New extensions of (2+1)-dimensional BLMP models with soliton solutions

Searching for soliton solutions of nonlinear partial differential equations is one of the most interesting and important areas of science in the field of nonlinear phenomena. Soliton is a localized wave with exponential wings or is a localized wave with an infinite support. In this work, we study two extensions of (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Based on the simplified Hirota’s method and the Cole-Hopf transformation method, new multiple front wave solutions are obtained for both versions.


Introduction
Soliton is a nonlinear wave with two important properties (Munteanu 2004). First, soliton is a localized wave which propagates without change of its shape, velocity, etc. Second, soliton is a stable localized wave against mutual collisions which retains its identities. From the second property, we conclude that soliton has a property of a particle.
Finding multiple-soliton solutions of nonlinear partial differential equations (NPDEs) plays a very important role in treating nonlinear phenomena. Plasma physics, nonlinear optics, optical fibers, biology, fluid state physics, quantum physics, hydrodynamics, etc, just show some different fields of nonlinear sciences which solitons have important significant.
Hirota's bilinear method has a powerful procedure which also produces multiple soliton solutions (Hereman and Nuseir 1997). In this paper, the following BLMP equation, which was derived by Boiti et al. (1986), is investigated Boiti et al. obtained this equation during their research on KdV equation through weak Lax pairs relations (Boiti et al. 1986). This equation describes the (2+1)-dimensional interaction of the Riemann wave propagated along the y -axis with a long wave propagated along the x-axis (Luo 2010(Luo , 2011. Based on the binary Bell polynomials, the bilinear form of the BLMP equation is found in Luo (2011). For y = x , and by integrating the resulting equation in (1), the equation is reduced to the KdV equation (Gilson et al. 1993). Besides, some stair and step soliton solutions are obtained by Darvishi et al. (2012) for (1). Solutions of (2+1)-dimensional BLMP equation using Wronskian techniques are obtained in Najafi et al. (2013). For this equation, the variable separable solutions obtained in Ma and Fang (2009). New solutions were obtained using the extended homoclinic testing (Tang and Zai 2015).
(2) (u yt + u xxxy − 3u xy u x − 3u xx u y ) x = − 2 u yyy , where the BLMP equation parameter 2 is a real constant. Also, we introduce a new extension of BLMP equation as where 2 is a real constant. We will obtain solitary wave solutions for both Eqs. (2) and (3). To do this, the HM and the CHT are used.
The remaining part of the paper is further arranged as follows: In Sect. 2, we introduce the algorithm of the Hirota's direct method. In Sects. 3 and 4, this method is applied to find the soliton solutions. Finally, a conclusion is reached in Sect. 5.

Analysis of the method
Hirota's direct method can be used to obtain N-soliton solutions for completely integrable equations (Hereman and Nuseir 1997;Zhaung 1980, 1991). Further, soliton solutions are just polynomials of exponentials which is proved by many researchers and this will be also verified in this paper. In order to solve a nonlinear PDE, we first substitute into the linear terms of any equation under discussion to determine the relation between the parameters k and c. After that, we substitute the following CHT in (4) where to determine R in (5), we define an auxiliary function f. Form of this auxiliary function depends on the solution which we look for. For example, the single soliton solution has the following form: Next, we must apply the HM. The steps of HM which has summarized in Hereman and Nuseir (1997); Hereman and Zhaung (1991) are as follows: (i) To find the relation between k i s and C i s, we use (ii) For single soliton, our auxiliary function is to determine R in (5). (iii) Similarly for two-soliton solutions the auxiliary function is: which we can determine the phase shift coefficient a 12 . This way can be generalized for finding phase shift coefficients a ij , 1 ≤ i < j ≤ 3.

The first extended BLMP equation
In this part, we investigate explicit formulas of soliton solutions for the first extension of (2+1)-dimensional BLMP equation as

Multiple soliton solutions
We follow steps (i)-(iv) of Hirota's method for Eq. (11). We first consider case C 1 = C 2 = C 3 = 1 . For this case we have into the linear terms of (11) yields hence i becomes To determine R, we substitute into (11) where by this we obtain R = −2.

Fig. 1 shows single-kink soliton solution (16) for some special values of its parameters.
For two-kink soliton solutions, we set the auxiliary function into (11), where 1 and 2 are given in (14). This results in This can be generalized for the phase shifts by .

Multiple singular soliton solutions
For C 1 = C 2 = C 3 = −1 , the multiple singular-kink solutions will be determined. In what follows we set only the results and the auxiliary functions that we used, because the procedure is similar to the procedure of previous parts. Auxiliary function for single singular-kink solutions is to find that Figure 4 shows single singular-kink soliton solutions (21) for some special values of its parameters.
Similarly setting the auxiliary function

The second extended BLMP equation
In this section, explicit formulas of soliton solutions for the following second extension of (2+1)-dimensional BLMP equation (22) (u yt + u xxxy − 3u xy u x − 3u xx u y ) x = − 2 u yyy − 2 u xxx .

Multiple soliton solutions
To follow steps (i)-(iv) of Hirota's method, we first consider C 1 = C 2 = C 3 = 1 . By into the linear terms of (22) gives From (24) the following values are obtained for i To determine R, we substitute into (22) with that by solving the resulting equation we obtain R = −2.
Page 11 of 16 568 or equivalently and proceed as before to find that For the three-kink soliton solutions, we substitute the last result for f(x, y, t) into (26). The higher level soliton solutions, for N ≥ 4 can be obtained in a similar manner. This confirms the fact that the (2+1)-dimensional BLMP Eq. (22) is completely integrable and gives rise to have multiple-soliton solutions of any order. Fig. 7 shows three-kink soliton solutions related to Eq. (31) for some special values of its parameters.

Multiple singular soliton solutions
Case C 1 = C 2 = C 3 = −1 determines the multiple singular-kink soliton solutions. In this case, the auxiliary function of single singular-kink soliton solutions is which finds that Figure 8 shows single singular-kink soliton solution of Eq. (31) for some special values of its parameters.

Conclusion
In this paper, the HM and the Cole-Hopf method are used to carry out the integrability of new extensions of (2+1)-dimensional BLMP equation. Various kinds of solutions are obtained for these extensions. We used the simplified Hirota's method and the CHM to formally derive the multiple soliton solutions for the extended equations. It can be concluded that HM with CHT is a powerful mathematical tool for a wide variety of another extensions of integrable multidimensional equations in different scientific areas. The obtained results in this study are helpful to explain some physical meanings of some nonlinear dynamics models in waves propagation. It has been noticed that this method is algorithmic and computational, and easy to implement.
Author Contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding
The authors have not disclosed any funding.

Data Availibility Statement
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of interest
The authors declare that they have no competing interests.

Consent to participate Not applicable.
Consent for publication Not applicable. b 123 = −a 12 a 13 a 23 .