On Solving the(2+1)-Dimensional Nonlinear Cubic-Quintic Ginzburg-Landau Equation Using Five Different Techniques

ZAYED Elsayed M.E.,Al-NOWEHY A.-G. and ELSHATER Mona E.M.

1,3Mathematics Department,Faculty of Sciences,Zagazig University,P.O.Box44519,Zagazig,Egypt.

2Mathematics Department,Faculty of Education and Science,Taiz University,Taiz,Yemen.

Abstract.In this article,we apply five differenttechniques,namely the(G′/G)-expansion method,an auxiliary equation method,the modi fied simple equation method,the if rst integral method and the Riccati equation method for constructing many new exact solutions with parameters as well as the bright-dark,singular and other soliton solutions of the(2+1)-dimensional nonlinear cubic-quintic Ginzburg-Landau equation.Comparing the solutions of this nonlinear equation together with each other are presented.Comparing our new resultsobtained in this articlewith the well-known results are given too.

Key Words:(G′/G)-expansion method;auxiliary equation method;modi fied simple equation method; first integral method;Riccati equation method;exact traveling wave solutions;solitary wave solutions;Cubic-quintic Ginzburg-Landau equation.

1 Introduction

Investigationsofexact traveling wave solutions,solitarywave solutions,singularsolitary wave solutions and periodic wave solutions to nonlinear evolution equations play an important role in the study of nonlinear physical phenomena.Nonlinear wave phenomena appears in various scienti fic and engineering fields,such as fl uid mechanics,plasma physics,optical fibers,biology,solid state physics,chemical physics and so on.In recent decades,many effective methods have been established to obtain the exact traveling wave solutions of the nonlinear evolution equations in mathematical physics,such as the inverse scattering transform[1],the Hirota method[2],the truncated Painlev′e expansion method[3],the B¨acklund transform method[1,4,5],the exp-function method[6-8],the simplest equation method[9,10],the Weierstrass elliptic function method[11],the Jacobi elliptic function method[12-16],the tan-function method[17-22],the sine-cosine method[23,24],the(G′/G)-expansion method[25-30],the modi fied simple equation method[31-36],the Kudryashov method[37-39],the multiple exp-function method[40,41],the transformed rational function method[42],a generalized new auxiliary equation method[43],an extended auxiliary equation method[44,45],the(G′/G,1/G)-expansion method[46-50],the first integral method[51,52],the soliton ansatz method[54-69],a special kind of(G′/G)-expansion method[70-74],a new mapping method[75],a new extended auxiliary equation method[76],the extended trial equation method[77-81],and so on.

The objective of this article is to apply five effective methods,namely,the(G′/G)-expansion method,an auxiliary equation method,the modi fied simple equation method,the first integral method and the Riccati equation method for solving the following(2+1)-dimensional nonlinear cubic-quintic Ginzburg-Landau equation[8,45,50,76]:

where the complex function u(x,z,τ)is slowly varying envelope of the electric field,β is a real constant,z and x are the propagation and transverse coordinates respectively,while r1and r2are constants.More over,

is the so-called reduced time,where t is the physical time,and V0is the group velocity of the carrier wave.Eq.(1.1)has been discussed in[8]using the exp-function method,in[50]using(G′/G,1/G)-expansion method,in[45]using an extended auxiliary equation methodand in[76]using a newextendedauxiliary equationmethodwhereits exact solutions have been obtained.To our knowledgement,Eq.(1.1)is not investigated elsewhere using the five methods proposed in this article.

This article is organized as follows:In Sections 2,3,4,we give the description of the(G′/G)-expansion method,the modi fied simple equation method and the first integral method,respectively.In Section 5,we solve Eq.(1.1)using the five suggested methods.In Section 6,some graphical representations of our results are presented.In Section 7,some conclusions are obtained.

2 Description of(G′/G)-expansion method

Consider the following nonlinear evolution equation:

where φ(x,t)is an unknown function,F is a polynomial in φ(x,t)and its partial derivatives of,in which the highest order derivatives and nonlinear terms are involved.With reference to the articles[70-74]we apply a special kind of the(G′/G)-expansion method which is different from the original(G′/G)-expansion method[25-30]to solve Eq.(2.1)through the following steps:

Step 1.We use the wave transformation

where k andw are nonzero constants,to reduce Eq.(2.1)into the following ordinary differential equation(ODE):

where P is a polynomial in φ(ξ)and its total derivatives φ′,φ′′and so on.

Step 2.Assume that Eq.(2.3)has non-integer balance number,”N”,then its formal solution is written in the form:

where G(ξ)satis fies the linear ODE:

and A1,λ,μ are constants to be determined,such that

Step 3.We determine the balance number”N”by balancing the highest order derivatives with the highest nonlinear term of Eq.(2.3).

Step 4.We substitute(2.4)along with Eq.(2.5)into Eq.(2.3),collect all the coefficients of powers of(G′/G)and set them to zero,to get a set of algebraic equations,which can be solved to find the values A1,λ,μ,k,w.

Step 5.Wesolve Eq.(2.5)to find theratio(G′/G).Now,we can getthe exact solutions and the solitary wave solutions of Eq.(2.1).

3 Description of the modi fied simple equation method

The main stepsof this method are well-known[31-36]and can be summarized as follows:

Step 1.Assume that Eq.(2.3)has the formal solution

where Aiare constants to be determined,such thatThe function ψ(ξ)is an unknown function to be determined later,such that

Step 2.We determined the positive integer N in(3.1)by balancing the homogeneous balance between the highest order derivatives and the highest nonlinear terms in Eq.(2.3).

Step 3.We substitute(3.1)into Eq.(2.3)and then we calculate all the necessary derivatives φ′,φ′′,...of the function φ(ξ).As a result of substitutions,we gather all the terms of the same powers of ψ?j(j=0,1,2,...),and equate them to zero.This operation yields a system of algebraic equations which can be solved to find Aiand ψ(ξ).Thus,we can get the exact solutions of Eq.(2.1).

4 Description of the first integral method

Feng[51]has presented this method originality,which is based on the ring theory of commutative algebra.The main steps of this method are summarized as follows:

Step 1.Assume that Eq.(2.3)has the solution

and introduce a new independent variable Y=Y(ξ)such that

Step 2.Under the conditions of Step 1,Eq.(2.3)can be converted into a system of nonlinear ODEs

where H is a polynomial in X and Y.

If we can find the integrals to Eqs.(4.3),the general solutions to Eqs.(4.3)can be solved directly.However,in general,it is really dif ficult for us to realize this even for one first integral because for a given plane autonomous system,there is no systematic theory that can tell us how to find its first integrals,nor there is a logical way for telling us what these first integrals are.

We will apply the so-called Division theorem to obtain one first integral to Eqs.(4.3)which reduces Eq.(2.3)to a first order integral ODE.Exact solutions to Eq.(2.1)are then obtained by solving this equation.

Division Theorem:Supposethat P(w,z)and Q(w,z)are polynomials in thecomplex domain C(w,z)and P(w,z)is irreducible in C(w,z).

If Q(w,z)vanishes at all zero points of P(w,z),thenthere existsa polynomial G(w,z)in C(w,z),such that

5 Mathematical analysis

Assume that Eq.(1.1)has the complex solution:

where φ(ξ)and ψ(η)are real functions of ξ and η,while k is a real constant.Here ξ=l0x?l1τ,η=h0x?h1τ,wherel0,l1,h0,h1are constants.

Substituting(5.1)into Eq.(1.1)and separating the real and imaginary parts,we have

respectively.

Let ψ(η)=η,then Eqs.(5.4)and(5.5)reduce to the two equations:

respectively.Eqs.(5.6)and(5.7)must be the same equation.Comparing theircoefficients,we get

and consequently,we have

where r1and r2satisfy the condition

Now,Eq.(5.6)or(5.7)reduces to the new equation

In the next subsections,we will solve Eq.(1.1)as follows:

5.1 On solving Eq.(1.1)using the(G′/G)-expansion method

To this aim,we first solve Eq.(5.11)using this method as follows.Balancing φ′′with φ5in Eq.(5.11)yields.With reference to Section 2,Eq.(5.11)has the formal solution:

where G satis fies Eq.(2.5)and A1,λ,μ are constants to be determined.Substituting(5.12)into Eq.(5.11),we have the equation:

According to the homogenous balance principle,we collect the coefficients of powers of(G′/G)in Eq.(5.13)and set them to zero,to get the following algebraic equations:

On solving the above algebraic equations(5.14)-(5.17)we have

Form(5.18)we have the results

From(5.12)and(5.19)we have the new exact solution of Eq.(5.11)as follows:

where c0,c1are arbitrary constants and r1＞0,r2＞0.

Form(5.1)and(5.20)we have the new exact traveling wave solution of Eq.(1.1)in the form

where λ is given by(5.19).

In particular,if we set c0=c1=1,in(5.21),then we have the dark soliton solution of Eq.(1.1)in the form

while,if we set c0=1,c1=?1,in(5.21),then we have the singular soliton solution:

Note that the exact traveling wave solution(5.21)is new,but the solutions(5.22)and(5.23)are equivalent to the solutions obtained in[50]using the(G′/G,1/G)-expansion method,in[45]using an extended auxiliary equation method and in[76]using the new extended auxiliary equation method.

5.2 On solving Eq.(1.1)using an auxiliary equation method

In this section,we apply two algebraic direct methods to solve Eq.(5.11).To this aim,we first multiply both sides of Eq.(5.11)by φ′(ξ)and integrate once with respect to ξ.We consider two cases:

Case 1.Suppose that the constant of integration is equal to zero,then we have the auxiliary equation:

where

With reference to[43],we see that Eq.(5.24)has many solutions.With the aid of these solutions,we can derive the following exact solutions of Eq.(1.1):

and the singular soliton solution

The solutions(5.28)and(5.29)are equivalent to the solutions(5.22)and(5.23),respectively.

(3)Since a1＞0,∈=±1,c1＞0,then we have the solitary wave solutions

Case 2.Suppose that the constant of integration is not zero,then we have the extended auxiliary ODE[75]:

where p,q,s,r are constants,which are given by

where A is the nonzero constant of integration and r2＞0,s＞0.

It is well-known[75]that Eq.(5.32)has many kinds of solutions.With the aid of some of them,we can derive some exact solution of Eq.(5.32)as follows:

Finally,the solutions(5.26),(5.27),(5.30),(5.31),(5.34)and(5.35)of Eq.(1.1)are new and not found elsewhere.

5.3 On solving Eq.(1.1)using the modi fied simple equation method

To this aim,we first multiply both sides of Eq.(5.11)by φ′(ξ)and integrate with respect to ξ with zero constant of integration,we get

Balancing φ′2with φ6yields.Now,we set

where V(ξ)is a new function of ξ.Substituting(5.37)into Eq.(3.36)we have equation:

Balancing V′2with V4yields N=1.Thus,Eq.(5.38)has the formal solution:

where A0and A1are constants to be determined,such that A16=0.Substituting(5.39)into Eq.(5.38)we have the equation

From(5.40)we have the following algebraic equations:

On solving the two algebraic equations(5.41),(5.45)we have the results:

Case 1.If A0=0.

In this case we deduce from Eq.(5.43)that

and from Eq.(5.44)that

From(5.47)and(5.48)we deduce that r1r2=0.Sincethen r2=0.This contradicts the condition r2＞0.Thus

Case 2.If

In this case we deduce from Eq.(5.43)that

and from(5.44)that

From(5.49)and(5.50)we have the condition

On solving Eq.(5.50)we have

where c1and c2are constants of integration.From(5.37),(5.39),(5.52)and(5.53),we have the new exact traveling wave solution of Eq.(1.1):

under the condition(5.51).

provided that r1and r2are positive constants satisfying the condition(5.51).On comparing the solutions(5.55)and(5.56)with the solutions(5.28)and(5.29)respectively,we deduce that they are equivalent under the condition(5.51).

5.4 On solving Eq.(1.1)using the first integral method

To this aim,we first solve Eq.(5.11)using this method as follows:We set

and then we have

According to the first integral method,we assume that X(ξ),Y(ξ)are nontrivial solution of Eqs.(5.58)and

is an irreducible polynomial in the complex domain C[X,Y]such that

where bk(X)are polynomials in X andDue to the Division theorem,there exist a polynomial[g(X)+h(X)Y(ξ)]in the complex domain C[X,Y]such that

Let us now consider the following two cases:

Case 1.If m=1.

In this case,we substitute(5.58),(5.60)into(5.61)and equate the coefficients of Yk(ξ)(k=0,1,2)we have the equations:

From(5.64)we deduce that b1(X)is a constant and h(X)=0.For simplicity,we take b1(X)=1.Balancing the degrees of g(X)and b0(X),we deduce that deg(g(X))=2 and deg(b0(X))=3.Thus we have

where A2,A1,A0,B0are constant to be determined,such that.Substituting(5.65),(5.66)into(5.64)and setting all the coefficients of powers X(ξ)to zero,we obtain the algebraic equations:

On solving the above algebraic equation(5.67),we have the results:

provided that AD＜0,AB＜0 and 16DB=3E2.Therefore,Eq.(5.66)becomes

From(5.57),(5.60)and(5.69)we have the equation:

In order to solve Eq.(5.70),we refer to the article[42]to get the solutions:

where r1＞0 and r2＞0.

The solutions(5.71)is equivalent to(5.22),(5.28)and(5.55)if ξ0=1,while the solution(5.72)is equivalent to(5.23),(5.29)and(5.56)if

Case 2.If m=2.

In this case,we deduce from(5.58),(5.60)and(5.61)that

and

Equating the coefficients of power of Y(ξ)in Eq.(5.75),we have

From(5.76)we deduce that b2(X)is a constant and h(X)=0.For simplicity,we take b2(X)=1.Balancing the degree of g(X),b1(X),b0(X),we conclude that degree(g(X))=2,degree(b1(X))=3,degree(b0(X))=6.Therefore,we get

where A2,A1,A0,B0are constants to be determined,such that

Substituting(5.80),(5.81)into Eq.(5.78)and integrating,we get

where d is a constant of integration.Substituting(5.80)-(5.82)into Eq.(5.79)and equating the coefficients of powers of X(ξ),we get a set of algebraic equations,which can be solved to obtain the results:

provided that AD＜0,AB＜0 and 16DB=3E2.Consequently,we deduce that

and

Substituting(5.84),(5.85)into Eq.(5.74)we have the same equation(5.70).Consequently,we have the same solutions(5.71)-(5.73).Finally,we note that the two case m=1 and m=2 give the same exact solutions of Eq.(1.1).

Remark 5.1.For the case m≥3,the discussions become more complicated,which is omitted here and has been left for the readers as an open problem.

5.5 On solving Eq.(1.1)using the Riccati equation method

To this aim,we assume that Eq.(5.38)has the formal solution:

where φ1[τ1(ξ)]=φ1(Bξ)is the amplitude component of the soliton satisfying the Riccati equation:

where α0,α1,α2are real constants and.Here A and B are respectively the amplitude and the width of the soliton to be determine later.

Substituting(5.86)along with Eq.(5.87)into Eq.(5.38)we have

From(5.88)we have the results:

With reference to[42],we deduce that the dark soliton solution of Eq.(1.1)is given by:

and the singular soliton solution of Eq.(1.1)is given by:

where ξ0is a constant.

The solutions(5.90)and(5.91)of Eq.(1.1)are new not found elsewhere.

Note that(5.90)and(5.91)are equivalent to(5.71)and(5.72)whenrespectively.Also(5.90)is equivalent to(5.28)whenand ξ0=1,while(5.91)is equivalent to(5.29)whenand ξ0=?1.Finally,(5.90)and(5.91)are equivalent to(5.55)and(5.56)whenunder the condition(5.51)and α1=?α2respectively.

6 Some graphical representations of some solutions

In this section,we have presented some of the trigonometric and hyperbolic solutions.For the established exact traveling wave solutions with trigonometric and hyperbolic solutions are special kinds of solitary waves solutions.Let us now examine Figures(1-4)as it illustrates some of our solutions obtained in this article.To this aim,we select some special values of the parameters obtained,for example,in some of the solutions(5.34),(5.35),(5.55)and(5.56)of the(2+1)-dimensional nonlinear cubic-quintic Ginzburg-Landau equation whit β=?1,?10＜x,τ＜10 respectively in the following:

Figure 1:Plot|u(x,z,τ)|of solution(5.34)with

Figure 2:Plot|u(x,z,τ)|of solution(5.35)with

Figure 3:Plot|u(x,z,τ)|of solution(5.55)with r1=1/3,r2=1/4.

Figure 4:Plot|u(x,z,τ)|of solution(5.56)with r1=1/3,r2=1/4.

7 Conclusions

In this article,based on a special kind of(G′/G)-expansion method described in Sec.2,the modi fied simple equation method described in Sec.3,the first integral method described in Sec.4,as well as the auxiliary equation method used in subsection 5.2 and the Riccati equation method used in subsection 5.4,we have obtained new exact traveling wave solutions of the(2+1)-dimensional nonlinear cubic-quintic Ginzburg-Landau equation(1.1).Comparing our results obtained in this article with the results obtained in[8,45,50,76],we conclude that our results are new and not found elsewhere.Finally,with the aid of Maple,we have checked that all solutions in this article satisfy the original equation(1.1).

Acknowledgement

The authors wish to thank the referees for their comments on this paper.