On some new soliton solutions of (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation using two different methods

Abstract In this article, a new solution for the dimensions Boiti–Leon–Manna–Pempinelli (BLMP) equation using the sine Gordon expansion method and the extended tanh function method is given. The methods were chosen very carefully due to the precision of their solutions. Also, some of the two- and three-dimensional figures and some of the contours plots of the obtained solutions were presented. Finally, a discussion of the results was given.


The extended tanh function method
We consider a partial differential equation: Pðu, u t , u x , u y , u z , u xt , u xx , u yt , u yy , :::Þ ¼ 0: (3) Step 1. Introduce the wave transformation: where a, b, d are constants and c is the velocity of the traveling wave. By using Eq.
(2), we get an ordinary differential equation of the form: FðU 0 , U 00 , U 000 , :::Þ ¼ 0: Step 2. The modified extended tanh method present the wave solution of Eq. (5) in the form of the finite series: where / ¼ /ðnÞ is a solution of the Riccati equation of the form The Riccati Eq. (7) has the general solutions: If w > 0 then Step 3. Obtaining N by balancing the highest order derivative term with the highest power nonlinear term in Eq. (5). Substituting Eqs. (7) and (6) into (5) and then set the coefficients of /ðnÞ i , we get a system of algebraic equations for w, a 0 , :::, a N , b 1 , :::, b N and we solve this system to find all constants.

Sine Gordon expansion method
Consider the sine-Gordon equation where u ¼ uðx, tÞ and m is a constant. Use the wave transformation uðx, tÞ ¼ UðnÞ, n ¼ xÀct in Eq. (11), we get the nonlinear ordinary differential equation: where U ¼ UðnÞ, n and c are the amplitude and velocity of the traveling waves. By integrating once and put the constant of integral equal to zero, we get: let wðnÞ ¼ U 2 and a 2 ¼ m 2 ð1Àc 2 Þ , so Eq. (13) becomes: Set a ¼ 1 in Eq. (14), we get: Solving Eq. (15), we obtain the two significant equations as: where p is the integral constant and non-zero.
We consider the solution of Eq. (5), which can be expressed as: Using Eqs. (16) and (17), we get: We applied the balance principle to determine the value of N as we did in the previous method. We put the summation of coefficients of sin i ðwÞ cos i ðwÞ with the same power equal to zero, we get a system of algebraic equations, which can be solved using Mathematica program.

Solution with the extended tanh function method
For the value N ¼ 1, the solution of (21) can be writen in the form: Substituting Eq. (22) into (21) and using Eq. (7), collecting the coefficients of / i ðnÞ, we obtain the following system: Put these coefficients equal to zero, and solving the system by the aid of Mathematica with Eqs. (22) and (4) we get more than one solution of Eq. (2) as follows: Case 1: Case 2: Case 3: Case 4: UðnÞ ¼ a 0 þ a 1 À1 n Àb 1 ðnÞ uðx, y, z, tÞ ¼ a 0 þ cðax þ by þ dzÀctÞ a 2 : (31)

Solution with sine Gordon expansion method
For the value N ¼ 1, Eq. (19) takes the form, substituting from Eq. (32) into the ordinary differential equation [Eq. (21)], we equate to zero the coefficients of the same power of the trigonometric functions, so we get the following algebraic system of equations: (33) By solving above system by the aid of Mathematica with Eqs. (32), (18) and (4), we get more than one solution of Eq. (2) as follows: Case 1: Case 2: Case 4: Case 5: uðx, y, z, tÞ ¼ A 0 À3ð2Þ Case 6: Case 7:

Conclusion
A new solution for the ð3 þ 1Þ dimensions of BLMP equation by using the sine Gordon expansion method and the extended tanh function method have been presented. These methods have been chosen very carefully due to their accuracy and ease of application. The accuracy has tested from presented some figures in two-and three dimensions and some of the contours for solutions that we have obtained. In the end, we can say that we have made a clear contribution to finding solutions to the proposed equation, and these solutions are satisfactory. Note that the solutions we obtained have soliton waves characteristics, and this is evident in Figures  1-8 in that it keeps its shape over time.