Numerical simulation of a generalized nonlinear derivative Schrödinger equation

Shasha Bian; Yitong Pei; Boling Guo; Shasha Bian; Yitong Pei; Boling Guo

doi:10.3934/era.2022159

Electronic Research Archive

2022, Volume 30, Issue 8: 3130-3152. doi: 10.3934/era.2022159

Previous Article Next Article

Research article

Numerical simulation of a generalized nonlinear derivative Schrödinger equation

Shasha Bian ^{1
,
,},
Yitong Pei ^{2
,
,},
Boling Guo ³

1.
Graduate School of China Academy of Engineering Physics, Beijing 100088, China
2.
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
3.
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Academic Editor: Jingzhi Li

Received: 04 May 2022 Revised: 31 May 2022 Accepted: 12 June 2022 Published: 16 June 2022

In this paper, three nonlinear finite difference schemes are proposed for solving a generalized nonlinear derivative Schrödinger equation which exposits the propagation of ultrashort pulse through optical fiber and has been illustrated to admit exact soliton-solutions. Two of the three schemes are two-level ones and the third scheme is a three-level one. It is proved that the two-level schemes only preserve the total mass or the total energy in the discrete sense and the three-level scheme preserves both the total mass and total energy. Furthermore, many numerical results are presented to test the conservative properties and convergence rates of the proposed schemes. Several dynamical behaviors including solitary-wave collisions and the first-order rogue wave solution are also simulated, which further illustrates the effectiveness of the proposed method for the generalized nonlinear derivative Schrödinger equation.
- nonlinear derivative Schrödinger equation,
- finite difference method,
- conservative properties,
- simulation of dynamics
Citation: Shasha Bian, Yitong Pei, Boling Guo. Numerical simulation of a generalized nonlinear derivative Schrödinger equation[J]. Electronic Research Archive, 2022, 30(8): 3130-3152. doi: 10.3934/era.2022159

Related Papers:

Abstract

In this paper, three nonlinear finite difference schemes are proposed for solving a generalized nonlinear derivative Schrödinger equation which exposits the propagation of ultrashort pulse through optical fiber and has been illustrated to admit exact soliton-solutions. Two of the three schemes are two-level ones and the third scheme is a three-level one. It is proved that the two-level schemes only preserve the total mass or the total energy in the discrete sense and the three-level scheme preserves both the total mass and total energy. Furthermore, many numerical results are presented to test the conservative properties and convergence rates of the proposed schemes. Several dynamical behaviors including solitary-wave collisions and the first-order rogue wave solution are also simulated, which further illustrates the effectiveness of the proposed method for the generalized nonlinear derivative Schrödinger equation.

References

[1]	M. Lakshmanan, K. Porsezian, M. Daniel, Effect of discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A., 133 (1988), 483–488. https://doi.org/10.1016/0375-9601(88)90520-8 doi: 10.1016/0375-9601(88)90520-8
[2]	L. H. Wang, K. Porsezian, J. S. He, Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation, Phys. Rev. E., 87 (2013), 053202. https://doi.org/10.1103/PhysRevE.87.053202 doi: 10.1103/PhysRevE.87.053202
[3]	X. Zhang, Y. Chen, Inverse scattering transformation for generalized nonlinear Schrödinger equation, Appl. Math. Lett., 98 (2019), 306–313. https://doi.org/10.1016/j.aml.2019.06.014 doi: 10.1016/j.aml.2019.06.014
[4]	Z. Zhao, L. He, Resonance-type soliton and hybrid solutions of a (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation, Appl. Math. Lett., 122 (2021), 107497. https://doi.org/10.1016/j.aml.2021.107497 doi: 10.1016/j.aml.2021.107497
[5]	Z. Zhao, L. He, Nonlinear superposition between lump waves and other waves of the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation, Nonlinear Dyn., 102 (2021), 555–568. https://doi.org/10.1007/s11071-022-07215-x doi: 10.1007/s11071-022-07215-x
[6]	Z. Zhao, L. He, Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV-mKdV equation, Theor. Math. Phys., 206 (2021), 142–162. https://doi.org/10.1134/S0040577921020033 doi: 10.1134/S0040577921020033
[7]	M. Jeli, B. Samet, C. Vetro, Nonexistence of solutions to higher order evolution inequalities with nonlocal source term on Riemannian manifolds, Complex. Var. Elliptic., (2022) 1–18. https://doi.org/10.1080/17476933.2022.2061474 doi: 10.1080/17476933.2022.2061474
[8]	M. Jeli, B. Samet, C. Vetro, On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain, Adv. Nonlinear Anal., 10 (2021), 1267–1283. https://doi.org/10.1515/anona-2020-0181 doi: 10.1515/anona-2020-0181
[9]	Q. Chang, E. Jia, W. Sun, Difference schemes for solving the generalized nonlinear equation, J. Comput. Phys., 148 (1999), 397–415. https://doi.org/10.1006/jcph.1998.6120 doi: 10.1006/jcph.1998.6120
[10]	M. Dehghan, A. Taleei, A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comput. Phys. Commun., 181 (2010), 43–51. https://doi.org/10.1016/j.cpc.2009.08.015 doi: 10.1016/j.cpc.2009.08.015
[11]	Z. Gao, S. Xie, Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations, Appl. Numer. Math., 61 (2011), 593–614. https://doi.org/10.1016/j.apnum.2010.12.004 doi: 10.1016/j.apnum.2010.12.004
[12]	S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16–42. https://doi.org/10.1016/0021-9991(92)90324-R doi: 10.1016/0021-9991(92)90324-R
[13]	M. Subasi, On the finite difference schemes for the numerical solution of two dimensional Schrödinger equation, Numer. Meth. Part. D. E., 18 (2002), 752–758. https://doi.org/10.1002/num.10029 doi: 10.1002/num.10029
[14]	T. Wang, B. Guo, Unconditional convergence of two conservative compact difference schemes for non-linear Schrödinger equation in one dimension, Sci. Sin. Math., 41 (2011), 207–233. https://doi.org/10.1360/012010-846 doi: 10.1360/012010-846
[15]	T. Wang, X. Zhao, Unconditional $L^{\infty}$ convergence of two compact conservative finite difference schemes for the nonlinear Schrödinger equation in multi-dimensions, Calcolo, 34 (2018). https://doi.org/10.1007/s10092-018-0277-0 doi: 10.1007/s10092-018-0277-0
[16]	T. Wang, B. Guo, Q. Xu, Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J. Comput. Phys., 243 (2013), 382–399. https://doi.org/10.1016/j.jcp.2013.03.007 doi: 10.1016/j.jcp.2013.03.007
[17]	T. Wang, B. Guo, L. Zhang, New conservative difference schemes for a coupled nonlinear Schrödinger system, Appl. Math. Comput., 217 (2010), 1604–1619. https://doi.org/10.1016/j.amc.2009.07.040 doi: 10.1016/j.amc.2009.07.040
[18]	T. Wang, T. Nie, L. Zhang, F. Chen, Numerical simulation of a nonlinearly coupled Schrödinger system: A linearly uncoupled finite difference scheme, Math. Comput. Simulat., 79 (2008), 607–621. https://doi.org/10.1016/j.matcom.2008.03.017 doi: 10.1016/j.matcom.2008.03.017
[19]	T. Wang, A linearized, decoupled, and energy-preserving compact finite difference scheme for the coupled nonlinear Schrödinger equations, Numer. Meth. Part. D. E., 33 (2017), 840–867. https://doi.org/10.1002/num.22125 doi: 10.1002/num.22125
[20]	T. Wang, L. Zhang, Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator, Appl. Math. Comput., 182 (2006), 1780–1794. https://doi.org/10.1016/j.amc.2006.06.015 doi: 10.1016/j.amc.2006.06.015
[21]	S. Xie, G. Li, S. Yi, Compact finite difference schemes with high accuracy for one-dimensional nonlinear Schrödinger equation, Comput. Method. Appl. M., 198 (2009), 1052–1060. https://doi.org/10.1016/j.cma.2008.11.011 doi: 10.1016/j.cma.2008.11.011
[22]	Z. Fei, V. M. Pérez-Grarc${\mathrm{\acute{i}}}$z, L. Vázquez, Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Appl. Math. Comput., 71 (1995), 165–177. https://doi.org/10.1016/0096-3003(94)00152-T doi: 10.1016/0096-3003(94)00152-T
[23]	Q. Chang, E. Jia, W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), 397–415. https://doi.org/10.1006/jcph.1998.6120 doi: 10.1006/jcph.1998.6120
[24]	J. Argyris, M. Haase, An engineer's guide to solitons phenomena: application of the finite element method, Comput. Method. Appl. M., 61 (1987), 71–122. https://doi.org/10.1016/0045-7825(87)90117-4 doi: 10.1016/0045-7825(87)90117-4
[25]	G. Akrivis, V. Dougalis, O. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31–53. https://doi.org/10.1007/BF01385769 doi: 10.1007/BF01385769
[26]	L. R. T. Gardner, G. A. Gardner, S. I. Zaki, Z. El Sahrawi, B-spline finite element studies of the non-linear Schrödinger equation, Comput. Method. Appl. M., 108 (1993), 303–318. https://doi.org/10.1016/0045-7825(93)90007-K doi: 10.1016/0045-7825(93)90007-K
[27]	O. Karakashian, C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method, Math. Comput., 67 (1998), 479–499. https://doi.org/10.1090/S0025-5718-98-00946-6 doi: 10.1090/S0025-5718-98-00946-6
[28]	Y. Xu, C. W. Shu, Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), 72–77. https://doi.org/10.1016/j.jcp.2004.11.001 doi: 10.1016/j.jcp.2004.11.001
[29]	M. Dehghan, D. Mirzaei, Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method, Int. J. Numer. Meth. Eng., 76 (2008), 501–520. https://doi.org/10.1002/nme.2338 doi: 10.1002/nme.2338
[30]	M. Dehghan, D. Mirzaei, The meshless local Petrov-Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation, Eng. Anal. Bound. Elem., 32 (2008), 747–756. https://doi.org/10.1016/j.enganabound.2007.11.005 doi: 10.1016/j.enganabound.2007.11.005
[31]	B. M. Caradoc-Davis, R. J. Ballagh, K. Burnett, Coherent dynamics of vortex formation in trapped Bose-Einstein condensates, Phys. Rev. Lett., 83 (1999), 895–898. https://doi.org/10.1103/PhysRevLett.83.895 doi: 10.1103/PhysRevLett.83.895
[32]	M. Dehghan, A. Taleei, Numerical solution of nonlinear Schrödinger equation by using time-space pseudo-spectral method, Numer. Meth. Part. D. E., 26 (2010), 979–992. https://doi.org/10.1002/num.20468 doi: 10.1002/num.20468
[33]	D. Pathria, J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations, J. Comput. Phys., 87 (1990), 108–125. https://doi.org/10.1016/0021-9991(90)90228-S doi: 10.1016/0021-9991(90)90228-S
[34]	W. Bao, D. Jaksch, P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), 318–342. https://doi.org/10.1016/S0021-9991(03)00102-5 doi: 10.1016/S0021-9991(03)00102-5
[35]	W. Bao, H. Li, J. Shen, A generalized-Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates, Siam. J. Sci. Comput., 31 (2009), 3685–3711. https://doi.org/10.1137/080739811 doi: 10.1137/080739811
[36]	W. Bao, J. Shen, A fourth-order time-splitting Laguerre-Hermite pseudo-spectral method for Bose-Einstein condensates, Siam. J. Sci. Comput., 26 (2005), 2010–2028. https://doi.org/10.1137/030601211 doi: 10.1137/030601211
[37]	M. Thalhammer, High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations, SIAM., 46 (2008), 2022–2038. https://doi.org/10.1137/060674636 doi: 10.1137/060674636
[38]	X. Antoine, C. Besse, V. Rispoli, High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger/Gross-Pitaevskii equations, J. Comput. Phys., 327 (2016), 252–269. https://doi.org/10.1016/j.jcp.2016.09.020 doi: 10.1016/j.jcp.2016.09.020
[39]	Y. Gong, Q. Wang, Y. Wang, J. Cai, A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation, J. Comput. Phys., 328 (2017), 354–370. https://doi.org/10.1016/j.jcp.2016.10.022 doi: 10.1016/j.jcp.2016.10.022
[40]	R. Zhang, J. Zhu, X. Yu, M. Li, A. F. D. Loula, A conservative spectral collocation method for the nonlinear Schrödinger equation in two dimensions, Appl. Math. Comput., 310 (2017), 194–203. https://doi.org/10.1016/j.amc.2017.04.035 doi: 10.1016/j.amc.2017.04.035
[41]	T. Wang, J. Wang, B. Guo, Two completely explicit and unconditionally convergent Fourier pseudo-spectral methods for solving the nonlinear Schrödinger equation, J. Comput. Phys., 404 (2020), 109116. https://doi.org/10.1016/j.jcp.2019.109116 doi: 10.1016/j.jcp.2019.109116
[42]	S. Wang, T. Wang, L. Zhang, Numerical computations for N-coupled nonlinear Schrödinger equations by split step spectral methods, Appl. Math. Comput., 222 (2013), 438–452. https://doi.org/10.1016/j.amc.2013.07.060 doi: 10.1016/j.amc.2013.07.060
[43]	M. Li, J. Zhao, N. Wang, S. Chen, Conforming and nonconforming conservative virtual element methods for nonlinear Schrödinger equation: A unified framework, Comput. Method. Appl. M., 380 (2021), 113793. https://doi.org/10.1016/j.cma.2021.113793 doi: 10.1016/j.cma.2021.113793
[44]	M. Li, Y. Zhao, A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator, Appl. Math. Comput., 338 (2018), 758–773. https://doi.org/10.1016/j.amc.2018.06.010 doi: 10.1016/j.amc.2018.06.010
[45]	L. Wang, M. Li, Galerkin finite element method for damped nonlinear Schrödinger equation, Appl. Numer. Math., 178 (2022), 216–247. https://doi.org/10.1016/j.apnum.2022.03.018 doi: 10.1016/j.apnum.2022.03.018
[46]	X. He, X. Lü, M-lump solution, soliton solution and rational solution to a (3+1)-dimensional nonlinear model, Math. Comput. Simulat., 197 (2022), 327–340. https://doi.org/10.1016/j.matcom.2022.02.014 doi: 10.1016/j.matcom.2022.02.014
[47]	X. Lü, S. Chen, Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: one-lump-multi-stripe and one-lump-multi-soliton types, Nonlinear Dyn., 103 (2021), 947–977. https://doi.org/10.1007/s11071-020-06068-6 doi: 10.1007/s11071-020-06068-6
[48]	P. Debnath, N. Konwar, S. Radenovic, Metric Fixed Point Theory: Applications in Science, Engineering and Behavioural Sciences, Springer Singapore, 2021. https://doi.org/10.1007/978-981-16-4896-0
[49]	V. Todorcevi, Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, Springer Cham, 2019. https://doi.org/10.1007/978-3-030-22591-9
[50]	V. Todorcevi, Subharmonic behavior and quasiconformal mappings, Anal. Math. Phys., 9 (2019), 1211–1225. https://doi.org/10.1007/s13324-019-00308-8 doi: 10.1007/s13324-019-00308-8
[51]	N. Fabiano, S. Radenovi, On scaling of Schrödinger equation and some results for heavy quarks mesons, Nat. Sci., 11 (2021), 49–53. https://doi.org/10.5937/bnsr11-31433 doi: 10.5937/bnsr11-31433
[52]	W. Bao, S. Jin, P. A. Markowich, On time-splitting spectral approximation for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487–524. https://doi.org/10.1006/jcph.2001.6956 doi: 10.1006/jcph.2001.6956
[53]	P. A. Markowich, P. Pietra, C. Pohl, Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit, Numer. Math., 81 (1999), 595–630. https://doi.org/10.1007/s002110050406 doi: 10.1007/s002110050406
[54]	M. Thalhammer, M. Caliari, C. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822–832. https://doi.org/10.1016/j.jcp.2008.10.008 doi: 10.1016/j.jcp.2008.10.008

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)