New exact solution for ( 2 + 1 ) and ( 3 + 1 ) dimensional nonlinear partial differential equations

In this paper, dark soliton solutions have been obtained for the (2+1)-dimensional reactiondiffusion equation, the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili (gBKP) equation and the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation using the solitary wave ansatz. Ansatz approach is utilized to carry out this integration. The constraint relations for each of the equations are given for the existence of dark soliton solutions.


INTRODUCTION
Nonlinear partial differential equations (NLPDEs) that are studied in the fields of physics and applied mathematics there are significant advances and across the globe many aspects of NLPDEs that are studied by various scientists.Some of them are the integrability aspects, symmetry issues, conservation laws.There has been quite thoroughly understood is the integrability aspect of the governing equations of the solitary waves in (2+1)-dimensional as well as in (3+1)-dimensional equations.For a long time, there are many new results related this aspects that are published in several area [1][2][3][4][5][6].More commonly used methods of integrability are the He's variational iteration method, the hirota bilinear method, the homotopy perturbation method, the first integral method, the modified tanh--coth method, the modified sine-cosine method, the ansatz method and many others [7][8][9][10][11][12].Among the methods mentioned above, the ansatz method is one of the most efficient method to determine solutions of NLPDEs [13][14][15][16][17][18][19][20][21][22].
In nonlinear dynamical systems, propagation of nonlinear waves has been a fundamental objects of nature.Nonlinear waves emerge to in a great array of contexts such as, solid state physics, hydrodynamics, nuclear physics, plasmas, nonlinear optics and many other nonlinear phenomena.In this paper, we apply the ansatz method to obtain dark soliton solutions of the following (2+1)-dimensional reaction-diffusion equation, the (3+1)-dimensional gBKP equation and the (3+1)-dimensional BLMP equation.Dark soliton solutions is one of the fastest growing research fields in the context of wave phenomena [26,27].
By equating the highest exponents of tanh 3p+1 τ and tanh p+3 τ terms in Eq (2.7), we The same value of p can yield when the exponents pairs 2p+1 and p+2, 2p-1 and p are equate with each other.Furthermore, set the coefficients of the linearly independent terms to zero.By setting the corresponding coefficients of tanh 3p+1 τ and tanh p+3 τ terms to zero we get Dλp(p+1)(p+2)a³ -3αλ³pa = 0, (2.9) after some calculations we obtain λ = ±a√ which implies that is necessary to have Dα>0.
Setting the coefficient of tanh p-1 τ terms in Eq. (2.12) Thus from (2.12) it is possible to conclude that the topological solitons will exist for γD<0.
In order to construct dark soliton solutions, we use ψ(x,y,z,t) = λtanh p τ, ( where the free parameter λ is given by (3.11) and velocity of the solitons v is given in (3.13).
In (3.13) v is dependent on the other free parameters a, b and c.

THE (3+1)-DIMENSIONAL BOITI-LEON-MANNA-PEMPINELLI EQUATION
For initial data decaying rapidly at infinity, Boiti, Leon, Manna and Pempinelli have developed an inverse scattering scheme to solve the Cauchy problem of a (2+1)-dimensional generalization of the shallow water wave equations: ψyt + ψxxxy -3ψxxψy -3ψxψxy = 0, ( Eq. (4.1), it is also known (2+1)-dimensional BLMP equation has been studied in detail by many scientists.For example, Song-Hua and Jian-Ping obtained various exact solutions which include solitary wave solution and by selecting appropriate functions, they investigated some novel localized excitations such as multi dromion-solitoffs and fractal-solitons.Eq. (4.1) has been considered as a (2+1)-dimensional generalization of the shallow water wave equations in [41].Painlevé analysis, Lax pair and some exact solutions of Eq. (4.1) have been given in [42] and some soliton-like solutions have been obtained through the symbolic-computation-based method in [43].Gao an Tian studied this equation and derived some new soliton-like solutions using the symbolic computation with the generalized tanh method [44].Now, we introduce an extension to Eq.  BLMP equations are obtained by ansatz method.The ansatz method was employed to carry out the integration.The dark soliton solutions also introduced several constraint conditions that must remain valid in order for these solutions to exist.