Meromorphic solutions of the (2 + 1)- and the (3 + 1)-dimensional BLMP equations and the (2 + 1)-dimensional KMN equation


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                  </jats:inline-formula>-dimension Kundu-Mukherjee-Naskar equation are investigated. Abundant new elliptic solutions, rational solutions and exponential solutions have been constructed.</jats:p>

In recent years, high-dimensional NLPDEs have attracted much attention of researchers. From the significance and surprising properties of higher-dimensional differential equations, it is very important to investigate the analytical exact solutions. The starting points of our considerations are the most famous ( + ) 2 1 -dimensional and the ( + ) 3 1 -dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equations which were given as follows: yt xxxy xx y x xy (1) where = ( ) u u x y t , , , and yt zt xxxy xxxz x xy xz xx y z (2) where = ( ) u u x y z t , , , . The ( + ) 2 1 -dimensional and ( + ) 3 1 -dimensional BLMP equations, being mathematical models of the incompressible fluid, which were investigated by employing the extended homoclinic test method, then several periodic solitary wave solutions and kink solutions were constructed in 2015 [18]. The transformed rational function method and the (− ( )) ξ exp Φ methods were used to construct several different analytical solutions in 2017 [19].

By using traveling wave transformation
where k l s τ , , , are constants, equation (1) reduces into the following ordinary differential equation (ODE): Integrating equation (3) with respect to the variable ξ , we have where c is a constant. Now consider equation (2), by using traveling wave transformation where k l m s τ , , , , are constants, equation (2) reduces into the following ODE: Integrating equation (5) with respect to the variable ξ , we have where c is a constant. One of our destinations is to investigate equations (4) and (6) by utilizing the complex method proposed by Yuan and collaborators [20][21][22][23]. The complex method is a systematic and powerful tool for constructing traveling wave exact solutions for some certain partial differential equations. To the best of our knowledge, our research is the first to use the complex method on the BLMP equations and the Kundu-Mukherjee-Naskar (KMN) equation. We must mention here, in this paper, we investigate meromorphic solutions of ODEs on the complex plane .
provided that = + k l g l s ck 12 1 2 3 are arbitrary constants.

The exponential solutions:
are arbitrary constants. 3. The elliptic solutions: provided that = + k g s ck 12 1 2 3 are arbitrary constants.
Then we will study the ( + ) 2 1 -dimensional KMN equation [24][25][26]: This equation depicts the oceanic rogue waves as well as hole waves [24] and optical wave propagation through coherently excited resonant waveguides [25]. The optical soliton solutions of the ( + ) 2 1 -dimensional KMN equation were intensively researched in [24] by the extended trial function method, and the first-order rogue wave solutions were studied by the Darboux transformation [26]. In [24], Ekici et al. applied the following reduction: are constants with respective physical means, and they reduced equation (7) into the ODE: 32400 , 3. The Jacobi elliptic solutions: where + = ω aκ κ 0 1 2 , A A , 4 5 are arbitrary constants.
This paper is prepared as follows. In Section 2, we introduce the preliminary lemmas and the methodology. In Section 3, we present the proof of the theorems. In Section 4, the conclusions will be given.   .
Definition 11. [20] If there are only p differential principle parts can be determined, we call equation (14) satisfies the weak ⟨ ⟩ p q , condition.
Lemma 12. [30,31] Giving the following kth order Briot-Bouquet equation: where P is a polynomial in f and ( ) f k with constant coefficients. If f is a meromorphic solution of equation (17) which with at least a pole, then f belongs to W .
where ∈ c 0 , − c ij come from (15), Any rational solution ( ) f z is of the form: In order to investigate the ( + ) 2 1 -dimensional and the ( + ) 3 1 -dimensional BLMP equations and the ( + ) 2 1dimensional KMN equation, we describe the broad outline of the complex method.
First, under the aforementioned conditions, we suppose that ′ = u v, then equations (4) and (6) reduce to and   The proof of Theorem 2 is similar to that of Theorem 1, it is therefore omitted. Remark 2. From (32), we can assume that Therefore, we have the following elliptic function solutions of equation (8): ( ) = ± ℘′( ) ℘( ) + u ξ aA A bκ ξ g g ξ g g A 1 2 , , , , ,