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Exact soliton-like solutions for the (3+1) dimensions Kadomtsev-Petviashvilli and Jimbo-Miwa equations by the extended tan and tanh methods.

1. Introduction

Nonlinear wave and evolution equations play an important role in many branches of science such as fluid dynamics as in [1], plasma and solid state physics as in [2], etc. Several methods were used to solve a variety of nonlinear wave and evolution equations, such as, the inverse scattering transform method as in [3], the tanh method as in [4]- [6], the extended tanh method as in [7]- [9], the Sine-Cosine method as in [10]. Our interest is to implement the extended tan and tanh methods to get exact soliton-like solutions for the (3+1) dimensions (KP) and (JM) equations. Our obtained solutions for the (3+1) dimensions (JM) equations are different from those obtained in [11] for the same equation but by using the further extended tanh method.

2. The Extended tanh Method

The main steps in these two methods are given in [8]- [10]: We are seeking a traveling wave solution u(x,y,z,t) to a given PDE in the form:

F (u,[u.sub.t],[u.sub.x],[u.sub.y],[u.sub.z],[u.sub.xx],[u.sub.yy],[u.sub.zz],...) = 0. (2.1)

If we assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

Then (2.1) will be transformed to an ordinary differential equation (ODE) in the form:

G(U, U',U",...) = 0. (2.3)

Where

U([xi]) = U(x,y,z,t). (2.4)

And the derivatives appear in (2.3) are with respect to [xi]. For the extended tanh method we introduce the dependent variable:

Y([xi]) = tanh([xi]). (2.5)

Then the derivatives appear in (2.3) become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

The extended tanh method admits solutions of the form

U([xi]) = U(x, y, z, t) = S(Y) = [n.summation over (i=0)][a.sub.i][(Y([xi])).sup.i] + [n.summation over (i=1)][b.sub.i] [(Y([xi])).sup.-i]. (2.7)

While the tanh method admits solutions of the same form in (2.7) but with [b.sub.i] = 0. In (2.7) n is an unknown positive integer, the determination of n is resulted from the obtained equation by balancing the linear and nonlinear terms of the highest (or lowest) orders. After determining n we substitute (2.7) in (2.3), the result is an equation contains algebraic coefficients of some powers of Y([xi]), then we set these algebraic coefficients with zero, and solve these equations to obtain ai and bi, after substituting these solutions for ai and bi in (2.7), we obtain exact solutions U(x,y,z, t) for the given PDE (2.1).

3. The Extended tan Method

This method is very similar to the extended tanh method. But, here we introduce the dependent variable

Y([xi]) = tan([xi]), (3.1)

instead of the assumption in (2.5), then the derivatives appear in (2.3) become:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

The extended tan method then admits solutions of the form given in (2.7), then we proceed as we did in the extended tanh method.

4. The Kadomtsev-Petviashvilli (KP) Equation

The (KP) equation is an example of a nonlinear dispersive weakly wave propagate on the surface of a fluid, it has many applications in many branches of science such as fluid dynamics and plasma physics. If we consider u(x,y,z,t) as the packet of a moving wave on the surface of a fluid, where the direction of propagation is the x-direction, (y z) are the transverse variables, and t represents the time, then the (KP) equation is

[u.sub.x,t] + 6[U.sub.x.sup.2] + 6u[u.sub.x,x] - [u.sub.x,x,x,x] - [u.sub.y,y] - [u.sub.z,z] = 0. (4.1)

Many solutions for (4.1) were found, such as soliton-like solutions as in [12]. Even though our solutions for (4.1) are familiar to the researchers, but to the best of our knowledge, the obtained solutions here are not written yet. To apply the above extended tanh method, ifwe start by the assumption in (2.2), then (4.1) becomes:

6[c.sub.1.sup.2][u.sup.2.sub.[xi]] - ([c.sub.2.sup.2] + [c.sup.2.sub.3] + [c.sub.1](-6[c.sub.1]u + [lambda]))[u.sub.[xi],[xi]] - [c.sub.1.sup.4] [u.sub.[xi],[xi],[xi],[xi]] = 0. (4.2)

Equation (4.2) is a 4th order ODE, if we assume that its solution has the form in (2.7), then after using (2.6) we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

The highest powers of the nonlinear terms in (4.3) are 2n + 2 and n + 4, by making them equal we get, n = 2, so (2.7) becomes:

S(Y) = [a.sub.0] + [a.sub.1]Y([xi]) + [a.sub.2]Y[([xi]).sup.2] + [b.sub.1]/Y([xi]) + [b.sub.2]/Y[([xi]).sup.2]. (4.4)

Substituting (4.4) in (4.3), making the coefficients of the powers of Y([xi]) equal zero, we get the following system of algebraic equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.5)

If we solve system (4.5), we get the following sets of solutions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.8)

Where {[a.sub.0],[c.sub.1],[c.sub.2],[c.sub.3]}are arbitrary nonzero real constants. Substituting these sets of solutions (4.6)-(4.8) in (2.7) and by using (2.5), we get the following solutions for (4.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.11)

In a similar way, if we apply the extended tan method, we get the following sets of solutions for (4.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.14)

5. The Jimbo-Miwa(JM) Equation

The (JM) equation is a member in the (KP) hierarchy, and is given by:

[u.sub.x,x,x,y] + 3[u.sub.x,y][u.sub.x] + 3[u.sub.y][u.sub.x,x] + 2[u.sub.y,t] - 3[u.sub.x,z] = 0, (5.1)

where U(x,y,z,t)is the packet of a wave propagates in the x-direction, (y, z) are the transverse variables, and t represents the time. Many methods were used to solve the (JM) equation, such as, the extended homogeneous balance method as in [13], the tanh-function method as in [14], the tanh-coth method as in [15], the exponential-function method as in [16]. Here, we use the extended tan and tanh methods to write two sets of solutions, even though those sets of solutions are familiar to researchers, but to the best of our knowledge, they are not written in the literature yet. First we will use the extended tanh method described above. If we use the assumption in (2.2), then (5.1) becomes:

[c.sub.1.sup.3][c.sub.2][u.sub.[xi],[xi],[xi],[xi]] + 6[c.sup.1.sup.2][c.sub.2][u.sub.[xi],[xi]][u.sub.[xi]] - (2 + [c.sub.2] [lambda] 3[c.sub.1][c.sub.3])[u.sub.[xi],[xi]]. (5.2)

Equation (5.2) is a 4th order ODE, if we assume that its solution has the form in (2.7), then after using (2.6) we get:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.3)

The highest powers of the nonlinear terms in (5.3) are 2n + 3 and n + 4, by making them equal we get, n = 1, so (2.7) becomes:

U([xi]) = U(x, y, z, t) = S(Y) = [a.sub.0] + [a.sub.1]Y([xi]) [b.sub.1]+ Y([xi]). (5.4)

Substituting (5.4) in (5.3), making the coefficients of the appeared powers of Y([xi]) equal zero, we get the following system of algebraic equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.5)

If we solve system (5.5), we get the following 3 sets of solutions:

Set 1: [a.sub.1] = [b.sub.1] = 2[c.sub.1],[c.sub.3] = 2(8[c.sub.1.sup.3][c.sub.2] - [c.sub.2][lambda])/3[c.sub.1], [c.sub.1] [not equal to] 0. (5.6)

Set 2: [a.sub.1] = 2[c.sub.1],[b.sub.1] = [b.sub.1] = 2(2[c.sub.1.sup.3][c.sub.2] - [c.sub.2][lambda])/3[c.sub.1], [c.sub.1] [not equal to] 0. (5.7)

Set 3: [a.sub.1] = 0[b.sub.1] = 2[c.sub.1][c.sub.3] = 2(2[c.sub.1.sup.3][c.sub.2] - [c.sub.2][lambda])/3[c.sub.1], [c.sub.1] [not equal to] 0. (5.8)

Where {[a.sub.0],[c.sub.1],[c.sub.2], [lambda]} are arbitrary nonzero real constants. Substituting the sets of solutions (5.6)-(5.8) in (5.4) and by using (2.5), we get the following solutions for (5.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.11)

In a similar way, if we apply the extended tan method, we get the following solutions for (5.1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.12)

[u.sub.5](x,y,z,t) = [a.sub.0] + 2[c.sub.1] tan ([c.sub.1]x + [c.sub.2]y + (2[c.sub.2](2[c.sub.1.sup.3] - [lambda]/3[c.sub.1])z - [lambda]t) (5.13)

[u.sub.6](x,y,z,t) = [a.sub.0] + 2[c.sub.1] cot ([c.sub.1]x + [c.sub.2]y + (2[c.sub.2](2[c.sub.1.sup.3] - [lambda]/3[c.sub.1])z - [lambda]t) (5.14)

6. Conclusion

We were able to prove the efficiency of the extended tan and tanh methods, in finding exact soliton like solutions for the (3+1) dimensions Kadomtsev-Petviashvilli (KP) and Jimbo-Miwa (JM) nonlinear evolution equations. These two powerful methods were also used by many other researchers to find exact solutions for many nonlinear evolution equations, some of them were mentioned in the references.

References

[1] Whitham, G. B. , 1974, Linear and Nonlinear Waves, John Wiley, New York.

[2] W. Malfliet, 1992, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60:650-654.

[3] Ablowitz, M. J., Segur, H., 1981 Solitons and the Inverse Scattering Transform, SIAM Stud. Appl. Math., 4, SIAM, Philadelphia.

[4] Malfliet, W., Hereman, W., 1996, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta., 54:563-568.

[5] Abdul-Majid Wazwaz, 2004, The tanh method for traveling wave solutions of nonlinear equations, Applied Math. and Computation, 154(3):713-723.

[6] Wazwaz, A.M., 2005, The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations. Chaos, Solitons and Fractals, 25(1):55-63.

[7] E. Yusufoglu, A. Bekir, 2007, On the Extended Tanh Method Applications of Nonlinear Equations, Int. J. of Nonlinear Science, 4(1):10-16.

[8] Wazwaz, A.M., 2007, The extended tanh method for new soliton solutions for many forms of the fifth order Kdv equations. Applied Mathematics and Computation, 184(2):1002-1014.

[9] Hassan, A. Zedan, 2009, New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations, Comput. Appl. Math., 28(1), Sao Carlos.

[10] Wazwaz, A. M., 2004, A sine-cosine method for handling nonlinear wave equations. Math. and Comput. Modeling, 40:499-508.

[11] Zhuosheng Lu, Hongqing Zhang, 2003, On a further extended tanh method, Physics Letters A, 307(5-6):269-273.

[12] Xu, G., 2006, The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in (3+1)-dimensions, Chaos Solitons and Fractals, 30(1):71-76.

[13] Xi-Qiang Liu, Song Jiang, 2004, New solutions of the 3+1 dimensional JimboMiwa equation, Applied Math. and Computation, 158(1):177-184.

[14] Parkes, E., Duffy, B., 1996, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys Commun., 98:288300.

[15] Abdul-Majid Wazwaz, 2008, New solutions of distinct physical structures to high-dimensional nonlinear evolution equations, Applied Mathematics and Computation, 196(1):363-370.

[16] Ji-Huan He, Xu-Hong Wu, 2006, Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 30(3):700-708.

Adeeb G. Talafha

Al-Hussein Bin Talal University, Department of Mathematics and Statistics, Ma'an Jordan.

E-mail: agtalafh@mtu.edu
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Author:Talafha, Adeeb G.
Publication:Global Journal of Pure and Applied Mathematics
Article Type:Report
Geographic Code:7JORD
Date:Dec 1, 2010
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