DENMDifferential Equations and Nonlinear Mechanics1687-41021687-4099Hindawi Publishing Corporation39589410.1155/2009/395894395894Research ArticleOn Perturbative Cubic Nonlinear Schrodinger Equations under Complex Nonhomogeneities and Complex Initial ConditionsEl-TawilMagdy A.1El-HazmyMaha A.2GrimshawRoger1Department of Engineering MathematicsFaculty of EngineeringCairo UniversityGiza 12613Egyptcu.edu.eg2Department of MathematicsGirls CollegeMedinaSaudi Arabia2009299200920091202200930052009080720092009Copyright © 2009This 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 perturbing nonlinear Schrodinger equation is studied under general complex nonhomogeneities and complex initial conditions for zero boundary conditions. The perturbation method together with the eigenfunction expansion and variational parameters methods are used to introduce an approximate solution for the perturbative nonlinear case for which a power series solution is proved to exist. Using Mathematica, the symbolic solution algorithm is tested through computing the possible approximations under truncation procedures. The method of solution is illustrated through case studies and figures.

1. Introduction

The nonlinear Schrodinger (NLS) equation is the principal equation to be analyzed and solved in many fields, see , for examples. In the last two decades, there are a lot of NLS problems depending on additive or multiplicative noise in the random case [6, 7] or a lot of solution methodologies in the deterministic case.

Wang et al.  obtained the exact solutions to NLS using what they called the subequation method. They got four kinds of exact solutions of iut+122ux2+α|u|pu+β|u|2pu=0, for which no sign to the initial or boundary conditions type is made. Xu and Zhang  followed the same previous technique in solving the higher-order NLS: iux-12α2ut2+β|u|2u+iε3ut3+iδ|u|2ut+iγu2u*t=0. Sweilam  solved iut+2ux2+q|u|2u=0,t>0,L0<x<L1,

with initial condition u(x,0)=g(x) and boundary conditions ux(L0,t)=ux(L1,t)=0, which gives rise to solitary solutions using variational iteration method. Zhu  used the extended hyperbolic auxiliary equation method in getting the exact explicit solutions to the higher-order NLS: iqz-β12qtt+γ1|q|2q=iβ26qttt+β324qtttt-γ2|q|4q, without any conditions. Sun et al.  solved the NLS: iψt+2ψx2+a|ψ|2ψ=0, with the initial condition ψ(x,0)=ψ0(x) using Lie group method. By using coupled amplitude phase formulation, Porsezian and Kalithasan  constructed the quartic anharmonic oscillator equation from the coupled higher-order NLS. Two-dimensional grey solitons to the NLS were numerically analyzed by Sakaguchi and Higashiuchi . The generalized derivative NLS was studied by Huang et al.  introducing a new auxiliary equation expansion method. Abou Salem and Sulem  studied the effective dynamics of solitons for the generalized Schrodinger equation in a random potential. El-Tawil  considered a nonlinear Schrodinger equation with random complex input and complex initial conditions. Colin et al.  considered three components of nonlinear Schrodinger equations related to the Raman amplification in a plasma. In , Jia-Min and Yu-Lu constructed an appropriate transformations and an extended elliptic subequation approach to find some exact solutions for variable coefficient cubic-quintic nonlinear Schrodinger equation with an external potential.

In this paper, a straight forward solution algorithm is introduced using the transformation from a complex solution to a coupled equations in two real solutions, eliminating one of the solutions to get separate independent and higher-order equations, and finally introducing a perturbative approximate solution to the system.

2. The Linear Case

Consider the nonhomogeneous linear Schrodinger equation: iu(t,z)z+α2u(t,z)t2=F1(t,z)+iF2(t,z),(t,z)(0,T)×(0,), where u(t,z) is a complex valued function which is subjected to I.C.:u(t,0)=f1(t)+if2(t),acomplexvaluedfunction,B.C.:uz(0,z)=0,uz(T,z)=0. Let u(t,z)=ψ(t,z)+iϕ(t,z), ψ,ϕ: real valued functions. Substituting (2.2) in (2.1), the following coupled equations are got as follows: ϕ(t,z)z=α2ψ(t,z)t2+G1(t,z),ψ(t,z)z=α2ϕ(t,z)t2+G2(t,z), where ψ(t,0)=f1(t), ϕ(t,0)=f2(t), G1(t,z)=-F1(t,z), G2(t,z)=F2(t,z), and all corresponding other I.C. and B.C. are zeros.

Eliminating one of the variables in (2.3), one can get the following independent equations: 4ψ(t,z)t4+1α22ψ(t,z)z2=1α2ψ̃1(t,z),4ϕ(t,z)t4+1α22ϕ(t,z)z2=1α2ψ̃2(t,z), where ψ̃1(t,z)=G2z-α2G1t2,ψ̃2(t,z)=αG2t2+G1z. Using the eigenfunction expansion technique , the following solutions for (2.4) are obtained: ψ(t,z)=n=0Tn(z)sin(nπT)t,ϕ(t,z)=n=0τn(z)sin(nπT)t, where Tn(z) and τn(z) can be got through the applications of initial conditions and then solving the resultant second-order differential equations using the method of the variational parameter . The final expressions can be got as follows Tn(z)=(C1+A1(z))sinβnz+(C2+B1(z))cosβnz,τn(z)=(C3+A2(z))sinβnz+(C4+B2(z))cosβnz, where βn=α(nπT)2,A1(z)=1βnψ̃1n(z;n)cos(βn)zdz,B1(z)=-1βnψ̃1n(z;n)sin(βn)zdz,A2(z)=1βnψ̃2n(z;n)cos(βn)zdz,B2(z)=-1βnψ̃2n(z;n)sin(βn)zdz, in which ψ̃1n(z;n)=2T0Tψ̃1(t,z)sin(nπTt)dt,ψ̃2n(z;n)=2T0Tψ̃2(t,z)sin(nπTt)dt. The following conditions should also be satisfied: C2=2T0Tf1(t)sin(nπT)dt-B1(0),C4=2T0Tf2(t)sin(nπT)dt-B2(0). Finally, the following solution is obtained: u(t,z)=(ψ(t,z)+iϕ(t,z)), or |u(t,z)|2=(ψ2(t,z)+ϕ2(t,z)).

3. The Nonlinear Case

Consider the homogeneous nonlinear Schrodinger equation: iu(t,z)z+α2u(t,z)t2+ε|u(t,z)|2u(t,z)+iγu(t,z)=F1(t,z)+iF2(t,z),(t,z)(0,T)×(0,), where u(t,z) is a complex-valued function which is subjected to the initial and boundary conditions (2.2).

Lemma 3.1.

The solution of (3.1) with the constraints (2.2) is a power series in ε if the solution exists.

Proof.

At ε=0, the following linear equation is got: iu(t,z)z+α2u(t,z)t2+iγu(t,z)=F(t,z),(t,z)(0,T)×(0,), which has the solution, see the previous section, u0(t,z)=e-γz(w0(t,z)+iv0(t,z)). Following Pickard approximation, (3.13) can be rewritten as iun(t,z)z+α2un(t,z)t2+iγun(t,z)=F(t,z)-ε|un-1(t,z)|2un-1(t,z),n1. At n=1, the iterative equation takes the following form: iu1(t,z)z+α2u1(t,z)t2+iγu1(t,z)=F(t,z)-ε|u0(t,z)|2u0(t,z)=F(t,z)+εh1(t,z), which can be solved as a linear case with zero initial and boundary conditions. The following general solution can be obtained: w1(t,z)=n=0(T0n+εT1n)sin(nπT)t,v1(t,z)=n=0(τ0n+ετ1n)sin(nπT)t,u1(t,z)=e-γz(w1(t,z)+iv1(t,z))=u1(0)+εu1(1). At n=2, the following equation is obtained: iu2(t,z)z+α2u2(t,z)t2+iγu2(t,z)=F(t,z)-ε|u1(t,z)|2u1(t,z)=F(t,z)+εh2(t,z), which can be solved as a linear case with zero initial and boundary conditions. The following general solution can be obtained: u2(t,z)=u2(0)+εu2(1)+ε2u2(2)+ε3u2(3)+ε4u2(4). Continuing like this, one can get un(t,z)=un(0)+εun(1)+ε2un(2)+ε3un(3)++ε(n+m)un(n+m). As n, the solution (if exists) can be reached as u(t,z)=limnun(t,z). Accordingly, the solution is a power series in ε.

According to the previous lemma, one can assume the solution of (3.1) as the following: u(t,z)=n=0εnun. Let u(t,z)=ψ(t,z)+iϕ(t,z), ψ,ϕ: real valued functions. The following coupled equations are got: ϕ(t,z)z=α2ψ(t,z)t2+ε(ψ2+ϕ2)ψ-γϕ-F1,ψ(t,z)z=-α2ϕ(t,z)t2-ε(ψ2+ϕ2)ϕ-γψ+F2, where ψ(t,0)=f1(t), ϕ(t,0)=f2(t), and all corresponding other I.C. and B.C. are zeros.

As a perturbation solution, one can assume that ψ(t,z)=ψ0+εψ1+ε2ψ2+,ϕ(t,z)=ϕ0+εϕ1+ε2ϕ2+, where ψ0(t,0)=f1(t), ϕ0(t,0)=f2(t), and all corresponding other I.C. and B.C. are zeros.

Substituting (3.12) into (3.11) and then equating the equal powers of ε, one can get the following set of coupled equations: ϕ0(t,z)z=α2ψ0(t,z)t2-γϕ0-F1,ψ0(t,z)z=-α2ϕ0(t,z)t2-γψ0+F2,ϕ1(t,z)z=α2ψ1(t,z)t2-γϕ1+(ψ03+ψ0ϕ02),ψ1(t,z)z=-α2ϕ1(t,z)t2-γψ1-(ϕ03+ϕ0ψ02),ϕ2(t,z)z=α2ψ2(t,z)t2-γϕ2+(3ψ02ψ1+2ψ0ϕ0ϕ1+ψ1ϕ02),ψ2(t,z)z=-α2ϕ2(t,z)t2-γψ2-(3ϕ02ϕ1+2ϕ0ψ0ψ1+ϕ1ψ02), and so on. The prototype equations to be solved are ϕi(t,z)z=α2ψi(t,z)t2-γϕi+Gi(1),i1,ψi(t,z)z=-α2ϕi(t,z)t2-γψi+Gi(2),i1, where ψi(t,0)=δi,0f1(t), ϕi(t,0)=δi,0f2(t) and all other corresponding conditions are zeros. The nonhomogeneity functions Gi(1) and Gi(2) are functions computed from previous steps.

Following the solution algorithm described in the previous section for the linear case, the general symbolic algorithm in Figure 39 can be simulated through the use of a symbolic package, mathematica-5 is used in this paper.

3.1. The Zero-Order Approximation

In this case, u(0)(t,z)=(ψ0+iϕ0), where ψ0(t,z)=e-γzn=0T0nsin(nπT)t,ϕ0(t,z)=e-γzn=0τ0nsin(nπT)t, in which T0n(z)=A01(z)sinβnz+(C02+B01(z))cosβnz,τ0n(z)=A02(z)sinβnz+(C̃02+B02(z))cosβnz, where the constants and variables A01(z),C02,B01(z), A02(z),C̃02, and B02(z) can be got by the aid of Section 2.

The absolute value of the zero-order approximation is got from |u(0)(t,z)|2=ψ02+ϕ02.

3.2. The First-Order Approximation

u(1)(t,z)=u(0)+ε(ψ1+iϕ1), where ψ1(t,z)=e-γzn=0T1n(z)sin(nπT)t,ϕ1(t,z)=e-γzn=0τ1n(z)sin(nπT)t, in which T1n(z)=A11(z)sinβnz+(C12+B11(z))cosβnz,τ1n(z)=A12(z)sinβnz+(C̃12+B12(z))cosβnz, where the constants and variables A11(z), B11(z), A12(z), and B12(z) can be evaluated in a similar manner as the zero-order approximation whereas C̃12=-B12(0) and C12=-B11(0).

The absolute value of the first-order approximation can be got using |u(1)(t,z)|2=|u(0)(t,z)|2+2ε(ψ0ψ1+ϕ0ϕ1)+ε2(ψ12+ϕ12).

3.3. The Second-Order Approximation

u(2)(t,z)=u(1)(t,z)+ε2(ψ2+iϕ2), where ψ2(t,z)=e-γzn=0T2n(z)sin(nπT)t,ϕ2(t,z)=e-γzn=0τ2n(z)sin(nπT)t, in which T2n(z)=A21(z)sinβnz+(C22+B21(z))cosβnz,τ2n(z)=A22(z)sinβnz+(C̃22+B22(z))cosβnz, where the constants and variables A21(z),C22,B21(z), A22(z),C̃22, and B22(z) can be evaluated similarly as the previous approximation.

The absolute value of the second-order approximation can be got using |u(2)(t,z)|2=|u(1)(t,z)|2+2ε2(ψ0ψ2+ϕ0ϕ2)+2ε3(ψ1ψ2+ϕ1ϕ2)+ε4(ψ22+ϕ22).

4. Case Studies

To examine the proposed solution algorithm, see Figure 39, some case studies are illustrated.

4.1. One Input Is OnCase Study 1

Taking f1(t)=0, f2(t)=0, F1(t,z)=1, F2(t,z)=0, and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 1, 2, and 3.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

Case Study 2

Taking f1(t)=0, f2(t)=0, F1(t,z)=0, F2(t,z)=1 and following the solution algorithm, it has been noticed that the same results for the case study 1 are got.

Case Study 3

Taking f1(t)=1, f2(t)=0, F1(t,z)=0, F2(t,z)=0 and following the solution algorithm, the selective results for the first-zero approximation are got in Figures 4, 5, and 6.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice the decrease of the solution level and its higher variability.

Case Study 4

Taking f1(t)=0, f2(t)=1, F1(t,z)=0, F2(t,z)=0 and following the solution algorithm, it has been noticed that the same results for the case study 3 are got:

4.2. Two Inputs Are OnCase Study 5

Taking f1(t)=0, f2(t)=0, F1(t,z)=1, F2(t,z)=1 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 7, 8, and 9.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice that the solution level becomes a little bit higher than that of case study 2.

Case Study 6

Taking f1(t)=1, f2(t)=1, F1(t,z)=0, F2(t,z)=0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 10, 11, and 12.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice the little increase of the solution level than that of case studies 3 and 4.

Case Study 7

Taking f1(t)=1, f2(t)=0, F1(t,z)=1, F2(t,z)=0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 13, 14, and 15.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice the small perturbations at small values of z.

4.3. Three Inputs Are OnCase Study 8

Taking f1(t)=1, f2(t)=1, F1(t,z)=1, F2(t,z)=0 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 16, 17, and 18.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice the increase of the depth of the perturbations.

Case Study 9

Taking f1(t)=1, f2(t)=0, F1(t,z)=1, F2(t,z)=1 and following the solution algorithm, the selective results for the zero-order approximation are got in Figures 19, 20, and 21.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice that the perturbations become smaller than that of case study 8.

4.4. Four Inputs Are OnCase Study 10

Taking the case of f1(t)=1, f2(t)=1, F1=1, F2=1, the following final results for the zero-order approximation are obtained in Figures 22, 23, and 24.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice the little increase in the solution level.

4.5. Exponential NonhomogeneityCase Study 11

Taking the case of f1(t)=1, f2(t)=0, F1(t,z)=e-t, F2(t,z)=0, the following final results for the zero-order approximation are obtained in Figures 25, 26, and 27.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice the low solution level and high perturbations.

4.6. Exponential Initial ConditionCase Study 12

Taking the case of f1(t)=e-t, f2(t)=0, F1(t,z)=1, F2(t,z)=0, the following final results for the zero-order approximation are obtained in Figures 28, 29, and 30.

The zero-order approximation of |u(0)| at α=1, T=10, γ=02 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=02 for different values of t, considering only one term in the series (M=1).

One can notice that a higher solution level is got compared with case study 11 and less perturbations are got at small values of z.

4.7. First-Order ApproximationCase Study 13

Taking the case of f1(t)=1, f2(t)=0, F1(t,z)=1, F2(t,z)=0, the following final results for the zero and first-order approximations are obtained in Figures 31, 32, and 33.

The zero-order approximation of |u(0)| at α=1, T=10, γ=0 with considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=0 for different values of z, considering only one term in the series (M=1).

The zero-order approximation |u(0)| at α=1, T=10, γ=0 for different values of t, considering only one term in the series (M=1).

The first-order approximation of |u(1)| at α=1, T=10, γ=0, ε=01 with considering only one term in the series (M=1).

The first-order approximation of |u(1)| at α=1, T=10, γ=0, ε=05 with considering only one term in the series (M=1).

The first-order approximation of |u(1)| at α=1, T=10, γ=0, ε=1 with considering only one term in the series (M=1).

The first-order approximation |u(1)| at α=1, T=10, γ=0, ε=05 for different values of z, considering only one term in the series (M=1).

The first-order approximation |u(1)| at α=1, T=10, γ=0, ε=05 for different values of t, considering only one term in the series (M=1).

The general solution algorithm.

One can notice the oscillations of the solution level compared with case 7.

One can notice that the solution level increases with the increase of ε.

5. Conclusions

The perturbation technique introduces an approximate solution to the NLS equation with a perturbative nonlinear term for a finite time interval. Using mathematica, the difficult and huge computations problems were fronted to some extent for limited series terms. To get more improved orders, it is expected to face a problem of computation. With respect to the solution level, the effect of the nonhomogeneity is higher than the effect of the initial condition. The initial conditions also cause perturbations for the solution at small values of the space variable. The solution level increases with the increase of ε.

AblowitzM. J.HerbstB. M.SchoberC. M.The nonlinear Schrödinger equation: asymmetric perturbations, traveling waves and chaotic structuresMathematics and Computers in Simulation1997431312MR1438816ZBL0869.68125AbdullaevF. Kh.BronskiJ. C.PapanicolaouG.Soliton perturbations and the random Kepler problemPhysica D20001353-4369386MR1731507ZBL0936.35171FewoS.AtanganaJ.Kenfack-JiotsaA.KofaneT. C.Dispersion-managed solitons in the cubic complex Ginzburg-Landau equation as perturbations of nonlinear Schrodinger equationOptics Communications2005252138149BiswasA.PorsezianK.Soliton perturbation theory for the modified nonlinear Schrödinger's equationCommunications in Nonlinear Science and Numerical Simulation2007126886903MR2309762CazenaveT.LionsP.-L.Orbital stability of standing waves for some nonlinear Schrödinger equationsCommunications in Mathematical Physics1982854549561MR677997ZBL0513.35007DebusscheA.Di MenzaL.Numerical simulation of focusing stochastic nonlinear Schrödinger equationsPhysica D20021623-4131154MR1886808ZBL0988.35156DebusscheA.Di MenzaL.Numerical resolution of stochastic focusing NLS equationsApplied Mathematics Letters2002156661669MR1913267ZBL1001.65006WangM.LiX.ZhangJ.Various exact solutions of nonlinear Schrödinger equation with two nonlinear termsChaos, Solitons and Fractals2007313594601MR2262293ZBL1138.35411XuL.-P.ZhangJ.-L.Exact solutions to two higher order nonlinear Schrödinger equationsChaos, Solitons and Fractals2007314937942MR2262186ZBL1143.35374SweilamN. H.Variational iteration method for solving cubic nonlinear Schrödinger equationJournal of Computational and Applied Mathematics20072071155163MR2332957ZBL1119.65098ZhuS.-D.Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation methodChaos, Solitons and Fractals200734516081612MR2335407ZBL1152.35502SunJ.-Q.MaZ.-Q.HuaW.QinM.-Z.New conservation schemes for the nonlinear Schrödinger equationApplied Mathematics and Computation20061771446451MR2234531ZBL1094.65095PorsezianK.KalithasanB.Cnoidal and solitary wave solutions of the coupled higher order nonlinear Schrödinger equation in nonlinear opticsChaos, Solitons and Fractals2007311188196MR2263277ZBL1138.35409SakaguchiH.HigashiuchiT.Two-dimensional dark soliton in the nonlinear Schrödinger equationPhysics Letters A20063596647651MR2288113HuangD.-J.LiD.-S.ZhangH.-Q.Explicit and exact travelling wave solutions for the generalized derivative Schrödinger equationChaos, Solitons and Fractals2007313586593MR2262292ZBL1139.35092Abou SalemW. K.SulemC.Stochastic acceleration of solitons for the nonlinear Schrödinger equationSIAM Journal on Mathematical Analysis2009411117152MR2505855El-TawilM.The average solution of a stochastic nonlinear Schrodinger equation under stochastic complex non-homogeneity and complex initial conditionsTransactions on Computational Science III20095300New York, NY, USASpringer143170Lecture Notes in Computer ScienceZBL1156.35470ColinM.ColinT.MohtaStability of solitary eaves for a system of nonlinear Schrodinger equations with three wave interactionsto appear in Annals de I'Institut Henri Poincare (c) Nonlinear AnalysisJia-MinZ.Yu-LuL.Some exact solutions of variable coefficient cubic quintic nonlinear Schrodinger equation with an external potentialCommunications in Theoretical Physics2009513391FarlowS. J.Partial Differential Equations for Scientists and Engineers1982New York, NY, USAJohn Wiley & Sonsix+402MR657763PipesL.HarvillL.Applied Mathematics for Engineers and Physicists1970Tokyo, JapanMcGraw-Hill