# Generalized Hyperbolic Function Solution to a Class of Nonlinear Schrödinger-Type Equations

Authors: Zeid I. A. Al-Muhiameed; Emad A.-B. Abdel-Salam
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

With the help of the generalized hyperbolic function, the subsidiary ordinary differential equation method is improved and proposed to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Liu equation are investigated and the exact solutions are derived with the aid of the homogenous balance principle and generalized hyperbolic functions. We study the effect of the generalized hyperbolic function parametersp and q in the obtained solutions by using the computer simulation.


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## Body

## 1. Introduction

The nonlinear Schrödinger (NLS) equation is a ubiquitous and significant model that naturally arises in many fields of physics, such as nonlinear optical systems, plasmas, fluid dynamics, and Bose-Einstein condensation. In the last three decades, great progress has been made on the construction of exact solutions of NLS equation. Many significant methods have been established by mathematicians and physicists to obtain special solutions of nonlinear partial differential equations (NLPDEs), including the inverse scattering method, Darboux transformation, Hirota’s bilinear method, homogeneous balance method, Jacobi elliptic function method, variational iteration method, the sine-cosine method, the(G′/G) expansion method, tanh-function method, F-expansion method, Lucas Riccati method, auxiliary equation method, algebraic method, and others  [1–36]. The last five methods mentioned above belong to a class of method called subsidiary ordinary differential equation (sub-ODE) method. The key ideas of the sub-ODE method are that the travelling wave solutions of the complicated NLPDE can be expressed as a polynomial, the variable of which is one of the solutions of simple and solvable ODE that called the sub-ODE. The sub-ODEs which were often used are the Riccati equation, Jacobi elliptic equation, projective Riccati equations, and so forth. With the development of computer science, recently, the sub-ODEs with nonlinear terms of high order have attracted much attention [34, 35]. This is due to the availability of symbolic computation systems like Mathematica or Maple which enable us to perform the complex and tedious computation on computers. Recently, Sirendaoreji [21–23] introduced the auxiliary equation method by using the six degree first-order nonlinear differential equation, Wang et al. [24] proposed new sub-ODE equation involving an arbitrary positive power to construct exact travelling wave solutions of NLPDEs in a unified way. In this study, we improve the method presented by Wang et al. [24] and introduce some solutions for the sub-ODE equation in terms of the generalized hyperbolic functions (GHFs) [36].The rest of this paper is organized as follows: in the following section, we introduce the improved sub-ODE method to construct exact solutions of some NLPDEs in terms of the GHFs. In Section3, we apply this method to the generalized Zakharov system, the Rangwala-Rao equation and the Chen-Lee-Liu equation. Finally, we conclude the paper and give some futures and comments.


## 2. The Improved Subsidiary Ordinary Differential Equation Method

The main idea of this method is to express the solutions of NLPDEs as polynomials in the solution of sub-ODE involving an arbitrary positive power of dependent variable that the GHFs satisfy (see the appendix). Consider a given NLPDE(2.1)H(u,ut,ux,utt,utx,uxx,…)=0.
The improved sub-ODE method for solving NLPDE (2.1) proceeds in the following four steps [36].Step 1.
We seek its traveling wave solution of (2.1) in the form
(2.2)u(x,t)=u(ξ),ξ=kx-ωt,
where k and ω are constants to be determined later. Substituting (2.2) into (2.1) yields an ordinary differential equation
(2.3)H̃(u,u′,u′′,u′′′,…)=0,u′=dudξ,…,
where H̃ is a polynomial of u and its various derivatives. If H̃ is not a polynomial of u and its various derivatives, then we may use new variables v=v(ξ) which makes H̃ become polynomial of v and its various derivatives.Step 2.
Suppose thatu(ξ) can be expressed by a finite power series of  F(ξ)(2.4)u(ξ)=a0+∑j=1najFj(ξ),an≠0,
where n is a positive integer which can be determined by balancing the highest derivative term with the highest nonlinear term in (2.3) and aj(j=0,1,…,n)are some parameters to be determined. The function F(ξ) satisfies the sub-ODE with an arbitrary positive power
(2.5)F′2(ξ)=AF2(ξ)+BF2+r(ξ)+CF2+2r(ξ),r>0,
where A,B, and C are parameters to be determined.Step 3.
Substituting (2.4) and (2.5) into the ODE (2.3), then the left-hand side of (2.3) can be converted into a polynomial in F(ξ). Setting all coefficients of the polynomial to zero yields system of algebraic equations for aj(j=0,1,…,n), k,and ω.Step 4.
Solving this system obtained in Step3, then aj(j=0,1,…,n), k, and ω can be expressed by A,B, and C. Substituting these results into (2.4), then the general formulae of the travelling wave solution of (2.1) can be obtained. Selecting the values of A,B,C, and the corresponding GHF solution F(ξ) of (2.5) given bellow to obtain the exact solutions of (2.1). The definition and proprieties of the GHFs are given in the appendix.
We list various types of exact solutions of (2.5) as follows.Case 1.
IfA>0,B=2σA,and C=(σ2-pq)A, then (2.5) admits the following positive solution:
(2.6)F(ξ)=[1cosh⁡pq⁡(rAξ)-σ]1/r,σ<1.Case 2.
IfA>0,B=2σA,and C=(σ2+pq)A, then (2.5) admits the following positive solution:
(2.7)F(ξ)=[1sinh⁡pq⁡(rAξ)-σ]1/r,σ<1.Case 3.
IfA>0,B=-2AC,and C>0, then (2.5) admits the following positive solution:
(2.8)F(ξ)=[12AC(1±tanhpq⁡(r2Aξ))]1/r,F(ξ)=[12AC(1±coth⁡pq⁡(r2Aξ))]1/r.Case 4.
IfA=0,B=4/r2, and C=-4σ/r2, then (2.5) admits the following positive solution:
(2.9)F(ξ)=[1ξ2+σ]1/r,σ>0.


## 3. Applications

In the following we use the improved sub-ODE method to seek exact traveling wave solutions of the class of nonlinear Schrödinger-type equations which are of interest in plasma physics, wave propagation in nonlinear optical fibers, Ginzburg-Landau theory of superconductivity, and so forth.


### 3.1. Generalized Zakharov System

In the interaction of laser-plasma the system of Zakharov equation plays an important role. This system has wide interest and attention for many scientists. Let us consider the generalized Zakharov system [18]:(3.1)utt-cs2uxx=β(|E|2)xx,iEt+αExx-δ1uE+δ2|E|2E+δ3|E|4E=0.
When δ2=δ3=0, the generalized Zakharov system reduce to the famous Zakharov system which describe the propagation Langmuir waves in plasmas. The real unknown function u(x,t) is the fluctuation in the ion density about its equilibrium value, and the complex unknown function E(x,t) is the slowly varying envelope of highly oscillatory electron field. The parameters δ1,δ2,δ3andcsare real numbers, wherecsis proportional to the ion acoustic speed (or electron sound speed). Here we seek its traveling wave solution in the form(3.2)E(x,t)=H(ξ)ei(kx-ωt),u(x,t)=u(ξ),ξ=x-ct,
where k,ω, and care constants and H(ξ) is real function. Therefore, system (3.1) is reduced to(3.3)(c2-cs2)u′′=β(H2)′′,αH′′+i(2αk-c)H′+(ω-αk2)H-δ1uH+δ2H3+δ3H5=0.
Integrating the first equation of (3.3) with respect to ξ and taking the integration constants to zero yields(3.4)u=βc2-cs2H2,c2-cs2≠0.
Substituting (3.4) into (3.3) results in(3.5)H′′+1α[(ω-αk2)H+(δ2-δ1βc2-cs2)H3+δ3H5]=0,c=2αk,α≠0.
Now, by balancing the higher order derivative term H′′ to the higher power nonlinear term H5in (3.5), we have n+2r=5n which gives2r=4n. Since n must be a positive integer, thus r=2,4,6,… this gives n=1,2,3,…. For simplicity, we take r=2 and n=1. We suppose that(3.6)H(ξ)=a0+a1F(ξ).
Substituting (3.6) with (2.5) into (3.5) and equating each of the coefficients of F(ξ) to zero, we obtain system of algebraic equations. Solving this system by using Maple, yields(3.7)a0=0,c=2αk,ω=α(k2-A),a1=±3C2δ3B(δ2-βδ14α2k2-cs2).
Now based on the solutions of (2.5), one can obtain new types of solitary wave solution of the generalized Zakharov system. The general formulae take the form(3.8)u(x,t)=3Cβ2δ3B(4α2k2-cs2)(δ2-βδ14α2k2-cs2)F2(x-2αkt),E(x,t)=±3C2δ3B(δ2-βδ14α2k2-cs2)F(x-2αkt)ei(kx-ωt).
By selecting the special values of the A,B,Cand the corresponding functionF(ξ), we have the following solutions of the generalized Zakharov system (3.1):(3.9)u1=3β(σ2-pq)4δ3σ(4α2k2-cs2)(δ2-βδ14α2k2-cs2)[1⁡cosh⁡pq(2A(x-2αkt))-σ],E1=±3(σ2-pq)4δ3σ(δ2-βδ14α2k2-cs2)[1cosh⁡pq⁡(2A(x-2αkt))-σ]ei(kx-ωt),(3.10)u2=3β(σ2+pq)4δ3σ(4α2k2-cs2)(δ2-βδ14α2k2-cs2)[1sinh⁡pq⁡(2A(x-2αkt))-σ],E2=±3(σ2+pq)4δ3σ(δ2-βδ14α2k2-cs2)[1sinh⁡pq⁡(2A(x-2αkt))-σ]1/2ei(kx-ωt),(3.11)u3=3βC8δ3A(4α2k2-cs2)(βδ14α2k2-cs2-δ2)[1±tanhpq⁡(A(x-2αkt))],E3=±123C2δ3A(βδ14α2k2-cs2-δ2)[1±tanhpq⁡(A(x-2αkt))]ei(kx-ωt),(3.12)u4=3βC8δ3A(4α2k2-cs2)(βδ14α2k2-cs2-δ2)[1±coth⁡pq⁡(A(x-2αkt))],E4=±123C2δ3A(βδ14α2k2-cs2-δ2)[1±⁡coth⁡pq(A(x-2αkt))]ei(kx-ωt),(3.13)u5=3σβ2δ3(4α2k2-cs2)(βδ14α2k2-cs2-δ2)[1(x-2αkt)2+σ],E5=±32δ3(βδ14α2k2-cs2-δ2)[1(x-2αkt)2+σ]ei(kx-ωt).
In order to understand the significance of these solutions expressed by (3.9)–(3.13), the main features of them are investigated by using direct computer simulations with the accuracy as high as 10-9. We study the effect of the parameters p and q of the GHF in the solution u1 given by the first equation of (3.9) by choosing different values of the parameters p and q, for A=δ1=δ2=δ3=α=β=k=1, σ=0.9, and cs=0.4, by means of Figures 1, 2, and 3 over same region {(x,t):∥x∥≤5,∥t∥≤5}. From Figures 1, 2 and 3, we see that the solution u1 describes dark soliton solution (increase, decrease, and have singularity with the change of the two parameters p and q). Also, we discuss u2 given by the first equation of (3.10), by taking the same value of the parameters by means of Figures 4, 5, and 6 over the region {(x,t):∥x∥≤5,∥t∥≤5}. From Figures 4, 5, and 6, we see that the solution u2 describes singular wave solution changed with different value of the two parameters p and q. Of course, we can plot the other figures of the exact solutions of (3.1), we omit them here for convenience. As a result, we find that the parameters p and q affect the solution structure.Evolutional behaviour ofu1 describes dark soliton: (a) p=q=1;(b) p=1/32,q=1/33, (singular wave solution); (c) p=23,q=32; (d) p=1/13,q=11; (e) p=13,q=1/11.
(a)(b)(c)(d)(e)Evolutional behaviour ofu1 describes dark soliton: (a) p=1,q=1/32 (singular wave solution); (b) p=1,q=3; (c) p=1,q=27.
(a)(b)(c)Evolutional behaviour ofu1 describes dark soliton: (a) p=1/32,q=1; (singular wave solution); (b) p=3,q=1; (c) p=27,q=1.
(a)(b)(c)Evolutional behaviour ofu2 describes singular wave solution: (a) p=q=1;(b) p=1/32,q=1/33; (c) p=23,q=32; (d) p=1/13,q=11; (e) p=13,q=1/11.
(a)(b)(c)(d)(e)Evolutional behaviour ofu2describes singular wave solution: (a) p=1,q=1/32; (b) p=1,q=3;(c) p=1,q=27.
(a)(b)(c)Evolutional behaviour ofu2describes singular wave solution: (a) p=1/32,q=1;(b) p=3,q=1; (c) p=27,q=1.
(a)(b)(c)


### 3.2. Rangwala-Rao Equation

The Rangwala-Rao equation [19](3.14)uxt-β1uxx+u+iTβ2|u|2ux=0,T=±1,
where β1and β2 are real constants. Rangwala and Rao introduced (3.14) as the integrability condition when they studied the mixed derivative nonlinear Schrödinger equations and looked for the Bäckland transformation and solitary wave solutions.Suppose the exact solutions of (3.14) is of the form(3.15)u(x,t)=e-ωteiψ(x-ct)H(x-ct),
where ω, c are constants determined later and ψ,H are undetermined real functions with one variable only. Set the relation of ψ,H as(3.16)ψ′(ξ)=ω2(c+β1)+Tβ24(c+β1)H2(ξ),′=ddξ,ξ=x-ct,
substituting (3.15) with (3.16) into (3.14) simultaneously yields(3.17)H′′-4(c+β1)-ω24(c+β1)2H-Tβ2ω2(c+β1)2H3+3T2β2216(c+β1)2H5=0.
Applying the homogeneous balance principle, to (3.17) we have r=2,4,6,… this gives n=1,2,3,…. For simplicity, we suppose(3.18)H(ξ)=a0+a1F(ξ).
Substituting (3.18) with (2.5) into (3.17) and equating each of the coefficients of F(ξ) to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:(3.19)a0=0,ω=±2(c+β1)[1-(c+β1)A],a1=±-8Cω3TBβ2.
The general formulae of the solution of Rangwala-Rao equation(3.20)u(x,t)=±-8Cω3TBβ2F(x-ct)e-iωteiψ(x-ct),
with ψ(ξ)=ω/(6B(c+β1))∫[3B+4CF2(ξ)]dξ,ω=±2(c+β1)[1-(c+β1)A]. By selecting the special values of the A,B,Cand the corresponding functionF(ξ)yields(3.21)|u1(x,t)|2=-4(σ-pq)ω3Tσβ2[⁡cosh⁡pq(2A(x-ct))-σ],(3.22)|u2(x,t)|2=-4(σ+pq)ω3Tσβ2[sinh⁡pq⁡(2A(x-ct))-σ].
We omitted the reminder solutions for simplicity. Now, we study the affect of the parameters p and q of the GHF in the intensity u1 given by the (3.21) by choosing different values of the parameters p and q, for A=δ=c=β2=T=1,ω=0.4, and σ=0.9, by means of Figure 7 over the region {(x,t):∥x∥≤5,∥t∥≤5}. We have bright soliton which are increase, decrease and have singularity with the different choose of the two parameters p and q. This means that the parameters p and q affect in the solution structure.Evolutional behaviour of the intensity|u1|2describes the bright soliton:  (a) p=1,q=1;(b) p=1/332,q=1/333 (singular wave solution); (c) p=2.132,q=1.723.
(a)(b)(c)


### 3.3. Chen-Lee-Liu Equation

The Chen-Lee-Liu equation [19](3.23)iut+uxx+iδ|u|2ux=0,
were δis a real constant. Similar as before, we suppose that the exact solution of (3.23)is of the form(3.24)u(x,t)=e-ωteiψ(x-ct)H(x-ct).
Set the relation of ψ,H as(3.25)ψ′(ξ)=c2+δ4H2(ξ),′=ddξ,ξ=x-ct,
substituting (3.25) with (3.24) into (3.23) simultaneously yields(3.26)H′′+14(4ω+c2)H-cδ2H3+3δ216H5=0.
According the homogeneous balance principle, we have r=2,4,6,… and n=1,2,3,…, we let(3.27)H(ξ)=a0+a1F(ξ).
Substituting (3.27) with (2.5) into (3.26) and equating each of the coefficients of F(ξ) to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:(3.28)a0=0,ω=A-c24,a1=±2-cCδβ2.
The general formulae of the solution of Chen-Lee-Liu equation(3.29)u(x,t)=±2-cCδβ2F(x-ct)e-iωteiψ(x-ct),
with ψ(ξ)=(c/2B)∫[B+2CF2(ξ)]dξ,ω=A-(c2/4). By selecting the special values of the A,B,Cand the corresponding function F(ξ)yields(3.30)|u1(x,t)|2=-2c(σ-pq)δσ[⁡cosh⁡pq(2A(x-ct))-σ],(3.31)|u2(x,t)|2=-2c(σ+pq)δσ[⁡sinh⁡pq(2A(x-ct))-σ].
We omitted the reminder solutions for simplicity. Now, we show the effect of the parameters p and q of the GHF in the intensity u1 given by the (3.30) by choosing different values of the parameters p and q, for A=δ=c=1, and σ=0.9, by means of Figure 8 over the region {(x,t):|x|≤5,|t|≤5}. We have bright soliton which are increase, decrease and have singularity with the different choose of the two parameters p and q. This means that the parameters p and q affect in the solution structure.Evolutional behaviour of the intensity|u1|2describes the bright soliton:  (a) p=1,q=1; (b) p=1/332,q=1/333 (singular wave solution); (c) p=2.132,q=1.723.
(a)(b)(c)Besides the solutions obtained above, the ODE equation (2.5), albeit with different parameters, has been studied in the different context [20–24]. It has been shown that this equation possesses abundant solutions, including Weierstrass function solutions, kink solutions and periodic solutions, and so forth. To the best of our knowledge, some of our explicit solutions are new.Notice that the GHFs are generalization of the hyperbolic functions as stated in the appendix. Also, the two parametersp and q describe the degree of the wave energy localization in the obtained solutions. Referring to [18, 19], it is easy to see that the obtained results in this paper are new and the improved method is valuable.


## 3.1. Generalized Zakharov System

In the interaction of laser-plasma the system of Zakharov equation plays an important role. This system has wide interest and attention for many scientists. Let us consider the generalized Zakharov system [18]:(3.1)utt-cs2uxx=β(|E|2)xx,iEt+αExx-δ1uE+δ2|E|2E+δ3|E|4E=0.
When δ2=δ3=0, the generalized Zakharov system reduce to the famous Zakharov system which describe the propagation Langmuir waves in plasmas. The real unknown function u(x,t) is the fluctuation in the ion density about its equilibrium value, and the complex unknown function E(x,t) is the slowly varying envelope of highly oscillatory electron field. The parameters δ1,δ2,δ3andcsare real numbers, wherecsis proportional to the ion acoustic speed (or electron sound speed). Here we seek its traveling wave solution in the form(3.2)E(x,t)=H(ξ)ei(kx-ωt),u(x,t)=u(ξ),ξ=x-ct,
where k,ω, and care constants and H(ξ) is real function. Therefore, system (3.1) is reduced to(3.3)(c2-cs2)u′′=β(H2)′′,αH′′+i(2αk-c)H′+(ω-αk2)H-δ1uH+δ2H3+δ3H5=0.
Integrating the first equation of (3.3) with respect to ξ and taking the integration constants to zero yields(3.4)u=βc2-cs2H2,c2-cs2≠0.
Substituting (3.4) into (3.3) results in(3.5)H′′+1α[(ω-αk2)H+(δ2-δ1βc2-cs2)H3+δ3H5]=0,c=2αk,α≠0.
Now, by balancing the higher order derivative term H′′ to the higher power nonlinear term H5in (3.5), we have n+2r=5n which gives2r=4n. Since n must be a positive integer, thus r=2,4,6,… this gives n=1,2,3,…. For simplicity, we take r=2 and n=1. We suppose that(3.6)H(ξ)=a0+a1F(ξ).
Substituting (3.6) with (2.5) into (3.5) and equating each of the coefficients of F(ξ) to zero, we obtain system of algebraic equations. Solving this system by using Maple, yields(3.7)a0=0,c=2αk,ω=α(k2-A),a1=±3C2δ3B(δ2-βδ14α2k2-cs2).
Now based on the solutions of (2.5), one can obtain new types of solitary wave solution of the generalized Zakharov system. The general formulae take the form(3.8)u(x,t)=3Cβ2δ3B(4α2k2-cs2)(δ2-βδ14α2k2-cs2)F2(x-2αkt),E(x,t)=±3C2δ3B(δ2-βδ14α2k2-cs2)F(x-2αkt)ei(kx-ωt).
By selecting the special values of the A,B,Cand the corresponding functionF(ξ), we have the following solutions of the generalized Zakharov system (3.1):(3.9)u1=3β(σ2-pq)4δ3σ(4α2k2-cs2)(δ2-βδ14α2k2-cs2)[1⁡cosh⁡pq(2A(x-2αkt))-σ],E1=±3(σ2-pq)4δ3σ(δ2-βδ14α2k2-cs2)[1cosh⁡pq⁡(2A(x-2αkt))-σ]ei(kx-ωt),(3.10)u2=3β(σ2+pq)4δ3σ(4α2k2-cs2)(δ2-βδ14α2k2-cs2)[1sinh⁡pq⁡(2A(x-2αkt))-σ],E2=±3(σ2+pq)4δ3σ(δ2-βδ14α2k2-cs2)[1sinh⁡pq⁡(2A(x-2αkt))-σ]1/2ei(kx-ωt),(3.11)u3=3βC8δ3A(4α2k2-cs2)(βδ14α2k2-cs2-δ2)[1±tanhpq⁡(A(x-2αkt))],E3=±123C2δ3A(βδ14α2k2-cs2-δ2)[1±tanhpq⁡(A(x-2αkt))]ei(kx-ωt),(3.12)u4=3βC8δ3A(4α2k2-cs2)(βδ14α2k2-cs2-δ2)[1±coth⁡pq⁡(A(x-2αkt))],E4=±123C2δ3A(βδ14α2k2-cs2-δ2)[1±⁡coth⁡pq(A(x-2αkt))]ei(kx-ωt),(3.13)u5=3σβ2δ3(4α2k2-cs2)(βδ14α2k2-cs2-δ2)[1(x-2αkt)2+σ],E5=±32δ3(βδ14α2k2-cs2-δ2)[1(x-2αkt)2+σ]ei(kx-ωt).
In order to understand the significance of these solutions expressed by (3.9)–(3.13), the main features of them are investigated by using direct computer simulations with the accuracy as high as 10-9. We study the effect of the parameters p and q of the GHF in the solution u1 given by the first equation of (3.9) by choosing different values of the parameters p and q, for A=δ1=δ2=δ3=α=β=k=1, σ=0.9, and cs=0.4, by means of Figures 1, 2, and 3 over same region {(x,t):∥x∥≤5,∥t∥≤5}. From Figures 1, 2 and 3, we see that the solution u1 describes dark soliton solution (increase, decrease, and have singularity with the change of the two parameters p and q). Also, we discuss u2 given by the first equation of (3.10), by taking the same value of the parameters by means of Figures 4, 5, and 6 over the region {(x,t):∥x∥≤5,∥t∥≤5}. From Figures 4, 5, and 6, we see that the solution u2 describes singular wave solution changed with different value of the two parameters p and q. Of course, we can plot the other figures of the exact solutions of (3.1), we omit them here for convenience. As a result, we find that the parameters p and q affect the solution structure.Evolutional behaviour ofu1 describes dark soliton: (a) p=q=1;(b) p=1/32,q=1/33, (singular wave solution); (c) p=23,q=32; (d) p=1/13,q=11; (e) p=13,q=1/11.
(a)(b)(c)(d)(e)Evolutional behaviour ofu1 describes dark soliton: (a) p=1,q=1/32 (singular wave solution); (b) p=1,q=3; (c) p=1,q=27.
(a)(b)(c)Evolutional behaviour ofu1 describes dark soliton: (a) p=1/32,q=1; (singular wave solution); (b) p=3,q=1; (c) p=27,q=1.
(a)(b)(c)Evolutional behaviour ofu2 describes singular wave solution: (a) p=q=1;(b) p=1/32,q=1/33; (c) p=23,q=32; (d) p=1/13,q=11; (e) p=13,q=1/11.
(a)(b)(c)(d)(e)Evolutional behaviour ofu2describes singular wave solution: (a) p=1,q=1/32; (b) p=1,q=3;(c) p=1,q=27.
(a)(b)(c)Evolutional behaviour ofu2describes singular wave solution: (a) p=1/32,q=1;(b) p=3,q=1; (c) p=27,q=1.
(a)(b)(c)


## 3.2. Rangwala-Rao Equation

The Rangwala-Rao equation [19](3.14)uxt-β1uxx+u+iTβ2|u|2ux=0,T=±1,
where β1and β2 are real constants. Rangwala and Rao introduced (3.14) as the integrability condition when they studied the mixed derivative nonlinear Schrödinger equations and looked for the Bäckland transformation and solitary wave solutions.Suppose the exact solutions of (3.14) is of the form(3.15)u(x,t)=e-ωteiψ(x-ct)H(x-ct),
where ω, c are constants determined later and ψ,H are undetermined real functions with one variable only. Set the relation of ψ,H as(3.16)ψ′(ξ)=ω2(c+β1)+Tβ24(c+β1)H2(ξ),′=ddξ,ξ=x-ct,
substituting (3.15) with (3.16) into (3.14) simultaneously yields(3.17)H′′-4(c+β1)-ω24(c+β1)2H-Tβ2ω2(c+β1)2H3+3T2β2216(c+β1)2H5=0.
Applying the homogeneous balance principle, to (3.17) we have r=2,4,6,… this gives n=1,2,3,…. For simplicity, we suppose(3.18)H(ξ)=a0+a1F(ξ).
Substituting (3.18) with (2.5) into (3.17) and equating each of the coefficients of F(ξ) to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:(3.19)a0=0,ω=±2(c+β1)[1-(c+β1)A],a1=±-8Cω3TBβ2.
The general formulae of the solution of Rangwala-Rao equation(3.20)u(x,t)=±-8Cω3TBβ2F(x-ct)e-iωteiψ(x-ct),
with ψ(ξ)=ω/(6B(c+β1))∫[3B+4CF2(ξ)]dξ,ω=±2(c+β1)[1-(c+β1)A]. By selecting the special values of the A,B,Cand the corresponding functionF(ξ)yields(3.21)|u1(x,t)|2=-4(σ-pq)ω3Tσβ2[⁡cosh⁡pq(2A(x-ct))-σ],(3.22)|u2(x,t)|2=-4(σ+pq)ω3Tσβ2[sinh⁡pq⁡(2A(x-ct))-σ].
We omitted the reminder solutions for simplicity. Now, we study the affect of the parameters p and q of the GHF in the intensity u1 given by the (3.21) by choosing different values of the parameters p and q, for A=δ=c=β2=T=1,ω=0.4, and σ=0.9, by means of Figure 7 over the region {(x,t):∥x∥≤5,∥t∥≤5}. We have bright soliton which are increase, decrease and have singularity with the different choose of the two parameters p and q. This means that the parameters p and q affect in the solution structure.Evolutional behaviour of the intensity|u1|2describes the bright soliton:  (a) p=1,q=1;(b) p=1/332,q=1/333 (singular wave solution); (c) p=2.132,q=1.723.
(a)(b)(c)


## 3.3. Chen-Lee-Liu Equation

The Chen-Lee-Liu equation [19](3.23)iut+uxx+iδ|u|2ux=0,
were δis a real constant. Similar as before, we suppose that the exact solution of (3.23)is of the form(3.24)u(x,t)=e-ωteiψ(x-ct)H(x-ct).
Set the relation of ψ,H as(3.25)ψ′(ξ)=c2+δ4H2(ξ),′=ddξ,ξ=x-ct,
substituting (3.25) with (3.24) into (3.23) simultaneously yields(3.26)H′′+14(4ω+c2)H-cδ2H3+3δ216H5=0.
According the homogeneous balance principle, we have r=2,4,6,… and n=1,2,3,…, we let(3.27)H(ξ)=a0+a1F(ξ).
Substituting (3.27) with (2.5) into (3.26) and equating each of the coefficients of F(ξ) to zero, we obtain system of algebraic equations. Solving this system with the aid of Maple, we obtain the following solution:(3.28)a0=0,ω=A-c24,a1=±2-cCδβ2.
The general formulae of the solution of Chen-Lee-Liu equation(3.29)u(x,t)=±2-cCδβ2F(x-ct)e-iωteiψ(x-ct),
with ψ(ξ)=(c/2B)∫[B+2CF2(ξ)]dξ,ω=A-(c2/4). By selecting the special values of the A,B,Cand the corresponding function F(ξ)yields(3.30)|u1(x,t)|2=-2c(σ-pq)δσ[⁡cosh⁡pq(2A(x-ct))-σ],(3.31)|u2(x,t)|2=-2c(σ+pq)δσ[⁡sinh⁡pq(2A(x-ct))-σ].
We omitted the reminder solutions for simplicity. Now, we show the effect of the parameters p and q of the GHF in the intensity u1 given by the (3.30) by choosing different values of the parameters p and q, for A=δ=c=1, and σ=0.9, by means of Figure 8 over the region {(x,t):|x|≤5,|t|≤5}. We have bright soliton which are increase, decrease and have singularity with the different choose of the two parameters p and q. This means that the parameters p and q affect in the solution structure.Evolutional behaviour of the intensity|u1|2describes the bright soliton:  (a) p=1,q=1; (b) p=1/332,q=1/333 (singular wave solution); (c) p=2.132,q=1.723.
(a)(b)(c)Besides the solutions obtained above, the ODE equation (2.5), albeit with different parameters, has been studied in the different context [20–24]. It has been shown that this equation possesses abundant solutions, including Weierstrass function solutions, kink solutions and periodic solutions, and so forth. To the best of our knowledge, some of our explicit solutions are new.Notice that the GHFs are generalization of the hyperbolic functions as stated in the appendix. Also, the two parametersp and q describe the degree of the wave energy localization in the obtained solutions. Referring to [18, 19], it is easy to see that the obtained results in this paper are new and the improved method is valuable.


## 4. Summary and Discussion

In this paper, a set of sub-ODEs are introduced by using GHFs. Also, we have obtained many families of exact traveling wave solutions of a class of nonlinear Schrödinger equations including the generalized Zakharov system, the Rangwala-Rao equation, and the chen-Lee-Liu equation. We study and analyze the properties of these solutions by taking different parameter values of the generalized hyperbolic functionsp and q. As a result, we find that these parameter values affect the solution structure. These solutions include the GHF solution, hyperbolic function solution, qdeformed hyperbolic function solution, and others. The obtained solutions which depend on p and q may be of important significance for the explanation of some practical physical problems. We believe that one can apply this method to many other nonlinear differential equations in mathematical physics.



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*Source: 265348-2012-03-25.xml*