We present an approach to the bright soliton solution of the nonlinear Schrödinger (NLS) equation from the standpoint of introducing a constant potential term in the equation. We discuss a “nongauge” bright soliton for which both the envelope and the phase depend only on the traveling variable. We also construct a family of generalized NLS equations with solitonic sechpsechp solutions in the traveling variable and find an exact equivalence with other nonlinear equations, such as the Korteveg–de Vries (KdV) and Benjamin–Bona–Mahony (BBM) equations when p=2

Gutierrez-Ruiz, D.

Mancas, S. C.

Reyes, M. A.

Rosu, H. C.

English

Embry-Riddle Aeronautical University

Publications 1-2016 Nongauge Bright Soliton of the Nonlinear Schrodinger (NLS) Equation and a Family of Generalized NLS Equations M. A. Reyes Universidad de Guanajuato D. Gutierrez-Ruiz Universidad de Guanajuato S. C. Mancas Embry-Riddle Aeronautical University, mancass@erau.edu H. C. Rosu Instituto Potosino de Invesitgacion Cientifica y Tecnologica Follow this and additional works at: https://commons.erau.edu/publication  Part of the Mathematics Commons Scholarly Commons Citation Reyes, M. A., Gutierrez-Ruiz, D., Mancas, S. C., & Rosu, H. C. (2016). Nongauge Bright Soliton of the Nonlinear Schrodinger (NLS) Equation and a Family of Generalized NLS Equations. Modern Physics Letters A, 31(3). https://doi.org/10.1142/S0217732316500206 This Article is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Publications by an authorized administrator of Scholarly Commons. For more information, please contact commons@erau.edu. Mod. Phys. Lett. A 31 (2016) 1650020Nongauge bright soliton of the nonlinear Schro¨dinger (NLS) equation and a family ofgeneralized NLS equationsM. A. Reyes and D. Gutie´rrez-RuizDepartamento de F´ısica, Universidad de Guanajuato,DCI Campus Leo´n, Apdo. Postal E143, 37150 Leo´n, Gto., Me´xicoS. C. MancasDepartment of Mathematics,Embry-Riddle Aeronautical University,Daytona Beach, FL. 32114-3900, U.S.A.H. C. Rosu∗IPICyT, Instituto Potosino de Investigacion Cientifica y Tecnologica,Camino a la presa San Jose´ 2055, Col. Lomas 4a Seccio´n, 78216 San Luis Potos´ı, S.L.P., MexicoWe present an approach to the bright soliton solution of the NLS equation from the standpointof introducing a constant potential term in the equation. We discuss a ‘nongauge’ bright soliton forwhich both the envelope and the phase depend only on the traveling variable. We also constructa family of generalized NLS equations with solitonic sechp solutions in the traveling variableand find an exact equivalence with other nonlinear equations, such as the Korteveg-de Vries andBenjamin-Bona-Mahony equations when p = 2.Keywords: Nonlinear Schro¨dinger equation, bright soliton, KdV equation, BBM equation1. IntroductionFor about fifty years now, the one-dimensional cubicNLS equation has been overwhelmingly used for the de-scription of wave propagation in nonlinear optics, fluids,and plasmas [1] and arguably it is the best studied nonlin-ear equation. On the other hand, linearly-extended formsof NLS equations are used by the majority of authors todescribe the Bose-Einstein condensates [2]. Perhaps thesimplest form of the latter class is [3]i~dqdt= − ~22md2qdx2− γ (|q|2 − q20) q , (1)where q0 = constant, γ > 0 is the nonlinear strengthparameter, and ~ and m are the Planck constant dividedby 2pi and the mass of the condensate, respectively. Thestandard NLS equation is obtained from (1) through thetrivial change of dependent variable (gauge)q(x, t) = eiγq20t/~ ψ(x, t) . (2)In this work, we have two goals. The first one isconcerned with the localized solutions of standard andlinearly-extended NLS equations. These localized solu-tions, of which solitons are a subset, have zero boundaryconditions as x → ±∞. The NLS soliton solutions areof course well known but we will point here to what wethink to be a rather ignored feature of these solutions,∗Electronic address: hcr@ipicyt.edu.mxnamely that there exist a bright soliton of equations oftype (1) which cannot be obtained from the standardbright soliton by the gauging (2). Secondly, we constructa family of generalized NLS equations with soliton so-lutions and find that it is possible to obtain an exactequivalence of these equations with the Korteveg-deVries(KdV) and Benjamin-Bona-Mahony (BBM) equations.2. The nongauge bright solitonBy applying (2) to (1), we obtain the standard NLSequation with no potentiali~∂ψ∂t= − ~22m∂2ψ∂x2− γ|ψ|2ψ , (3)The localized solution of this standard NLS equation isgiven byψ(x, t) =√2Aγsech[√2mA~2(z + z0)]ei[mv~z+φ0+αt] .(4)This solution has been derived by Remoissenet in a dif-ferent context and notation [4], see also below, and canbe obtained as a particular case from the solution givenby Katyshev et al [5] for NLS equations with arbitrarypower nonlinearities of the type |ψ|νψ, ν > 1. In (4), Ais an energy-like quantity given byA = ~α− mv22,z = x− vt is the traveling variable with speed v, z0 andφ0 are real integration constants, while α is an angular2frequency parameter used to obtain the localized solu-tion. Since γ > 0, and for A > 0, (4) can be identifiedwith a bright soliton solution [6]. Soliton (4) is plottedin Fig. 1 for the values of the parameters given in thecaption, where the oscillatory real and imaginary partsare displayed as well as its squared amplitude.By including a constant potential V0, equation (3)changes toi~∂ψ∂t= − ~22m∂2ψ∂x2− γ|ψ|2ψ + V0ψ , (5)whose solution can be obtained by using the ansatzψ(x, t) = A(z)ei[φ(z)+αt] , (6)where A and φ are functions only of the traveling variablez. Using (6) in (5) and separating the real and imaginaryparts we obtain the system of equationsd2Adz2−A(dφdz)2−2m(~α+ V0)~2A+2mγ~2A3+2mv~Adφdz= 0 ,(7)Ad2φdz2+ 2dAdz(dφdz− mv~)= 0 . (8)Employing a linear phase φ(z) = mv~z+φ0, which satisfies(8) identically, the amplitude equation (7) becomesd2Adz2− 2mB~2A+2mγ~2A3 = 0 . (9)This elliptic equation is easily integrated by assumingzero boundary conditions to obtainψ(x, t) =√2Bγsech[√2mB~2(z + z0)]ei[mv~z+φ0+αt] ,(10)whereB = ~α+∆Q , ∆Q = V0 − 12mv2 .From (10), one can see that setting V0 = 0 leads to solu-tion (4).We stress that traveling soliton wave solutions withoutphase factors that depend only on x or t, cannot occur for(3), while in the constant potential case this is possible.Indeed, one can see that setting α = 0 in the solution (4)destroys the solitonic feature of the amplitude by turningit into a secant function, while this is possible in the caseof the solution (10) because the potential V0 maintainsthe solitonic profile of the amplitude.In other words, only linearly-extended NLS equationscan have traveling soliton solutions of the formψ(x, t) =√2∆Qγsech[√2m∆Q~2(z + z0)]ei[mv~z+φ0] .(11)FIG. 1: (Color online) Plots of the oscillatory real and imag-inary parts of the wave function for α = 2, for the case ofdimensionless variables, with z0 = φ0 = 0, and ~ = γ = m =1, v = ∓1 , and its squared modulus. For v = −1, the waveis traveling to the left, and for v = 1 to the right.This solution, which is plotted in Fig. (2), has an inter-esting particular feature that does not occur in the caseof the standard solution. It is the solution for a nonzero potential which depends exclusively on the travelingvariable in both amplitude and phase and in this senseit is also different from the solution (2) since it cannotbe obtained from the standard solution by applying thegauge. Therefore we call (11) a nongauge bright soliton.Both its amplitude and wave width depend on the excesspotential energy ∆Q with respect to the kinetic energywhich keeps into motion a classical particle of mass m atconstant speed v.3. One-parameter family of generalized NLSequationsWe turn now to the second goal of this work, and showthat there exists a one-parameter family of generalizedNLS equations with solitonic solutions, where the param-eter of the family is the exponent of the nonlinear term.The NLS equations we consider are linearly extended bya constant potential as previously but the nonlinearity isexpressed as an arbitrary function f(|ψ|)i~∂ψ∂t= − ~22m∂2ψ∂x2− γf(|ψ|)ψ + V0ψ . (12)3FIG. 2: (Color online) Plots of the real and imaginary partsas well as the squared modulus of the traveling soliton solu-tion (11) for the case of dimensionless variables and the sameparameters as in the previous figure and V0 = 1. In this solu-tion, the real and imaginary parts of the wave function havealmost fixed shapes during the propagation.We pose the same problem of whether the constant po-tential term allows us to find soliton solutions. Let us fixthe phase as before, φ(z) = mv~z + φ0, and assume thatthe amplitude has the formA(z) = c1 sechp(c2z) , (13)where c1 and c2 are real constants, and p is a strictly pos-itive integer or rational number. Recently, this type ofansatz has been used in an alternative form of the varia-tional approximation for solitons of equations with higherdegree of polynomial nonlinearities [7, 8]. Differentiatingtwice, one can find that A(z) satisfiesd2Adz2− p2c22A+ p(p+ 1)(c2c1/p1)2A2pA = 0 . (14)On the other hand, if we insert the ansatz (6) togetherwith the linear phase φ(z), into (12), we find that theamplitude also satisfiesd2Adz2− 2mB~2A+2m~2γf(A)A = 0 . (15)Clearly, if we want to relate equations (14) and (15),we must have thatc2 =√2mBp2~2(16)andf(|ψ|) ≡ f(A) = p+ 1pBγ(Ac1) 2p. (17)Since p is arbitrary, we have constructed an infinitenumber of generalized NLS equations with soliton solu-tions. This family includes the NLS equation of constantpotential for p = 1 and other nonlinear equations withsoliton solutions for other values of p, as we shall shownext for the case p = 2.4. The p = 2 generalized NLS equation: Connec-tion with KdV and BBM equationsA lot of work has been developed in order to find con-nections between the standard NLS equation and othernonlinear differential equations that possess solitonic so-lutions. That is the case of the KdV [9] and the relatedBBM equation [10], which are well-established mathe-matical models of waves on shallow water surfaces andknown to be connected to the standard NLS equationonly by approximations [11–14]. Here, we show thatthere exists an exact relationship between these equationsand the equation of parameter p = 2 from the family ofgeneralized NLS equationsd2Adz2− 4c22A+ 6(c2√c1)2A2 = 0 . (18)For the KdV equationBt + 6BBx +Bxxx = 0 , (19)which, additionally to shallow water waves, is used inelectric transmission lines, optical fibers, and other fields[4], we use the traveling wave ansatz (6) with zero phaseand integrate once to findd2Bdz2+ 3B2 − vB = 0 , (20)where the integration constant has been taken as zero.Assuming B and Bz both going to zero when z → ±∞,one can get the well-known solitonB(x, t) =v2sech2[√v2(x− vt)], (21)which is plotted in Fig. 3 (top).4Equations (20) and (18) are the same whenc1 ≡ v2 c2 ≡√v2~22m ≡ 1 , (22)which exactly match the solution (13) with (10) if further(16) and (17) are taken into account.Finally, we refer to the regularized long wave regimeof the KdV equation, or the BBM equation. It can beviewed as an alternative to KdV by replacing the dis-persion term Bxxx with −Bxxt to reflect a bounded dis-persion relation, see [10, 15]. Thus, we write the BBMequation as followsCt + 6CCx − Cxxt = 0 . (23)FIG. 3: (Color online) Plots of KdV (top) and BBM (bottom)solutions, for v = 2. The width of the soliton is bigger by√2in the BBM case.Changing again to the traveling variable, the solutionis found from the elliptic equationd2Cdz2− 3vC2 + C = 0 , (24)where again the integration constant is zero. Then viathe identificationsc1 ≡ v2 c2 ≡ 12 ~22m ≡ −1 (25)in (13) and (18), and zero boundary conditions as above,we obtain the known solitonC(x, t) =v2sech2[12(x− vt)], (26)see Fig. 3 (bottom).For complicated cases of the BBM equation modifiedby viscosity, where the modified elliptic equation (20) issolved via the Weierstrass ℘ functions, see [16] togetherwith the references cited therein.5. ConclusionWe have discussed the ‘nongauge’ bright soliton of aNLS equation extended by a constant potential that can-not occur for the standard NLS equation (3). A one-parameter family of generalized NLS equations with con-stant potential having the family parameter determinedby the order of nonlinearity is also introduced in thiswork. Furthermore, it is shown that the KdV and BBMsolutions represent the amplitude of the solution of gen-eralized NLS equation with nonlinearity f(|ψ|) ≡ f(A).This points to the possibility of finding relationships be-tween these generalized NLS equations and other nonlin-ear equations with soliton solutions.Acknowledgments: The second author wishes tothank the Red de Gravitacio´n CONACyT for a fellow-ship.[1] B. Malomed, Nonlinear Schro¨dinger Equations, Ency-clopedia of Nonlinear Science, A. Scott ed., New York:Routledge, 639-643 (2005).[2] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa-tion, Oxford, U.K., Clarendon Press (2003); C. J. Pethickand H. Smith, Bose-Einstein Condensation in DiluteGases, Cambridge University Press, 2ndEd. (2008).[3] G. Biondini and G. Kovacˇicˇ, J. Math. Phys. 55, 031506(2014).[4] M. Remoissenet, Waves Called Solitons, Springer, 3rdEd.(1999), pp. 98-100.[5] Yu. V. Katyshev, N. V. Makhaldiani, and V. G.Makhankov, Phys. Lett. 66A, 456 (1978).[6] D. Anderson, Phys. Rev. A 27, 3135-3145 (1983).[7] J. Zeng and B. A. Malomed Phys. Rev. E 86, 036607(2012).[8] A. S. Reyna, B. A. Malomed, and C. B. de Araujo, Phys.Rev. A 92, 033810 (2015).[9] D.J. Korteweg and G. de Vries, Phil. Mag. 39, 422-443(1895).[10] T.B. Benjamin, J.L. Bona, and J.J. Mahony, Phil. Trans.Royal Soc. London Series A, Math. and Phys. Sci. 272,48-78 (1972).[11] Y. S. Kivshar, D. Anderson, and M. Lisak, Phys. Scr. 47,679-681 (1993).[12] F. Be´thuel, P. Gravejat, J.-C. Saut, and D. Smets, Intern.5Math. Res. Notices 2009, 2700-2748 (2009).[13] D. Chiron and F. Rousset, SIAM J. Math. Anal., 42(1),64-96 (2010).[14] J. P. Boyd and G.-Y. Chen, Math. Comput. Simulat. 55,317-328 (2001).[15] D.H. Peregrine, J. Fluid Mech. 25, 321-330 (1966).[16] S. C. Mancas, G. Spradlin, and H. Khanal, J. Math.Phys. 54, 081502 (2013).

Nongauge Bright Soliton of the Nonlinear Schrodinger (NLS) Equation and a Family of Generalized NLS Equations

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Nongauge Bright Soliton of the Nonlinear Schrodinger (NLS) Equation and a Family of Generalized NLS Equations

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