Abstract
In this study, the nonautonomous variable coefficients Schrödinger equation describes rogon waves in ocean dynamics and optics, is reduced to the nonlinear ordinary differential equation by using the direct similarity technique. The reduced equation is a Riccati equation of Jacobi elliptic wave type solutions. Therefore, many new Jacobi elliptic wave, periodic and hyperbolic wave solutions are obtained for the nonautonomous variable coefficients Schrödinger equation with some constraints between the variable coefficients. Moreover, a rational solution is given. Finally, many plots for the new rogon wave solutions are investigated.
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1 Introduction
Schrödinger equation is a famous equation in many fields of science, therefore, it attracts a lot of mathematicians trying to solve it in all forms. Recently, more attention were focusing on solving the variable coefficients nonlinear version of Schrödinger equation (vc-NLS) for different reasons, the first one, is that solving the vc-NLS equation is covering its constant coefficient version, the second reason, it is rely reflect the real physical situation more than the constant version. Moreover, the vc-NLS equation can cover many physical situations in different branches like optics, ocean dynamics and quantum mechanics ..etc.
In this paper, we are interested in studying and finding new solitary wave solutions for nonautonomous variable coefficients Schrödinger equation (Serkin et al. 2007)
where \(\alpha (z)\) is the group-velocity dispersion, \(\phi (z,t)\) is the linear potential, \(\beta (z)\) is the gain/loss term and \(\gamma (z)\) is the nonlinearity term. Equation (1) describes self-similar waves which can be used for amplification and focusing of spatial solitons in nonlinear optics (Guo et al. 2011; Ponomarenko and Agrawal 2006; Tian et al. 2005; Yan 2010). Additionally, if \(\phi (z,t)\) is a function on z only, then the vc-NLS Eq. (1) represents many physical backgrounds in dusty plasma, nonlinear optics, ocean dynamics and arterial mechanics (El-Shiekh 2019; El-Shiekh and Al-Nowehy 2013; El-Shiekh and Gaballah 2020a, c; El-Shiekh 2019).
2 Direct similarity reduction
Recently, many new methods have been constructed to obtain new solutions for nonlinear partial differential equations like symmetry groups, tanh method, trial equation method, sin-Gordon equation method, etc. (Chen et al. 2019, 2020; Hua et al. 2019; Liu et al. 2019; Mao et al. 2019; Xia et al. 2020; Xu et al. 2020) (El-Sayed et al. 2015, 2014; El-Shiekh 2018, 2021, 2018a; El-Shiekh and Rehab 2018b; El-Shiekh 2017, 2015, 2013; El-Shiekh and Gaballah 2020b; Moatimid et al. 2013; Moussa et al. 2012; Moussa and El-Shiekh 2010, 2011; Moatimid and El Shikh 2008; Moussa and El Shikh 2006; Chen et al. 2021; He et al. 2021; Lü and Chen 2021; Lü et al. 2021; Lü and Ma 2016; Xia et al. 2020; Yin et al. 2020).
In the following we are going to apply one of the similarity techniques, the direct similarity reduction method (El-Shiekh 2019, 2017, 2015, 2013, 2018, 2018a; El-Shiekh and Gaballah 2020b; Moussa and El-Shiekh 2008, 2011), it used to transform the nonlinear partial differential equation into ordinary differential equation as follows:
Assume
where \(\varsigma (z,t)\) and \(\eta (z,t)\) are two arbitrary real functions and \(U(\varsigma )\) is a new dependent real variable.
where \(\prime\) denotes the derivative with respect to \(\varsigma .\) Collect the U coefficient and its derivatives, also, the real and imaginary parts together, assuming that \(e^{i(\eta (z,t))}\ne 0,\) we have
Assume that the imaginary part is finished, then we get
where \(h_{1}(z)\) and \(h_{2}(z)\) are two arbitrary functions in z. According to the direct similarity reduction method (El-Shiekh 2018a; He et al. 2021; Hua et al. 2019; Liu et al. 2019; Lü et al.2021; Moatimid et al. 2013; Moussa and El-Shiekh 2011, Moussa and El Shikh 2008, Moussa et al. 2012), the main target is to transform Eq. (2) into real nonlinear ordinary differential equation with constants coefficients in \(\varsigma\), therefore, by taking the dispersive term \(U^{\prime \prime }\) as a normalized coefficient . We get the following nonlinear system of partial differential equations
where \(\digamma _{1}(\varsigma ),\digamma _{2}(\varsigma )\) and \(\digamma _{3}(\varsigma )\) are three arbitrary real functions in \(\varsigma\) . By solving Eqs. (6)–(9) together using the direct reduction assumptions, we obtain
where k is an integration constant and \(\digamma _{1}(\varsigma )=0,\digamma _{2}(\varsigma )=c_{2}\) and \(\digamma _{3}(\varsigma )=c_{3}\) with \(c_{2}\) and \(c_{3}\) as two arbitrary non-zero constants. Moreover, Eq. (4) transforms into the following nonlinear ordinary differential equation
To solve Eq. (13) , integrate it firstly with respected to \(\varsigma\)
where \(c_{4}\) is an integration constant. Now, two cases arises for solutions of Eq. (14)
Case 1: If \(c_{2},c_{3}\) and \(c_{4}\) are nonzero constants, then Eq. (14) has many Jacobi elliptic wave solutions (El-Shiekh 2019), by using those solutions, many new Jacobi periodic wave solutions are obtained for the inhomogeneous nonlinear Schrödinger equation with variable coefficients as follows:
where the variables \(\eta\) and \(\varsigma\) given by Eqs. (5) and (10) respectively. The Jacobi elliptic wave solutions transformed to hyperbolic if the modulus of it becomes 1 and the following new solutions are given
where the variables \(\eta\) and \(\varsigma\) given by Eqs. (5) and (10) respectively. If the modulus of the Jacobi functions on solutions (15–29) approch zero, the following new periodic wave solutions obtained
where the variables \(\eta\) and \(\varsigma\) given by Eqs. (5) and (10) respectively.
Case 2: If \(c_{2}=0\) and \(c_{4}=0\) but \(c_{3}\ne 0\), then Eq. (14) has the following rational solution
By back substitution from (41) into (2) using Eqs. (5) and (10–12) we get the rational solution
3 Application in nonlinear optics
Yan (2010) defind ”the Rogon waves” as Rough waves if they reappear virtually unaffected in size or shape shortly after interactions therefore, we can say those waves appear in optics as optical rogon waves. In the following we are going to show the dynamical behavior of the intensity \(\left| \Psi \right| ^{2},\) by fixing the parameters \(h_{1}\left( z\right) =\frac{1}{z},\alpha \left( z\right) =z\), and \(k=\frac{1}{2}.\)
We can see that in Figs. 1, 2 and 3, two fixed values for the gain/loss term \(\beta (z)\) are given as tan(z) and \(\frac{-1}{z}\) chosen as positive and negative functions for the gain (+) and the loss (−) sign respectively. In Fig. 1, the rogon wave intensity propagation \(\left| \Psi _{1}\right| ^{2}\)effected with periodic jacobi sn wave and we could see it like a snake in figure (b). Moreover, in Fig. 2, the propagation of \(\left| \Psi _{16}\right| ^{2}\) was like a dark rogon wave in both figures (c) and (d) especially, in figure (d) it was so sharp and high. Finally, in Fig. 3, intensity propagation behavior \(\left| \Psi _{17}\right| ^{2}\) was as a bright rogon wave in both (e) and (f).
4 Conclusion
In this paper, the nonautonomous variable coefficients Schrödinger equation is reduced to nonlinear Riccati equation by using direct similarity reduction method. The Riccati equation has two cases for solution, the first case gives new Jacobi, hyperbolic, and periodic wave solutions. In other case, only rational solutions is obtained. From the obtained solutions we have the following concluding remarks:
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1.
The Direct similarity reduction method is an easy methodology for transforming nonlinear partial differential equations with variable coefficients to nonlinear ordinary differential equation with constant coefficients.
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2.
The obtained solutions cover other solutions obtained before in litrature (Serkin et al. 2007) additionaly, other new solutions were obtained.
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3.
Abundant novel exact travelling wave solutions including periodic Jacobi elliptic waves, solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering and physics fields.
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4.
The obtained first Jacobi elliptic wave solution \(\left| \Psi _{1}\right| ^{2}\) was plotted in Fig. 1 and its limit solution when \(\hbox {m}-{>}1\), corresponding to kink wave solution \(\left| \Psi _{16}\right| ^{2}\) in Fig. 2 so we have shown that the Rogon wave shape was different in both figures. Moreover, for the soliton type orgon waves we take \(\left| \Psi _{17}\right| ^{2}\) and plot it in Fig. 3, finally we can see in all figures how the type of solution could affect the shape of the rogon wave propagation.
References
Chen, S.-J., Yin, Y.-H., Ma, W.-X., Lü, X.: Abundant exact solutions and interaction phenomena of the (2 + 1)-dimensional YTSF equation. Anal. Math. Phys. 9, 2329–2344 (2019). https://doi.org/10.1007/s13324-019-00338-2
Chen, S.J., Lü, X., Ma, W.X.: Bäcklund transformation, exact solutions and interaction behaviour of the (3+1)-dimensional Hirota-Satsuma-Ito-like equation. Commun. Nonlinear Sci. Numer. Simul. 83, 105135 (2020). https://doi.org/10.1016/j.cnsns.2019.105135
Chen, S.J., Lü, X., Tang, X.F.: Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 95, 105628 (2021). https://doi.org/10.1016/j.cnsns.2020.105628
El-Sayed, M.F., Moatimid, G.M., Moussa, M.H.M., El-Shiekh, R.M., El-Satar, A.A.: Symmetry group analysis and similarity solutions for the (2+1)-dimensional coupled Burger’s system. Math. Methods Appl. Sci. 37, 1113–1120 (2014). https://doi.org/10.1002/mma.2870
El-Sayed, M.F., Moatimid, G.M., Moussa, M.H.M., El-Shiekh, R.M., El-Shiekh, F.A.H., El-Satar, A.A.: A study of integrability and symmetry for the (p + 1)th Boltzmann equation via Painlevé analysis and Lie-group method. Math. Methods Appl. Sci. 38, 3670–3677 (2015). https://doi.org/10.1002/mma.3307
El-Shiekh, R.M.: Novel solitary and shock wave solutions for the generalized variable-coefficients (2+1)-dimensional KP-Burger equation arising in dusty plasma. Chinese J. Phys. 71, 341–350 (2021). https://doi.org/10.1016/j.cjph.2021.03.006
El-Shiekh, R.M.: Classes of new exact solutions for nonlinear Schrö dinger equations with variable coefficients arising in optical fiber. Results Phys. (2019). https://doi.org/10.1016/j.rinp.2019.102214
El-Shiekh, R.M.: New similarity solutions for the generalized variable-coefficients KdV equation by using symmetry group method. Arab J. Basic Appl. Sci. 25, 66–70 (2018a). https://doi.org/10.1080/25765299.2018.1449343
El-Shiekh, R.M.: Painlevé rest, Bäcklund transformation and consistent Riccati expansion solvability for two generalised Cylindrical Korteweg-de Vries equations with variable coefficients. Zeitschrift fur Naturforsch. Sect. A J. Phys. Sci. 73, 207–213 (2018b). https://doi.org/10.1515/zna-2017-0349
El-Shiekh, R.M.: Jacobi elliptic wave solutions for two variable coefficients cylindrical Korteweg-de Vries models arising in dusty plasmas by using direct reduction method. Comput. Math. Appl. 75, 1676–1684 (2018)
El-Shiekh, R.M.: Periodic and solitary wave solutions for a generalized variable-coefficient Boiti-Leon-Pempinlli system. Comput. Math. Appl. 73 1414–1420 (2017). https://doi.org/10.1016/j.camwa.2017.01.008
El-Shiekh, R.M.: Direct similarity reduction and new exact solutions for the variable-coefficient kadomtsev-petviashvili equation. Zeitschrift fur Naturforsch. Sect. A J. Phys. Sci. 70 445–450 (2015). https://doi.org/10.1515/zna-2015-0057
El-Shiekh, R.M.: New exact solutions for the variable coefficient modified KdV equation using direct reduction method. Math. Methods Appl. Sci. 36 1–4 (2013). https://doi.org/10.1002/mma.2561
El-Shiekh, R.M., Al-Nowehy, A.-G.: Integral methods to solve the variable coefficient nonlinear Schrödinger equation. Zeitschrift fur Naturforsch. Sect. C J. Biosci. 68A 255–260 (2013). https://doi.org/10.5560/ZNA.2012-0108
El-Shiekh, R.M., Gaballah, M.: Bright and dark optical solitons for the generalized variable coefficients nonlinear Schrödinger equation. Int. J. Nonlinear Sci. Numer. Simul. 21(7–8), 675–681 (2020a). https://doi.org/10.1515/ijnsns-2019-0054
El-Shiekh, R.M., Gaballah, M.: Novel solitons and periodic wave solutions for Davey-Stewartson system with variable coefficients. J. Taibah Univ. Sci. 14, 783–789 (2020b). https://doi.org/10.1080/16583655.2020.1774975
El-Shiekh, R.M., Gaballah, M.: Solitary wave solutions for the variable-coefficient coupled nonlinear Schrödinger equations and Davey-Stewartson system using modified sine-Gordon equation method. J. Ocean Eng. Sci. 5, 180–185 (2020c). https://doi.org/10.1016/j.joes.2019.10.003
Guo, S., Mei, L., Zhou, Y., Li, C.: The extended Riccati equation mapping method for variable-coefficient diffusion-reaction and mKdV equations. Appl. Math. Comput. 217, 6264–6272 (2011). https://doi.org/10.1016/j.amc.2010.12.116
He, X.J., Lü, X., Li, M.G.: Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions to the (3 + 1) -dimensional generalized Kadomtsev-Petviashvili equation. Anal. Math. Phys. 11, 4 (2021). https://doi.org/10.1007/s13324-020-00414-y
Hua, Y.-F., Guo, B.-L., Ma, W.-X., Lü, X.: Interaction behavior associated with a generalized (2+1)-dimensional Hirota bilinear equation for nonlinear waves. Appl. Math. Model. 74, 184–198 (2019). https://doi.org/10.1016/j.apm.2019.04.044
Liu, J.-G., Eslami, M., Rezazadeh, H., Mirzazadeh, M.: Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev-Petviashvili equation. Nonlinear Dyn. 95, 1027–1033 (2019). https://doi.org/10.1007/s11071-018-4612-4
Lü, X., Chen, S.J.: Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types. Nonlinear Dyn. 103, 947–977 (2021). https://doi.org/10.1007/s11071-020-06068-6
Lü, X., Hua, Y.F., Chen, S.J., Tang, X.F.: Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws. Commun. Nonlinear Sci. Numer. Simul. 95, 105612 (2021). https://doi.org/10.1016/j.cnsns.2020.105612
Lü, X., Ma, W.X.: Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation. Nonlinear Dyn. 85, 1217–1222 (2016). https://doi.org/10.1007/s11071-016-2755-8
Mao, J.-J., Tian, S.-F., Zou, L., Zhang, T.-T., Yan, X.-J.: Bilinear formalism, lump solution, lumpoff and instanton/rogue wave solution of a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation. Nonlinear Dyn. 95 3005–3017 (2019). https://doi.org/10.1007/s11071-018-04736-2
Moatimid, G.M., El-Shiekh, R.M., Al-Nowehy, A.-G.A.A.H.: Exact solutions for Calogero-Bogoyavlenskii-Schiff equation using symmetry method. Appl. Math. Comput. 220 455–462 (2013). https://doi.org/10.1016/j.amc.2013.06.034
Moussa, M.H.M., El-Shiekh, R.M.: Direct reduction and exact solutions for generalized variable coefficients 2D KdV equation under some integrability conditions. Commun. Theor. Phys. 55, 551–554 (2011). https://doi.org/10.1088/0253-6102/55/4/03
Moussa, M.H.M., El-Shiekh, R.M.: Similarity solutions for generalized variable coefficients zakharov-kuznetso equation under some integrability conditions. Commun. Theor. Phys. 54, 603–608 (2010). https://doi.org/10.1088/0253-6102/54/4/04
Moussa, M.H.M., El Shikh, R.M.: Auto-Bäcklund transformation and similarity reductions to the variable coefficients variant Boussinesq system. Phys. Lett. Sect. A Gen. At. Solid State Phys. 372 1429–1434 (2008). https://doi.org/10.1016/j.physleta.2007.09.056
Moussa, M.H.M., El Shikh, R.M.: Similarity Reduction and similarity solutions of Zabolotskay-Khoklov equation with a dissipative term via symmetry method. Phys. A Stat. Mech. Appl. 371 325–335 (2006). https://doi.org/10.1016/j.physa.2006.04.044
Moussa, M.H.M., Omar, R.A.K., El-Shiekh, R.M., El-Melegy, H.R.: Nonequivalent similarity reductions and exact solutions for coupled burgers-type equations. Commun. Theor. Phys. 57 1–4 (2012). https://doi.org/10.1088/0253-6102/57/1/01
Ponomarenko, S.A., Agrawal, G.P.: Do solitonlike self-similar waves exist in nonlinear optical media? Phys. Rev. Lett. 97, 013901 (2006). https://doi.org/10.1103/PhysRevLett.97.013901
Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007). https://doi.org/10.1103/PhysRevLett.98.074102
Tian, B., Shan, W.-R., Zhang, C.-Y., Wei, G.-M., Gao, Y.-T.: Transformations for a generalized variable-coefficient nonlinear Schrö dinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation. Eur. Phys. J. B 47, 329–332 (2005). https://doi.org/10.1140/epjb/e2005-00348-3
Xia, J.W., Zhao, Y.W., Lü, X.: Predictability, fast calculation and simulation for the interaction solutions to the cylindrical Kadomtsev-Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 90, 105260 (2020). https://doi.org/10.1016/j.cnsns.2020.105260
Xu, H.-N., Ruan, W.-Y., Zhang, Y., Lü, X.: Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior. Appl. Math. Lett. 99, 105976 (2020). https://doi.org/10.1016/J.AML.2019.07.007
Yan, Z.: Nonautonomous “rogons” in the inhomogeneous nonlinear Schrödinger equation with variable coefficients. Phys. Lett. Sect. A Gen. At. Solid State Phys. 374, 672–679 (2010). https://doi.org/10.1016/j.physleta.2009.11.030
Yin, Y.H., Chen, S.J., Lü, X.: Localized characteristics of lump and interaction solutions to two extended Jimbo-Miwa equations. Chinese Phys. B 29, 120502 (2020). https://doi.org/10.1088/1674-1056/aba9c4
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The author would like to thank the Deanship of Scientific Research, Majmaah University, Saudi Arabia, for funding this work under Project No. R-2021-128.
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El-Shiekh, R.M., Gaballah, M. New rogon waves for the nonautonomous variable coefficients Schrö dinger equation. Opt Quant Electron 53, 431 (2021). https://doi.org/10.1007/s11082-021-03066-9
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DOI: https://doi.org/10.1007/s11082-021-03066-9