In this paper, the exact travelling wave solution of the Schr¨odinger equation with power law nonlinearity is studied by the Sub-ODE method. It is shown that the method is one of the most effective approaches for finding exact solutions of nonlinear differential equations

Akbari, Mozhgan

Esmaeelnejhad, Parirokh

Taghizadeh, Nasir

English

Digital Commons @ PVAMU (Prairie View A&M Univ)

Applications and Applied Mathematics: An International Journal (AAM) Volume 12 Issue 1 Article 38 6-2017 Application of Bernoulli Sub-ODE Method For Finding Travelling Wave Solutions of Schrodinger Equation Power Law Nonlinearity Nasir Taghizadeh University of Guilan Mozhgan Akbari University of Guilan Parirokh Esmaeelnejhad University of Guilan Follow this and additional works at: https://digitalcommons.pvamu.edu/aam Recommended Citation Taghizadeh, Nasir; Akbari, Mozhgan; and Esmaeelnejhad, Parirokh (2017). Application of Bernoulli Sub-ODE Method For Finding Travelling Wave Solutions of Schrodinger Equation Power Law Nonlinearity, Applications and Applied Mathematics: An International Journal (AAM), Vol. 12, Iss. 1, Article 38. Available at: https://digitalcommons.pvamu.edu/aam/vol12/iss1/38 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. Available athttp://pvamu.edu/aamAppl. Appl. Math.ISSN: 1932-9466Applications and AppliedMathematics:An International Journal(AAM)Vol. 12, Issue 1 (June 2017), pp. 596 – 603Application of Bernoulli Sub-ODE MethodFor Finding Travelling Wave Solutionsof Schrödinger Equation Power Law NonlinearityNasir Taghizadeh, Mozhgan Akbari, & Parirokh EsmaeelnejhadDepartment of Pure MathematicsFaculty of Mathematical SciencesUniversity of GuilanP.O. Box. 1914Rasht, Irantaghizadeh@guilan.ac.ir, m akbari@guilan.ac.ir, parirokh89@gmail.comReceived: June 21, 2016; Accepted: February 17, 2017AbstractIn this paper, the exact travelling wave solution of the Schrödinger equation with power lawnonlinearity is studied by the Sub-ODE method. It is shown that the method is one of the mosteffective approaches for finding exact solutions of nonlinear differential equations.Keywords: Bernoulli Sub-ODE Method; Travelling Wave Solutions; Nonlinear EvolutionEquation; Schrödinger Equation with Power Law NonlinearityMSC 2010 No.: 35C07, 35C08, 35L05, 35Q551. IntroductionThe Schrödinger equation is one of the important partial differential equations and plays a vitalrole in various areas of physical, biological, and engineering sciences. It appears in the studyof nonlinear optics, plasma physics, mathematical bioscience, quantum mechanics, and severalother disciplines. One of the considerable cases of the nonlinear Schrödinger equations is powerlaw nonlinearity which was studied by Wazwaz (2009).5961Taghizadeh et al.: Application of Bernoulli Sub-ODE Method For Finding TravellingPublished by Digital Commons @PVAMU, 2017AAM: Intern. J., Vol. 12, Issue 1 (June 2017) 597Recently, there has been a growing interest in finding exact analytical solutions to nonlinear waveequations by using appropriate techniques. The investigation of exact travelling wave solutionsfor nonlinear partial differential equations (NLPDEs) plays an important role in the study ofnonlinear physical phenomena. Many powerful methods to construct exact solutions of NLPDEshave been established and developed, which lead to one of the most excited advances of nonlinearscience and theoretical physics. In fact, many kinds of exact soliton solutions have been obtainedby using, for example, the homogeneous balance method (see Wang (1995), Zayed et al. (2004)),the hyperbolic tangent expansion method (see Yang et al. (2001), Zayed et al. (2004)), the trialfunction method (see Inc and Evans (2004)), the tanh-method (see Abdou (2007), Fan (2000),Malfliet (1992)), the non-linear transform method (see Hu (2004)), the inverse scattering transform(see Ablowitz and Clarkson (1991)), the Backlund transform (see Miura (1978), Rogers andShadwick (1982)), the Hirotas bilinear method (see Hirota (1973), Hirota and Satsuma (1981)),the generalized Riccati equation method (see Yan and Zhang (2001), Porubov (1996)), the Jacobielliptic functions method (see Liu et al. (2001), Yan (2003)), F-expansion method (see Wangand Li (2005), Wang and Li (2005)), and so on. The rest of the paper is organized as follows.In Section 2, we describe the Bernoulli Sub-ODE method to obtain travelling wave solutions ofnonlinear evolution equations, and give the main steps of the method. In the subsequent section,we will apply the method for finding exact travelling wave solutions for the nonlinear Schrödingerequation with power law nonlinearity. In the last section, some conclusions are presented.2. Description of the Bernoulli Sub-ODE MethodIn this section we present the solutions of the following ODE,G′ + λG = µG2, (1)where λ 6= 0, G = G(ξ). When µ 6= 0, equation (1) is the type of Bernoulli equation, and wecan obtain the solution as:G =1µλ+ deλξ, (2)where d is an arbitrary constant.Suppose that a nonlinear equation, say in three independent variables x, y and t,, is given byP (u, ut, ux, uy, utt, uxt, uyt, uxx, uyy, . . .) = 0, (3)where u = u(x, y, t) is an unknown function, P is a polynomial in u = u(x, y, t) and its variouspartial derivatives, in which the highest order derivatives and nonlinear terms are involved. Byusing the solutions of equation (1), we can construct a serials of exact solutions of nonlinearequations:Step 1. We suppose thatu(x, y, t) = u(ξ), ξ = ξ(x, y, t), (4)the travelling wave variable (4) permits us reducing equation (3) to an ODE for u = u(ξ)P (u, u′, u′′, . . .) = 0. (5)2Applications and Applied Mathematics: An International Journal (AAM), Vol. 12 [2017], Iss. 1, Art. 38https://digitalcommons.pvamu.edu/aam/vol12/iss1/38598 N. Taghizadeh et al.Step 2. Suppose the solution of (5) can be expressed by a polynomial in G as follows:u(ξ) = αmGm + αm−1Gm−1 + · · · , (6)where G = G(ξ) satisfies equation (3), and αm, αm−1, ... are constants to be determined later,αm 6= 0, m can be determined by considering the homogeneous balance between the highestorder derivatives and nonlinear terms appearing in (5).Step 3. Substituting (6) into (5) and using (1), collecting all the terms with the same orderof G together, the left hand side of equation (5) is converted into another polynomial in G.Equating each coefficient of this polynomial to zero yields a set of algebraic equations forαm, αm−1, ..., λ, µ.Step 4. Solving the algebraic equations system in step 3, and by using the solutions of equation(1), we can construct the travelling wave solutions of the nonlinear evolution equations (5).3. Application of the Bernoulli Sub-ODE Method For Schrödinger equation with powerlaw nonlinearityIn this section, we will consider the following NLS equation with power law nonlinearityiwt + wxx + a|w|2n+1w = 0, (7)where a is a real parameter and w = w(x, t) is a complex-valued function of two real variablex, t. We use the transformationw(x, t) = φ(ξ)exp[i(αx+ βt)], ξ = k(x− 2αt), (8)where k, α and β are constants, all of them are to be determined.Substitution (8) into (7), we obtain ordinary differential equation:−(β + α2)φ+ k2φξξ + aφ2n+1 = 0. (9)Due to the difficulty in obtaining the Sub-ODE of equation (9), we suppose a transformationdenoted by u = v1n . Therefore, equation (9) is converted tonv′′v + (1− n)(v′)2 − n2Av2 + n2Bv4 = 0, (10)where A = β+α2k2, B = ak2.Suppose that the solution of (10) can be expressed by a polynomial in G as follows:v(ξ) =L∑i=0biGi, (11)where bi are constants and G = G(ξ) satisfies equation (1).Balancing the order of v4 with the order of (v′)2 in equation (11), we find L = 1. So the solutiontakes on the formv(ξ) = b1G+ b0, b1 6= 0, (12)3Taghizadeh et al.: Application of Bernoulli Sub-ODE Method For Finding TravellingPublished by Digital Commons @PVAMU, 2017AAM: Intern. J., Vol. 12, Issue 1 (June 2017) 599where b0 and b1 are constants to be determined later.Substituting (12) into the ODE (10) and collecting all the terms with the same power of Gtogether and equating each coefficient to zero, yields a set of the simultaneous algebraic equationsas follows:G4 : 2nb21µ2 + b21µ2 − nb21µ2 + n2Bb41 = 0,G3 : −3b21nµλ+ 2b0b1nµ2 − 2b21µλ+ 2b21nµλ+ 4n2Bb0b31 = 0,G2 : nb21λ2 − 3nb0b1µλ+ b21λ2 − nb21λ2 − n2Ab21 + 6b20b21n2B = 0,G1 : nb0b1λ2 − 2n2Ab0b1 + 4b30b1n2B = 0,G0 : −n2Ab20 + n2Bb40 = 0.Solving the algebraic equations above, yields:Case 1b0 =√β + α2α, b1 =√n+ 1βµn, β = b20α− α2. (13)Substituting (13) into (12), we havev1(ξ) =√n+ 1βµnG+√β + α2α, ξ = k(x− 2αt). (14)Combining with equation (2) and considering φ = v1n , we can obtain the travelling wave solutionsof NLS equation with power law nonlinearity as follows:φ1(ξ) =(√n+ 1βµn( 1µλ+ deλξ)+√β + α2α) 1n, ξ = k(x− 2αt), (15)where d is an arbitrary constant.Substituting (15) into (8), we havew1(ξ) =(√n+ 1βµn( 1µλ+ deλξ)+√β + α2α) 1nexp[i(αx+ βt)], ξ = k(x− 2αt).Then we havew1(x, t) =(√n+ 1βµn( 1µλ+ deλk(x−2αt))+√β + α2α) 1nexp[i(αx+ (b20α− α2)t)].Case 2b0 = −√β + α2α, b1 =√n+ 1βµn, β = b20α− α2. (16)Substituting (16) into (12), we havev2(ξ) =√n+ 1βµnG−√β + α2α, ξ = k(x− 2αt). (17)4Applications and Applied Mathematics: An International Journal (AAM), Vol. 12 [2017], Iss. 1, Art. 38https://digitalcommons.pvamu.edu/aam/vol12/iss1/38600 N. Taghizadeh et al.Combining with equation (2) and considering φ = v1n , we can obtain the travelling wave solutionsof NLS equation with power law nonlinearity as follows:φ2(ξ) =(√n+ 1βµn( 1µλ+ deλξ)−√β + α2α) 1n, ξ = k(x− 2αt), (18)where d is an arbitrary constant.Substituting (18) into (8), we havew2(ξ) =(√n+ 1βµn( 1µλ+ deλξ)−√β + α2α) 1nexp[i(αx+ βt)], ξ = k(x− 2αt).Then we havew2(x, t) =(√n+ 1βµn( 1µλ+ deλk(x−2αt))−√β + α2α) 1nexp[i(αx+ (b20α− α2)t)].Case 3b0 =√β + α2α, b1 = −√n+ 1βµn, β = b20α− α2. (19)Substituting (19) into (12), we havev3(ξ) = −√n+ 1βµnG+√β + α2α, ξ = k(x− 2αt). (20)Combining with equation (2) and considering φ = v1n , we can obtain the travelling wave solutionsof NLS equation with power law nonlinearity as follows:φ3(ξ) =(−√n+ 1βµn( 1µλ+ deλξ)+√β + α2α) 1n, ξ = k(x− 2αt), (21)where d is an arbitrary constant.Substituting (21) into (8), we havew3(ξ) =(−√n+ 1βµn( 1µλ+ deλξ)+√β + α2α) 1nexp[i(αx+ βt)], ξ = k(x− 2αt).Then we havew3(x, t) =(−√n+ 1βµn( 1µλ+ deλk(x−2αt))+√β + α2α) 1nexp[i(αx+ (b20α− α2)t)].Case 4b0 = −√β + α2α, b1 = −√n+ 1βµn, β = b20α− α2. (22)Substituting (22) into (20), we havev4(ξ) = −√n+ 1βµnG−√β + α2α, ξ = k(x− 2αt). (23)5Taghizadeh et al.: Application of Bernoulli Sub-ODE Method For Finding TravellingPublished by Digital Commons @PVAMU, 2017AAM: Intern. J., Vol. 12, Issue 1 (June 2017) 601Combining with equation (2) and considering φ = v1n , we can obtain the travelling wave solutionsof NLS equation with power law nonlinearity as follows:φ4(ξ) =(−√n+ 1βµn( 1µλ+ deλξ)−√β + α2α) 1n, ξ = k(x− 2αt), (24)where d is an arbitrary constant.Substituting (24) into (8), we havew4(ξ) =(−√n+ 1βµn( 1µλ+ deλξ)−√β + α2α) 1nexp[i(αx+ βt)], ξ = k(x− 2αt).Then we havew4(x, t) =(−√n+ 1βµn( 1µλ+ deλk(x−2αt))−√β + α2α) 1nexp[i(αx+ (b20α− α2)t)].4. ConclusionIn this paper, we have seen that some new traveling wave solutions of NLS equation with powerlaw nonlinearity are successfully found by using the Bernoulli sub-ODE method. This method isconcise and effective and can be fulfilled by the aid of the mathematical software Maple. Alsothis method can be applied to many other nonlinear problems.AcknowledgmentsThe authors are very grateful to the referees for their valuable suggestions and opinions. 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Application of Bernoulli Sub-ODE Method For Finding Travelling Wave Solutions of Schrodinger Equation Power Law Nonlinearity

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Application of Bernoulli Sub-ODE Method For Finding Travelling Wave Solutions of Schrodinger Equation Power Law Nonlinearity

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