dark soliton solutions [20,21] and so on.
In 2009, Dai proposed the homoclinic test approach and extended homoclinic test approach to search solitary-wave solution of high dimensional nonlinear equation in [22]. Then Xu proposed the homoclinic breather limit method for searching rogue wave solution to nonlinear evolution equation in [23]. In 2011, Ma constructed the resonant multi-soliton solutions of nonlinear equations by introducing introducing the linear superposition principle [24]. Afterwards the linear superposition principle was used to establish the resonant multiple wave solutions of a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation by Lin in [25]. By comparing the linear superposition principle and the velocity resonant condition, Kuo [26,27] concluded that the linear superposition principle is more effective, although they could result in the same results. In recent years, the complex conjugate method was applied to N-soliton solutions to construct M-breather solutions and corresponding hybrid solutions in [28][29][30]. As early as 1977, lump wave solutions were first proposed by Manakov et al. [31] to indicate that the type of wave does not decrease in the direction of (x, y)-plane. Later, Satsuma and Ablowitz [32] proposed the long wave limit method for seeking M-lump solutions based on the collision effects of lump and N-soliton. And it is proved to be the most effective method to construct M-lump solutions of nonlinear evolution equationsİn the past few years, a lot of literature on M-lump solutions and hybrid solutions between lump and solitons of nonlinear evolution equations has been appeared such as the (2+1)-dimensional combined pKP-BKP equation [33], the (3+1)dimensional potential-YTSF equation [34], a (2+1)-dimensional generalized KDKK equation [35], the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation [36], (2+1)-dimensional HSI equation [37], (3+1)-dimensional Kadomtsev-Petviashvili equation [38], (3+1)-dimensional gCH-KP equation [39], the (4+1)-dimensional Boiti-LeonManna-Pempinelli equation [40] and so on.
Recently, Zhang etc [41] considered a (3 +1)dimensional nonlinear evolution equation which was written by − 4u xt + u xxxz + 3αu yy + 4u x u xz + 2u xx u z = 0. (1.1) Its M-lump and interactive solutions of Eq (1.1) were given in [41] by using long wave limit method. when α = 1, Eq (1.1) can be reduced to the potential-YTSF equation [34], whose nontravelling wave solutions were discussed by Yan in [42]. Based on the above literature, we aim to consider the breather wave solutions and resonant multi-soliton solutions, M-breather solutions of Eq (1.1). The rest of this paper is organized as follows. In Section 2, the bilinear formalism of the (3 + 1)dimensional equation (1.1) is derived via introducing variable transformation; In Section 3, breather wave solutions are derived by using the extend homoclinic test approach. Meanwhile, a rouge wave solution is derived by using the homoclinic breather limit method ; In Section 4, resonant multi-soliton solutions are obtained by using linear superposition principle; In Section 5, multi-soliton solutions are derived by Hirota bilinear method, then its M-breather solutions and hybrid solutions are constructed based upon the obtained multi-soliton solutions. Finally, related remarks are given. Through simple calculation, we can see that it is impossible to establish the Hirota bilinear formalism of the (3 + 1)-dimensional equation (1.1) directly. By introducing a dependent variable transformation ξ = x + kz in (1.1), the (3 + 1)-dimensional equation (1.1) can be transformed into the following form − 4u ξt + ku ξξξξ + 3αu yy + 6ku ξ u ξξ = 0, where k is a real constant. In what follows, we introduce another potential transformation By substituting (2.2) into (2.1), we can get Integrating the Eq (2.3) with respect to ξ once, we have − 4cq ξt + ckq ξξξξ + 3cαq yy + 3c 2 kq 2 ξξ = 0. (2.4) According to the results between Bell polynomials and bilinear formalism in Ref. [43] and taking c = 1, the above expression (2.4) leads to the following bilinear form of (2.1) with the help of the following transformation relationship
in which δ 2 , p, m 1 , m 2 are arbitrary real numbers with δ 2 < 0. For Case 1, substituting (3.3) along with ξ = x + kz into (3.1), we obtain the following results where (3.5) can be reduced to the following equation According to the variable transformation u = 2ln(ϕ) ξ , we obtain the exact wave solutions for (1.1) as where u 1 and u 2 are two homoclinic breather wave solutions of (1.1). Their dynamical properties are much similar, so we only discuss u 2 for an example. By selecting appropriate parameters, the profile of u 2 is shown in Figure 1, which is of great value in understanding the dynamical behaviors of the breathers (3.7) and (3.8).
For u 2 in Eq (3.8) at p = 0, taking δ 2 = 1, we can obtain the following Taylor expansions Supposing an arbitrary constant u 0 is the equilibrium solution of Eq (1.1), substituting above Taylor expansions into Eq (3.8), then the following rouge wave solution for Eq (1.1) can be obtained (3.10) By choosing the same parameters as in Figure 1, we exhibit the profile of rouge wave solution (3.10) in Figure 2. We can see that there is one peak and one trough in Figure 2(a). For any given time t, when x 2 + y 2 + z 2 → +∞, u 3 tends to the equilibrium solution u 0 . If taking u 0 = 0, when p tends to 0, u 3 is exactly the limit of u 2 . From the mathematical point of view, the rogue wave solution (3.10) is a limit behavior of breather wave solution (3.8). From the physical point of view, we may think perhaps the energy collection and superposition of breather waves in many periods generate a rogue wave.
For Case 2, when δ 2 < 0, (3.5) can be reduced to the following equation ± Ω 2 cos(p 2 (x + m 2 y + kz + Ω 3 t)). (3.11) According to the variable transformation u = 2ln(ϕ) ξ , we obtain the exact wave solutions for (1.1) as , (3.12) where u 4 and u 5 are also two homoclinic breather wave solutions of (1.1). They also have similar dynamical properties , here we only choose u 4 for an example. By selecting proper parameters, we present the profile of u 5 in Figure 3, which will help us better understand the dynamical behaviors of the breather wave solutions (3.12) and (3.13).  Substituting (3.4) along with ξ = x + kz into (3.1), we have the following expression ϕ =exp(−p(x + m 1 y + kz + Φ 1 t)) ± 2 −δ 2 cos(p(x + m 2 y + kz + Φ 2 t) + δ 2 exp(p 1 (ξ + m 1 y + kz + Φ 1 t)), (3.14) where When δ 2 < 0, (3.14) can be reduced to the following equation ϕ = −2 −δ 2 sinh(p(x + m 1 y + kz + Φ 1 t)) ± 2 −δ 2 cos(p(x + m 2 y + kz + Φ 2 t). (3.15) According to the variable transformation u = 2ln(ϕ) ξ , we obtain the breather wave solution for (1.1) as where u 6 and u 7 are another two homoclinic breather wave solutions of (1.1). They also have similar dynamical properties , here we only choose u 7 for an example. By selecting appropriate parameters, the profile of u 7 is presented in Figure 4. By comparing the other two pairs of breather wave solutions, we can better comprehending the dynamical behaviors of the breather wave solutions (3.16) and (3.17).  In what follows, we would like to construct the resonant multi-soliton solution of Eq (1.1). For the bilinear formula Eq (2.5), we choose N-exponential wave function as the following form in which the resonant multi-soliton variables ζ j = h j ξ + l j y + r j t and µ j are nonzero (1 ≤ j ≤ N). On the basis of the properties of differential operator to exponential functions, substituting the expression (4.1) into the bilinear equation (2.5), we have therefore the Eq (4.2) holds if and only if P(h j − h i , l j − l i , r j − r i ) = 0. Hence the resonant multi-soliton condition can be written as By balancing the power of h i , l j , r j , we conjecture the following wave related numbers Substituting (4.4) into (4.2), we can obtain the following solution for bilinear equation (2.5) with the condition c 1 = k α , c 2 = k. Through the transformation u = 2ln(ϕ) ξ and ξ = x + kz, we can derive the resonant multi-soliton solution of the equation (1.1), which can be written as (4.6) Especially, the 2-kink solution is written as . (4.7) In Figure 5
In order to describe their evolution process, appropriate parameters for two-breather are selected with N = 4, k = 1, α = 2, The motion pattern of the wave is shown in Figure 6. By the similar way, we select suitable parameters for three-breather solution with N = 6, k = 2, α = 3, Its fluctuation form is shown in Figure 7.
By analyzing these pictures, several column waves emerge for two-breather while many column waves emerge for three-breather. In Figure 6, when t = −5, it appears three peaks, two of them are taller than the third one. Then their heights are different from each other at t = 0. At t = 5, it appears four or more peaks. As can be seen from Figure 7, when t = −30, it appears many peaks and troughs with different heights. Then their numbers decrease at t = 0. At t = 30, it appears many peaks and troughs, but their height differences are not obvious. They maintain a froward posture for both the two-breather and three-breather waves.  In the next, we would like to construct hybrid solutions based on multi-soliton solutions. Taking N = 3 in (5.5), the function ϕ can be written as the following form ϕ = 1 + exp(η 1 ) + exp(η 2 ) + exp(η 3 ) + exp(η 1 + η 2 + A 12 ) + exp(η 1 + η 3 + A 13 ) + exp(η 2 + η 3 + A 23 ) + exp(η 1 + η 2 + η 3 + A 12 + A 13 + A 23 ), (5.10) which is exactly the three-soliton solution for bilinear equation (2.5). By selecting l 1 = l * 3 = a 1 + b 1 i, m 1 = m * 3 = c 1 + d 1 i in (5.10), we can derive the hybrid solution between one-soliton and onebreather. Similarly, by selecting N = 5 and (5.4), the hybrid solution between one-soliton and two-breather can also be obtained. As their expressions are very long, we omit them here in order to save space.
In order to depict the evolution progress of hybrid solutions between breather and soliton, proper parameters are selected with N = 3, k = 2, α = 4, a 1 = 1 2 , b 1 = 1 3 , c 1 = 1 4 , d 1 = 1 3 at t = −20, 0, 20 to exhibit its wave motions. Meanwhile, proper parameters are selected with N = 5, k = 1, α = 3, a 1 = 1 10 , a 2 = 1 20 , b 1 = 1 2 , b 2 = 1 4 , c 1 = 3 5 , c 2 = 3 10 , d 1 = 4 5 , d 2 = 9 10 , k 5 = 3 2 , l 5 = 3 at t = −5, 0, 10 to depict its fluctuations. As is shown in Figure 8, it appear many breathers at t = −20. When colliding occurs at t=0 between the breather and soliton, we can see that the breather changes a lot at t = 20. It keeps moving forward over time. It can be seen from Figure 9, two parts constitute the hybrid wave solution. It is clear that one part is soliton wave and the other part is two parallel breathers. Both the two parallel breathers maintain their original propagation path. They gradually separate from each other after colliding with solitons.

Conclusions
In order to establish the bilinear formalism of a (3+1)-dimensional equation, a dependent variable transformation was introduced. This bilinear expression plays a key role in this paper. Then we derived three pairs of breather solutions by employing the homoclinic test method. Among these breather solutions, the rouge wave solution can only be derived from a from u 2 . Next we obtained resonant multi-soliton solutions by employing the linear superposition principle. Obviously, the resonant multisoliton solutions which we obtained don't depend on dispersion relation. By choosing N = 2, 4, 6, the physical shape of the waves were shown graphically. In what follows, M-breather solutions including one-breather, two-breather and three-breather were obtained by applying the complex conjugate method to the multi-soliton solutions, and their evolution progress were exhibited by choosing appropriate parameters. Compared their dynamical behaviors, we can see that breather solutions which were derived by using the homoclinic test method are different from the ones which were derived by using the complex conjugate method. Finally, the hybrid solutions between breathers and solitons were constructed and their dynamic properties were exhibited in the form of plotting.Hence, these solutions would amend the existing literature on the exact solutions of nonlinear evolution equation. Furthermore, the method in this paper can be more effectively used in other nonlinear evolution equations.