JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 960798 10.1155/2013/960798 960798 Research Article The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations Zhao Yun-Mei He Ying-Hui Long Yao Biswas Anjan Department of Mathematics Honghe University Mengzi Yunnan 661100 China uoh.edu.cn 2013 1 12 2013 2013 17 09 2013 07 11 2013 07 11 2013 2013 Copyright © 2013 Yun-Mei Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

1. Introduction

Nonlinear phenomena exist in all areas of science and engineering, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, and chemical physics. It is well known that many nonlinear partial differential equations (NLPDEs) are widely used to describe these complex physical phenomena. The exact solution of a differential equation gives information about the construction of complex physical phenomena. Therefore, seeking exact solutions of NLPDEs has long been one of the central themes of perpetual interest in mathematics and physics. With the development of symbolic computation packages, like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as the homogeneous balance method [1, 2], the auxiliary equation method , the sine-cosine method , the Jacobi elliptic function method , the exp-function method , the tanh-function method [7, 8], the Darboux transformation [9, 10], and the (G/G)-expansion method [11, 12].

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [13, 14] and used successfully by many authors for finding exact solutions of ODEs in mathematical physics .

In this paper, we first apply the simplest equation method and the modified simplest equation method to derive the exact solutions of the elliptic-like equation, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained.

2. Description of Methods 2.1. The Simplest Equation Method Step 1.

Suppose that we have a nonlinear partial differential equation (PDE) for u(x,t) in the form (1)N(u,ut,ux,utt,uxt,uxx,)=0, where N is a polynomial in its arguments.

Step 2.

By taking u(x,t)=u(ξ),ξ=x-ct, we look for traveling wave solutions of (1) and transform it to the ordinary differential equation (ODE) (2)N(u,-cu,u,c2u′′,-cu′′,u′′,)=0.

Step 3.

Suppose the solution u of (2) can be expressed as a finite series in the form (3)u=i=0nAi(H(ξ))i, where H(ξ) satisfies the Bernoulli or Riccati equation, n is a positive integer that can be determined by balancing procedure, and Ai(i=0,1,2,,n) are parameters to be determined.

The Bernoulli equation we consider in this paper is (4)H(ξ)=aH(ξ)+bH2(ξ), where a and b are constants. Its solutions can be written as (5)H(ξ)=-aC1b(C1+cosh(a(ξ+ξ0))-sinh(a(ξ+ξ0))),H(ξ)=-a(cosh(a(ξ+ξ0))+sinh(a(ξ+ξ0)))b(C2+cosh(a(ξ+ξ0))+sinh(a(ξ+ξ0))), where C1,C2, and ξ0 are constants.

For the Riccati equation (6)H(ξ)=aH2(ξ)+bH(ξ)+s, where a,b, and s are constants, we will use the solutions (7)H(ξ)=-b2a-θ2atanh[θ2(ξ+ξ0)],H(ξ)=-b2a-θ2atanh(θ2ξ)+sech(θξ/2)Ccosh(θξ/2)-(2a/θ)sinh(θξ/2), where θ2=b2-4as.

Step 4.

Substituting (3) into (2) with (4) (or (6)), then the left hand side of (2) is converted into a polynomial in H(ξ); equating each coefficient of the polynomial to zero yields a set of algebraic equations for Ai,a,b(i=0,1,2,,n). Solving the algebraic equations by symbolic computation, we can determine those parameters explicitly.

Step 5.

Assuming that the constants Ai,a,b(i=0,1,2,,n) can be obtained in Step 4 and substituting the results into (3), then we obtain the exact traveling wave solutions for (1).

2.2. The Modified Simplest Equation Method

In the modified version, one makes an ansatz for the solution u(ξ) as (8)u=i=0nai(ψ(ξ)ψ(ξ))i, where ai(i=0,1,2,,n) are arbitrary constants to be determined, such that an0 and ψ(ξ) is an unspecified function to be determined afterward.

Substitute (8) into (2) and then we account the function ψ(ξ). As a result of this substitution, we get a polynomial of ψ(ξ)/ψ(ξ) and its derivatives. In this polynomial, we equate the coefficients of the same power of ψ-j(ξ) to zero, where j0. This procedure yields a system of equations which can be solved to find ai(i=0,1,2,,n),ψ(ξ), and ψ(ξ). Then the substitution of the values of ai(i=0,1,2,,n),ψ(ξ), and ψ(ξ) into (8) completes the determination of exact solutions of (1).

3. Solutions of the Elliptic-Like Equation

Now, let us choose the following elliptic-like equation (9)Aϕ′′(ξ)+Bϕ(ξ)+Cϕ3(ξ)=0, where A,B, and C are arbitrary constants. Equation (9) is one of the most important auxiliary equations, because many nonlinear evolution equations can be converted to (9) using the travelling wave reduction.

3.1. Using Simplest Equation Method 3.1.1. Solutions of (<xref ref-type="disp-formula" rid="EEq11">9</xref>) Using the Bernoulli Equation as the Simplest Equation

Considering the homogeneous balance between ϕ(ξ), and ϕ3(ξ) we get n=1, so the solution of (9) is the form (10)ϕ(ξ)=A0+A1H(ξ).

Substituting (10) into (9) and making use of the Bernoulli equation (4) and then equating the coefficients of the functions Hi(ξ) to zero, we obtain an algebraic system of equations in terms of Ai(i=0,1),a, and b. Solving this system of algebraic equations, with the aid of Maple, we obtain (11)A0=±iBC,A1=±i2AC,a=2BA,A0=±iBC,A1=i2AC,a=-2BA.

Therefore, using solutions (5) of (4) and ansatz (10), we obtain the following exact solution of (9): (12)ϕ1(ξ)=±iBC(-sinh(2BA(ξ+ξ0)))-11-(2C1)hhhhhhh×(C1+cosh(2BA(ξ+ξ0))hhhhhhhhhh-sinh(2BA(ξ+ξ0)))-1),ϕ2(ξ)=±iBC(-sinh(2BA(ξ+ξ0)))-11-(2C1)hhhhhhh×(C1+cosh(2BA(ξ+ξ0))hhhhhhhhhh+sinh(2BA(ξ+ξ0)))-1),ϕ3(ξ)=±iBC(-sinh(2BA(ξ+ξ0)))-11-(2(cosh(2BA(ξ+ξ0))hhhhhhhhhhhh+sinh(2BA(ξ+ξ0))))hhhhhhh×(C2+cosh(2BA(ξ+ξ0))hhhhhhhhhh+sinh(2BA(ξ+ξ0)))-1),ϕ4(ξ)=±iBC(-sinh(2BA(ξ+ξ0)))-11-(2(cosh(2BA(ξ+ξ0))hhhhhhhhhhhh-sinh(2BA(ξ+ξ0))))hhhhhhh×(C2+cosh(2BA(ξ+ξ0))hhhhhhhhhh-sinh(2BA(ξ+ξ0)))-1).

3.1.2. Solutions of (<xref ref-type="disp-formula" rid="EEq11">9</xref>) Using Riccati Equation as the Simplest Equation

Suppose the solutions of (9) are the form (13)ϕ(ξ)=B0+B1H(ξ).

Substituting (13) into (9) and making use of the Riccati equation (6) and then equating the coefficients of the functions Hi(ξ) to zero, we obtain an algebraic system of equations in terms of Bi(i=0,1),a,b, and s. Solving this system of algebraic equations, with the aid of Maple, one possible set of values of Bi(i=0,1),a,b, and s is (14)B0=±iA2Cb,B1=±i2ACa,s=Ab2-2B4Aa.

Therefore, using solutions (7) of (6) and ansatz (13), we obtain the following exact solution of (9): (15)ϕ5(ξ)=iBCtanh(B2A(ξ+ξ0)),ϕ6(ξ)=iB2C((C2BAsinh(B2Aξ)hhhhhhhhhh-2acosh(B2Aξ))hhhhhhhh×(CBAcosh(B2Aξ)hhhhhhhhhh-a2sinh(B2Aξ))-1).

3.2. Using Modified Simplest Equation Method

Suppose the solution of (9) is the form (16)ϕ(ξ)=a0+a1(ψ(ξ)ψ(ξ)), where a0 and a1 are constants, such that a10, and ϕ(ξ) is an unspecified function to be determined. It is simple to calculate that (17)ϕ=a1(ψ′′ψ-(ψψ)2),ϕ=a1(ψ′′′ψ)-3a1(ψ′′ψψ2)  +2a1(ψψ)3,ϕ3=a13(ψψ)3+3a12a0(ψψ)2+3a1a02(ψψ)+a03.

Substituting the values of ϕ,ϕ, and ϕ3 into (9) and equating the coefficients of ψ0,ψ-1,ψ-2, and ψ-3 to zero yield (18)ψ0:Ca03+Ba0=0,(19)ψ-1:(3Ca02+B)ψ+Aψ′′′=0,(20)ψ-2:-3Aψ′′+3Ca0a1ψ=0,(21)ψ-3:(Ca13+2Aa1)(ψ)3=0.

Solving (18), we obtain (22)a0=0,a0=±iBC.

And solving (21), we obtain (23)a1=±i2AC,sincea10.

Case 1.

When a0=0, we obtain trivial solution; therefore, the case is rejected.

Case 2.

When a0=±iB/C  anda1=±i2A/C, (19) and (20) yield (24)ψ′′′ψ′′+2BA=0. Integrating (24) with respect to ξ, we obtain (25)ψ′′=C1exp(-2BAξ). Using (25), from (20), we obtain (26)ψ=-A2BC1exp(-2BAξ).

Upon integration, we obtain (27)ψ=AC12Bexp(-2BAξ)+C2, Where C1 and C2 are constants of integration. Therefore, the exact solution of (9) is (28)ϕ7(ξ)=±iBC(1-2AC1exp(-(2B/A)ξ)AC1exp(-(2B/A)ξ)+2BC2).

From (28), we obtain the exact solution of (9) which is (29)ϕ8(ξ)=±iBC(1-(2AC1(cosh(B2Aξ)hhhhhhhhhhhhhhhhh-sinh(B2Aξ)))hhhhhhh×((2BC2+AC1)cosh(B2Aξ)hhhhhhhhhh+(2BC2-AC1)sinh(B2Aξ))-1).

We can arbitrarily choose the parameters C1 and C2. Therefore, if we set C1=2BC2/A, (29) reduces to (30)ϕ9(ξ)=±iBCtanh(B2Aξ).

Again setting C1=-(2BC2/A), (29) reduces to, (31)ϕ10(ξ)=±iBCcoth(B2Aξ).

Using hyperbolic function identities, from (30) and (31), we obtain the following periodic solutions (32)ϕ11(ξ)=±BCtan(-B2Aξ),ϕ12(ξ)=±BCcot(-B2Aξ).

4. Exact Solutions of Some Class of NLPDEs 4.1. The Perturbed Nonlinear Schrödinger's Equation (NLSE) in the Form [<xref ref-type="bibr" rid="B20">20</xref>]

Using (33)iut+uxx+αu|u|2+i[γ1uxxx+γ2|u|2ux+γ3u(|u|2)x]=0, where γ1 is the third order dispersion, γ2 is the nonlinear dispersion, while γ3 is also a version of nonlinear dispersion. We assume that (33) has exact solution in the form (34)u(x,t)=ϕ(ξ)exp(i(λx-ωt)),ξ=k(x-ct), where λ,ω,k, and c are arbitrary constant to be determined. Substituting (34) into (33), removing the common factor exp(i(λx-ωt)), we have (35)i(γ1k3ϕ′′-k(c-2λ+3γ1λ2)ϕ+k(γ2+2γ3)ϕ2ϕ)+k2(1-3γ1λ)ϕ+(ω-λ2+γ1λ3)ϕ+(α-γ2λ)ϕ3=0, where γi(i=1,2, and 3),α,  and  k are positive constants and the prime means differentiation with respect to ξ. Then we have two equations as follows (36)γ1k2ϕ′′-(c-2λ+3γ1λ2)ϕ+(γ2+2γ3)ϕ2ϕ=0,(37)k2(1-3γ1λ)ϕ+(ω-λ2+γ1λ3)ϕ+(α-γ2λ)ϕ3=0.

Integrating (36) with respect to ξ once and setting the integration constant to be zero, then we have (38)γ1k2ϕ+(2λ-c-3γ1λ2)ϕ+(13γ2+23γ3)ϕ3=0.

As (37) and (38) have the same solutions, we have the following equation:(39)γ11-3γ1λ=2λ-c-3γ1λ2ω-λ2+γ1λ3=Cα-γ2λ, where C=(1/3)γ2+(2/3)γ3.

From (39), we can obtain (40)ω=(α-γ2λ)(2λ-c-3γ1λ2)C+λ2-γ1λ3,hhhhhhhhhhhhhhhhhhhhλ=C-αλ13Cγ1-γ1γ2.

Based on the conclusion just mentioned, we only solve (38) or (37), instead of both (37) and (38), provided that (37) and (36) are replaced by (40), respectively, we get (41)Aϕ′′(ξ)+Bϕ(ξ)+Cϕ3(ξ)=0.

Equation (41) is identical to (9) and A,B, and C are defined by (42)A=γ1k2,B=2λ-c-3γ1λ2,C=13γ2+23γ3.

Then, solutions of (33) are defined as follows: (43)u(x,t)=ϕ(ξ)exp(i(λx-ωt)),ξ=k(x-ct),ω=(α-γ2λ)(2λ-c-3γ1λ2)C+λ2-γ1λ3,hhhhhhhhhhhhhhhhhλ=C-αλ13Cγ1-γ1γ2, where ϕ(ξ), appearing in these solutions, is given by relations (12), (15), and (28)–(32). A,B, and C are defined by (42).

4.2. The Klein-Gordon-Zakharov (KGZ) System [<xref ref-type="bibr" rid="B21">21</xref>]

Consider(44)Ett-Exx+E-αNE-β|E|2E=0,Ntt-Nxx=γ(|E|2)xx, wherein the complex valued unknown function E=E(x,t) denotes the fast time scale component of electric field raised by electrons, and the real valued unknown function N=N(x,t) represents the deviation of ion density. α,β and γ are some real parameters.

We assume that (45)E(x,t)=ϕ(ξ)exp(i(kx+ωt)),N=N(ξ),hξ=ωx+kt.

Substituting (45) into (44), we have (46)-αNϕ-βϕ3+(k2-ω2+1)ϕ+(k2-ω2)ϕ=0,(47)(k2-ω2)N′′-γω2(2ϕ2  +2ϕϕ)=0.

Integrating (47) with respect to ξ twice and setting the integration constant to be zero, then we have (48)N=γω2k2-ω2ϕ2.

Substituting (48) into (46), we have (49)Aϕ′′(ξ)+Bϕ(ξ)+Cϕ3(ξ)=0.

Equation (49) is identical to (9) and A,B, and C are defined by (50)A=k2-ω2,B=k2-ω2+1,C=-(β+αγω2k2-ω2).

Then, solutions of the Klein-Gordon-Zakharov (KGZ) system are defined as follows: (51)E(x,t)=ϕ(ξ)exp(i(kx+ωt)),N=γω2k2-ω2ϕ2,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhξ=ωx+kt, where ϕ(ξ) appearing in these solutions is given by relations (12), (15), and (28)–(32). A,B, and C are defined by (50).

4.3. A Class of Nonlinear Partial Differential Equations (NPDEs)

We consider a class of NLPDEs with constant coefficients  (52)iut+μ(uxx+D1uyy)+E1|u|2u+A1un=0,D2ntt+(nxx-E2uyy)+A2(|u|2)xx=0, where μ,Di,Ei,andAi(i=1,2) are real constants and μ0,D10,A10,andA20. Equations (52) are a class of physically important equations. In fact, if one takes (53)μ=12κ2,D1=2μ,E1=α,A1=-1,D2=0,E2=D1,A2=-2α,κ2=±1, then (52) represent the Davey-Stewartson (DS) equations  (54)iut+12κ2(uxx+κ2uyy)+α|u|2u-un=0,nxx-κ2nyy-2α(|u|2)xx=0.

If one takes (55)n=n(x,t),thatis,ny=0,μ=1,D1=0,E1=-2λ,E2=-1,A2=-1,A1=2, then (52) become generalized Zakharov (GZ) equations  (56)iut+uxx-2λ|u|2u+2un=0,ntt-nxx+(|u|2)xx=0.

Since u is a complex function, we assume that (57)u(x,t)=ϕ(ξ)exp(i(kx+ly-Ωt)),hhhhhhhn=n(ξ)ξ=px+qy-ωt, where both ϕ(ξ) and n(ξ) are real functions and k,l,p,q,Ω, and ω are constants to be determined later. Substituting (57) into (52), we have the following ODE for ϕ(ξ) and n(ξ): (58)E1ϕ3+(Ω-μ(k2+D1l2)+A1n)ϕ+μ(D1q2+p2)ϕ+i(2μ(pk+D1ql)-ω)ϕ=0,(59)(D2ω2+p2-E2q2)n′′+A2p2(2ϕ2+2ϕϕ)=0.

If we set (60)ω=2μ(pk+D1ql), then (58) reduces to (61)E1ϕ3+(Ω-μ(k2+D1l2)+A1n)ϕ+μ(D1q2+p2)ϕ=0.

Integrating (59) twice to ξ, we get (62)n=DD2ω2+p2-E2q2-A2p2D2ω2+p2-E2q2ϕ2, where D is the integrating constant and Substituting (62) into (61) yields (63)Aϕ′′+Bϕ+Cϕ3=0.

Equation (63) is identical to (9) and A,B, and C are defined by (64)A=μ(D1q2+p2),B=Ω-μ(k2+D1l2)+A1DD2ω2+p2-E2q2,C=E1-A1A2p2D2ω2+p2-E2q2.

Then, solutions of (52) are defined as follows: (65)u(x,y,t)=ϕ(ξ)exp(i(kx+ly-Ωt)),ξ=px+qy-ωt,ω=2μ(pk+D1ql),n(x,y,t)=DD2ω2+p2-E2q2-A2p2D2ω2+p2-E2q2ϕ2, where ϕ(ξ) appearing in these solutions is given by relations (12), (15), and (28)–(32) and A,B, and C are defined by (64).

We may obtain from (54) that (66)ω=κ2(pk+κ2ql),u(x,y,t)=ϕ(ξ)exp(i(kx+ly-Ωt)),n(x,y,t)=Dω2-κ2q2+2αp2ω2-κ2q2ϕ2, where D is the integrating constant and then (54) reduce to (67)Aϕ′′+Bϕ+Cϕ3=0.

This equation coincides also with (9), where A,B, and C are defined as follows: (68)A=12κ2(p2+κ2q2),B=Ω-12κ2(k2+κ2l2)-Dω2-κ2q2,C=α-2αp2ω2-κ2q2, where ξ=px+qy-ωt, ϕ(ξ) appearing in these solutions is given by relations (12), (15), and (28)–(32) and A,B, and C are defined by (68).

We may obtain from (56) that (69)ω=2pk,u(x,t)=ϕ(ξ)exp(i(kx-Ωt)),n(x,t)=Dp2-ω2+p2p2-ω2ϕ2, where D is the integrating constant and then (56) reduce to (70)Aϕ′′+Bϕ+Cϕ3=0.

This equation coincides also with (9), where A,B, and C are defined as follows: (71)A=p2,B=Ω-k2+2Dp2-ω2,C=2(p2p2-ω2-λ), where ξ=px-ωt, ϕ(ξ) appearing in these solutions is given by relations (12), (15), and (28)–(32) and A,B, and C are defined by (71).

5. Conclusions

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations, and the elliptic-like equation is one of the most important auxiliary equations because many nonlinear evolution equations, such as the perturbed nonlinear Schrödinger's equation, the Klein-Gordon-Zakharov system, the generalized Davey-Stewartson equations, the Davey-Stewartson equations, the generalized Zakharov equations, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system, and the generalized ZK-BBM equation, can be converted to this equation using the travelling wave reduction.

In this paper, we apply the simplest equation method and the modified simplest equation method to derive the exact solutions of the elliptic-like equation. The exact solutions of the perturbed nonlinear Schrödinger's equation, the Klein-Gordon-Zakharov system, the generalized Davey-Stewartson equations, the Davey-Stewartson equations, and the generalized Zakharov equations are derived. Comparing the currently proposed method with other methods, such as the (G/G)-expansion method, the various extended hyperbolic methods, and the exp-function method, we might conclude that some exact solutions that we obtained can be investigated using these methods with the aid of the symbolic computation software, such as Matlab, Mathematica, and Maple to facilitate the complicated algebraic computations. But, by means of the simplest equation method and the modified simplest equation method the exact solutions to these equations have been gained in this paper without using the symbolic computation software since the computations are simple. This study shows that the simplest equation method and the modified simplest equation method are much more simple than the other methods and can be applied to many other nonlinear evolution equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11161020; 11361023), the Natural Science Foundation of Yunnan Province (2011FZ193; 2013FZ117), and the Natural Science Foundation of Education Committee of Yunnan Province (2012Y452; 2013C079).

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