doi: 10.3934/dcdsb.2021149
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Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system

South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China

* Corresponding author: Gengen Zhang

Received  October 2020 Early access May 2021

Fund Project: The author is supported by the National Natural Science Foundation of China (Grant No.11701110), the China Postdoctoral Science Foundation (Grant No.2020M682746)

In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.

Citation: Gengen Zhang. Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021149
References:
[1]

W. Bao and X. Dong, Analysis and comparison of numerical methods for Klein-Gordon equation in nonrelativistic limit regime, Numer. Math., 120 (2012), 189-229.  doi: 10.1007/s00211-011-0411-2.  Google Scholar

[2]

W. Bao, X. Dong and X. Zhao, An exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, SIAM J. Sci. Comput., 35 (2013), A2903–A2927. doi: 10.1137/110855004.  Google Scholar

[3]

W. BaoX. Dong and X. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study, 47 (2014), 111-150.  doi: 10.4208/jms.v47n2.14.01.  Google Scholar

[4]

W. Bao and C. Su, Uniform error bounds of a finite difference method for the Zakharov system in the subsonic limit regime via an asymptotic consistent formulation, Multiscale Model. Simul., 15 (2017), 977-1002.  doi: 10.1137/16M1078112.  Google Scholar

[5]

W. Bao and C. Su, A uniformly and optically accurate method for the Zakharov system in the subsonic limit regime, SIAM J. Sci. Comput., 40 (2018), A929–A953. doi: 10.1137/17M1113333.  Google Scholar

[6]

W. Bao and F. Sun, Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM J. Sci. Comput., 26 (2005), 1057-1088.  doi: 10.1137/030600941.  Google Scholar

[7]

W. BaoF. Sun and G. W. Wei, Numerical methods for the generalized Zakharov system, J. Comput. Phys., 190 (2003), 201-228.  doi: 10.1016/S0021-9991(03)00271-7.  Google Scholar

[8]

W. Bao and X. Zhao, A uniformly accurate multiscale time integrator spectral method for the Klein-Gordon-Zakharov system in the high-plasma-frequency limit regime, J. Comput. Phys., 327 (2016), 270-293.  doi: 10.1016/j.jcp.2016.09.046.  Google Scholar

[9]

W. Bao and X. Zhao, Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, J. Comput. Phys., 398 (2019), 108886, 30 pp. doi: 10.1016/j.jcp.2019.108886.  Google Scholar

[10]

Y. Cai and Y. Yuan, Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime, Math. Comp., 87 (2018), 1191-1225.  doi: 10.1090/mcom/3269.  Google Scholar

[11]

Q. Chang, B. Guo and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comput., 64 (1995), 537–553, S7–S11. doi: 10.1090/S0025-5718-1995-1284664-5.  Google Scholar

[12]

B. J. Choi, Global well-posedness of the adiabatic limit of quantum Zakharov system in 1D, preprint, (2019), arXiv: 1906.10807v2. Google Scholar

[13]

A. S. Davydov, Solitons in molecular systems, Phys. Scr., 20 (1979), 387-394.  doi: 10.1088/0031-8949/20/3-4/013.  Google Scholar

[14]

L. M. DegtyarevV. G. Nakhankov and L. I. Rudakov, Dynamics of the formation and interaction of Langmuir solitons and strong turbulence, Sov. Phys. JETP, 40 (1974), 264-268.   Google Scholar

[15]

Y. FangH. Shih and K. Wang, Local well-posedness for the quantum Zakharov system in one spatial dimension, J. Hyperbolic Differ. Equ., 14 (2017), 157-192.  doi: 10.1142/S0219891617500059.  Google Scholar

[16]

Y. FangJ. Segata and T. Wu, On the standing waves of quantum Zakharov system, J. Math. Anal. Appl., 458 (2018), 1427-1448.  doi: 10.1016/j.jmaa.2017.10.033.  Google Scholar

[17]

Y. Fang and K. Nakanishi, Global well-posedness abd scattering for the quantum Zakharov system in $L^2$, Proc. Amer. Math. Soc., 6 (2019), 21-32.  doi: 10.1090/bproc/42.  Google Scholar

[18]

Y. FangH. KuoH. Shih and K. Wang, Semi-classical limit for the quantum Zakharov system, Taiwan. J. Math., 23 (2019), 925-949.  doi: 10.11650/tjm/180806.  Google Scholar

[19]

L. G. Garcia, F. Haas, L. P. L. de Oliveira and J. Goedert, Modified Zakharov equations for plasmas with a quantum correction, Phys. Plasmas, 12 (2005), 012302. doi: 10.1063/1.1819935.  Google Scholar

[20]

L. Gauckler, On a splitting method for the Zakharov system, Numer. Math., 139 (2018), 349-379.  doi: 10.1007/s00211-017-0942-2.  Google Scholar

[21]

R. T. Glassey, Approximate solutions to the Zakharov equations via finite differences, J. Comput. Phys., 100 (1992), 377-383.  doi: 10.1016/0021-9991(92)90243-R.  Google Scholar

[22]

V. Grimm, On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations, Numer. Math., 100 (2005), 71-89.  doi: 10.1007/s00211-005-0583-8.  Google Scholar

[23]

V. Grimm, A note on the Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 102 (2005), 61-66.  doi: 10.1007/s00211-005-0639-9.  Google Scholar

[24]

Y. GuoJ. Zhang and B. Guo, Global well-posedness and the classical limit of the solution for the quantum Zakharov system, Z. Angew. Math. Phys., 64 (2013), 53-68.  doi: 10.1007/s00033-012-0215-y.  Google Scholar

[25] B. GuoZ. GanL. Kong and J. Zhang, The Zakharov System and its Soliton Solutions, Science Press, Beijing, 2016.  doi: 10.1007/978-981-10-2582-2.  Google Scholar
[26]

F. Haas, Variational approach for the quantum Zakharov system, Phys. Plasmas, 14 (2007), 042309. Google Scholar

[27]

F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets, Phys. Rev. E, 79 (2009), 066402. Google Scholar

[28]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer Series on Atomic, Optical, and Plasma Physics, 65, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8.  Google Scholar

[29]

H. HofstätterO. Koch and M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross-Pitaevskii equations, Numer. Math., 127 (2014), 315-364.  doi: 10.1007/s00211-013-0586-9.  Google Scholar

[30]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286.  doi: 10.1017/S0962492910000048.  Google Scholar

[31]

M. Hochbruck and C. H. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 83 (1999), 403-426.  doi: 10.1007/s002110050456.  Google Scholar

[32]

J.-C. JiangC. K. Lin and S. Shao, On one dimensional quantum Zakharov system, Discrete Contin. Dyn. Syst., 36 (2016), 5445-5475.  doi: 10.3934/dcds.2016040.  Google Scholar

[33]

S. JinP. A. Markowich and C. Zheng, Numerical simulation of a generalized Zakharov system, J. Comput. Phys., 201 (2004), 376-395.  doi: 10.1016/j.jcp.2004.06.001.  Google Scholar

[34]

S. Jin and C. Zheng, A Time-splitting spectral method for the generalized Zakharov system in multi-dimensions, J. Sci. Comput., 26 (2006), 127-149.  doi: 10.1007/s10915-005-4929-2.  Google Scholar

[35]

X. Li and L. Zhang, Error estimates of a trigonometric integrator sine pseudo-spectral method for the extended Fisher-Kolmogorov equation, Appl. Numer. Math., 131 (2018), 39-53.  doi: 10.1016/j.apnum.2018.04.010.  Google Scholar

[36]

F. LiaoL. Zhang and S. Wang, Time-splitting combined with exponential wave integrator fourier pseudospectral method for Schrödinger-Boussinesq system, Commun. Nonlinear Sci. Numer. Simulat., 55 (2018), 93-104.  doi: 10.1016/j.cnsns.2017.06.033.  Google Scholar

[37]

M. Marklund, Classical and quantum kinetics of the Zakharov system, Phys. Plasmas, 12 (2005), 082110, 5 pp. doi: 10.1063/1.2012147.  Google Scholar

[38]

V. Masselin, A result of the blow-up rate for the Zakharov system in dimension 3, SIAM J. Math. Anal., 33 (2001), 440-447.  doi: 10.1137/S0036141099363687.  Google Scholar

[39]

A. P. MisraD. Ghosh and A. R. Chowdhury, A novel hyperchaos in the quantum Zakharov system for plasmas, Phys. Lett. A, 372 (2008), 1469-1476.  doi: 10.1016/j.physleta.2007.09.054.  Google Scholar

[40]

A. P. Misra and P. K. Shukla, Pattern dynamics and spatiotemporal chaos in the quantum Zakharov equations, Phys. Rev. E, 79 (2009), 056401. doi: 10.1103/PhysRevE.79.056401.  Google Scholar

[41]

G. C. PapanicolaouC. SulemP. L. Sulem and X. P. Wang, Singular solutions of the Zakharov equations for Langmuir turbulence, Phys. Fluids B, 3 (1991), 969-980.  doi: 10.1063/1.859852.  Google Scholar

[42]

C. Su and X. Zhao, A uniformly first-order accurate method for Klein-Gordon-Zakharov system in simultaneous high-plasma-frequency and subsonic limit regime, J. Comput. Phys., 428 (2021), 110064, 22 pp. doi: 10.1016/j.jcp.2020.110064.  Google Scholar

[43]

A. Taleei and M. Dehghan, Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations, Comput. Phys. Commun., 185 (2014), 1515-1528.  doi: 10.1016/j.cpc.2014.01.013.  Google Scholar

[44]

S. WangT. Wang and L. Zhang, Numerical computations for N-coupled nonlinear Schrödinger equations by split step spectral methods, Appl. Math. Comput., 222 (2013), 438-452.  doi: 10.1016/j.amc.2013.07.060.  Google Scholar

[45]

Y. Wang and X. Zhao, Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, Int. J. Numer. Anal. Mod., 15 (2018), 405-427.   Google Scholar

[46]

Y. XiaY. Xu and C. Shu, Local discontinuous Galerkin methods for the generalized Zakharov system, J. Comput. Phys., 229 (2010), 1238-1259.  doi: 10.1016/j.jcp.2009.10.029.  Google Scholar

[47]

A. XiaoC. Wang and J. Wang, Conservative linearly-implicit difference scheme for a class of modified Zakharov systems with high-order space fractional quantum correction, Appl. Numer. Math., 146 (2019), 379-399.  doi: 10.1016/j.apnum.2019.07.019.  Google Scholar

[48]

S. Yao, J. Sun and T. Wu, Stationary quantum Zakharov systems involving a higher competing perturbation, Electron. J. Differential Equations, 2020 (2020), 18 pp.  Google Scholar

[49]

V. E. Zakharov, Collapse of langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.   Google Scholar

[50]

G. Zhang and C. Su, A conservative linearly-implicit compact difference scheme for the quantum Zakharov system, J. Sci. Comput., 87 (2021), 71. doi: 10.1007/s10915-021-01482-3.  Google Scholar

[51]

X. Zhao, On error estimates of an exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, Numer. Meth. Part. D. E., 32 (2016), 266-291.  doi: 10.1002/num.21994.  Google Scholar

show all references

References:
[1]

W. Bao and X. Dong, Analysis and comparison of numerical methods for Klein-Gordon equation in nonrelativistic limit regime, Numer. Math., 120 (2012), 189-229.  doi: 10.1007/s00211-011-0411-2.  Google Scholar

[2]

W. Bao, X. Dong and X. Zhao, An exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, SIAM J. Sci. Comput., 35 (2013), A2903–A2927. doi: 10.1137/110855004.  Google Scholar

[3]

W. BaoX. Dong and X. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study, 47 (2014), 111-150.  doi: 10.4208/jms.v47n2.14.01.  Google Scholar

[4]

W. Bao and C. Su, Uniform error bounds of a finite difference method for the Zakharov system in the subsonic limit regime via an asymptotic consistent formulation, Multiscale Model. Simul., 15 (2017), 977-1002.  doi: 10.1137/16M1078112.  Google Scholar

[5]

W. Bao and C. Su, A uniformly and optically accurate method for the Zakharov system in the subsonic limit regime, SIAM J. Sci. Comput., 40 (2018), A929–A953. doi: 10.1137/17M1113333.  Google Scholar

[6]

W. Bao and F. Sun, Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM J. Sci. Comput., 26 (2005), 1057-1088.  doi: 10.1137/030600941.  Google Scholar

[7]

W. BaoF. Sun and G. W. Wei, Numerical methods for the generalized Zakharov system, J. Comput. Phys., 190 (2003), 201-228.  doi: 10.1016/S0021-9991(03)00271-7.  Google Scholar

[8]

W. Bao and X. Zhao, A uniformly accurate multiscale time integrator spectral method for the Klein-Gordon-Zakharov system in the high-plasma-frequency limit regime, J. Comput. Phys., 327 (2016), 270-293.  doi: 10.1016/j.jcp.2016.09.046.  Google Scholar

[9]

W. Bao and X. Zhao, Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, J. Comput. Phys., 398 (2019), 108886, 30 pp. doi: 10.1016/j.jcp.2019.108886.  Google Scholar

[10]

Y. Cai and Y. Yuan, Uniform error estimates of the conservative finite difference method for the Zakharov system in the subsonic limit regime, Math. Comp., 87 (2018), 1191-1225.  doi: 10.1090/mcom/3269.  Google Scholar

[11]

Q. Chang, B. Guo and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comput., 64 (1995), 537–553, S7–S11. doi: 10.1090/S0025-5718-1995-1284664-5.  Google Scholar

[12]

B. J. Choi, Global well-posedness of the adiabatic limit of quantum Zakharov system in 1D, preprint, (2019), arXiv: 1906.10807v2. Google Scholar

[13]

A. S. Davydov, Solitons in molecular systems, Phys. Scr., 20 (1979), 387-394.  doi: 10.1088/0031-8949/20/3-4/013.  Google Scholar

[14]

L. M. DegtyarevV. G. Nakhankov and L. I. Rudakov, Dynamics of the formation and interaction of Langmuir solitons and strong turbulence, Sov. Phys. JETP, 40 (1974), 264-268.   Google Scholar

[15]

Y. FangH. Shih and K. Wang, Local well-posedness for the quantum Zakharov system in one spatial dimension, J. Hyperbolic Differ. Equ., 14 (2017), 157-192.  doi: 10.1142/S0219891617500059.  Google Scholar

[16]

Y. FangJ. Segata and T. Wu, On the standing waves of quantum Zakharov system, J. Math. Anal. Appl., 458 (2018), 1427-1448.  doi: 10.1016/j.jmaa.2017.10.033.  Google Scholar

[17]

Y. Fang and K. Nakanishi, Global well-posedness abd scattering for the quantum Zakharov system in $L^2$, Proc. Amer. Math. Soc., 6 (2019), 21-32.  doi: 10.1090/bproc/42.  Google Scholar

[18]

Y. FangH. KuoH. Shih and K. Wang, Semi-classical limit for the quantum Zakharov system, Taiwan. J. Math., 23 (2019), 925-949.  doi: 10.11650/tjm/180806.  Google Scholar

[19]

L. G. Garcia, F. Haas, L. P. L. de Oliveira and J. Goedert, Modified Zakharov equations for plasmas with a quantum correction, Phys. Plasmas, 12 (2005), 012302. doi: 10.1063/1.1819935.  Google Scholar

[20]

L. Gauckler, On a splitting method for the Zakharov system, Numer. Math., 139 (2018), 349-379.  doi: 10.1007/s00211-017-0942-2.  Google Scholar

[21]

R. T. Glassey, Approximate solutions to the Zakharov equations via finite differences, J. Comput. Phys., 100 (1992), 377-383.  doi: 10.1016/0021-9991(92)90243-R.  Google Scholar

[22]

V. Grimm, On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations, Numer. Math., 100 (2005), 71-89.  doi: 10.1007/s00211-005-0583-8.  Google Scholar

[23]

V. Grimm, A note on the Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 102 (2005), 61-66.  doi: 10.1007/s00211-005-0639-9.  Google Scholar

[24]

Y. GuoJ. Zhang and B. Guo, Global well-posedness and the classical limit of the solution for the quantum Zakharov system, Z. Angew. Math. Phys., 64 (2013), 53-68.  doi: 10.1007/s00033-012-0215-y.  Google Scholar

[25] B. GuoZ. GanL. Kong and J. Zhang, The Zakharov System and its Soliton Solutions, Science Press, Beijing, 2016.  doi: 10.1007/978-981-10-2582-2.  Google Scholar
[26]

F. Haas, Variational approach for the quantum Zakharov system, Phys. Plasmas, 14 (2007), 042309. Google Scholar

[27]

F. Haas and P. K. Shukla, Quantum and classical dynamics of Langmuir wave packets, Phys. Rev. E, 79 (2009), 066402. Google Scholar

[28]

F. Haas, Quantum Plasmas: An Hydrodynamic Approach, Springer Series on Atomic, Optical, and Plasma Physics, 65, Springer, New York, 2011. doi: 10.1007/978-1-4419-8201-8.  Google Scholar

[29]

H. HofstätterO. Koch and M. Thalhammer, Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross-Pitaevskii equations, Numer. Math., 127 (2014), 315-364.  doi: 10.1007/s00211-013-0586-9.  Google Scholar

[30]

M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numer., 19 (2010), 209-286.  doi: 10.1017/S0962492910000048.  Google Scholar

[31]

M. Hochbruck and C. H. Lubich, A Gautschi-type method for oscillatory second-order differential equations, Numer. Math., 83 (1999), 403-426.  doi: 10.1007/s002110050456.  Google Scholar

[32]

J.-C. JiangC. K. Lin and S. Shao, On one dimensional quantum Zakharov system, Discrete Contin. Dyn. Syst., 36 (2016), 5445-5475.  doi: 10.3934/dcds.2016040.  Google Scholar

[33]

S. JinP. A. Markowich and C. Zheng, Numerical simulation of a generalized Zakharov system, J. Comput. Phys., 201 (2004), 376-395.  doi: 10.1016/j.jcp.2004.06.001.  Google Scholar

[34]

S. Jin and C. Zheng, A Time-splitting spectral method for the generalized Zakharov system in multi-dimensions, J. Sci. Comput., 26 (2006), 127-149.  doi: 10.1007/s10915-005-4929-2.  Google Scholar

[35]

X. Li and L. Zhang, Error estimates of a trigonometric integrator sine pseudo-spectral method for the extended Fisher-Kolmogorov equation, Appl. Numer. Math., 131 (2018), 39-53.  doi: 10.1016/j.apnum.2018.04.010.  Google Scholar

[36]

F. LiaoL. Zhang and S. Wang, Time-splitting combined with exponential wave integrator fourier pseudospectral method for Schrödinger-Boussinesq system, Commun. Nonlinear Sci. Numer. Simulat., 55 (2018), 93-104.  doi: 10.1016/j.cnsns.2017.06.033.  Google Scholar

[37]

M. Marklund, Classical and quantum kinetics of the Zakharov system, Phys. Plasmas, 12 (2005), 082110, 5 pp. doi: 10.1063/1.2012147.  Google Scholar

[38]

V. Masselin, A result of the blow-up rate for the Zakharov system in dimension 3, SIAM J. Math. Anal., 33 (2001), 440-447.  doi: 10.1137/S0036141099363687.  Google Scholar

[39]

A. P. MisraD. Ghosh and A. R. Chowdhury, A novel hyperchaos in the quantum Zakharov system for plasmas, Phys. Lett. A, 372 (2008), 1469-1476.  doi: 10.1016/j.physleta.2007.09.054.  Google Scholar

[40]

A. P. Misra and P. K. Shukla, Pattern dynamics and spatiotemporal chaos in the quantum Zakharov equations, Phys. Rev. E, 79 (2009), 056401. doi: 10.1103/PhysRevE.79.056401.  Google Scholar

[41]

G. C. PapanicolaouC. SulemP. L. Sulem and X. P. Wang, Singular solutions of the Zakharov equations for Langmuir turbulence, Phys. Fluids B, 3 (1991), 969-980.  doi: 10.1063/1.859852.  Google Scholar

[42]

C. Su and X. Zhao, A uniformly first-order accurate method for Klein-Gordon-Zakharov system in simultaneous high-plasma-frequency and subsonic limit regime, J. Comput. Phys., 428 (2021), 110064, 22 pp. doi: 10.1016/j.jcp.2020.110064.  Google Scholar

[43]

A. Taleei and M. Dehghan, Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations, Comput. Phys. Commun., 185 (2014), 1515-1528.  doi: 10.1016/j.cpc.2014.01.013.  Google Scholar

[44]

S. WangT. Wang and L. Zhang, Numerical computations for N-coupled nonlinear Schrödinger equations by split step spectral methods, Appl. Math. Comput., 222 (2013), 438-452.  doi: 10.1016/j.amc.2013.07.060.  Google Scholar

[45]

Y. Wang and X. Zhao, Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime, Int. J. Numer. Anal. Mod., 15 (2018), 405-427.   Google Scholar

[46]

Y. XiaY. Xu and C. Shu, Local discontinuous Galerkin methods for the generalized Zakharov system, J. Comput. Phys., 229 (2010), 1238-1259.  doi: 10.1016/j.jcp.2009.10.029.  Google Scholar

[47]

A. XiaoC. Wang and J. Wang, Conservative linearly-implicit difference scheme for a class of modified Zakharov systems with high-order space fractional quantum correction, Appl. Numer. Math., 146 (2019), 379-399.  doi: 10.1016/j.apnum.2019.07.019.  Google Scholar

[48]

S. Yao, J. Sun and T. Wu, Stationary quantum Zakharov systems involving a higher competing perturbation, Electron. J. Differential Equations, 2020 (2020), 18 pp.  Google Scholar

[49]

V. E. Zakharov, Collapse of langmuir waves, Sov. Phys. JETP, 35 (1972), 908-914.   Google Scholar

[50]

G. Zhang and C. Su, A conservative linearly-implicit compact difference scheme for the quantum Zakharov system, J. Sci. Comput., 87 (2021), 71. doi: 10.1007/s10915-021-01482-3.  Google Scholar

[51]

X. Zhao, On error estimates of an exponential wave integrator sine pseudospectral method for the Klein-Gordon-Zakharov system, Numer. Meth. Part. D. E., 32 (2016), 266-291.  doi: 10.1002/num.21994.  Google Scholar

Figure 1.  Convergence of $ E $ (left) and $ N $ (right) between the QZS and classical ZS in Example $ 4.1 $
Figure 2.  Inelastic collision between two solitons in Example $ 4.2 $ under case (i)
Figure 3.  Inelastic collision between two solitons in Example $ 4.2 $ under case (ii)
Figure 4.  Inelastic collision between two solitons in Example $ 4.2 $ under case (iii)
Figure 5.  Pattern dynamics of QZS in Example $ 4.3 $, contours of $ |E| $
Figure 6.  Convergence of $ E $ (left) and $ N $ (right) between the QZS and classical ZS in Example $ 4.4 $
Figure 7.  Plot of energy density $ |E|^2 $ and ion density fluctuation $ N $ in Example $ 4.5 $, $ \mu = \nu = 20 $, $ \varepsilon = \frac{1}{2^5} $
Table 1.  Spatial errors of the scheme at $ T = 1 $ for Example 4.1 with different $ \varepsilon $, $ \tau = 10^{-5} $
$ e_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 1.41e-2 6.83e-5 3.78e-9 4.77e-12
$ \varepsilon =\frac{1}{2^3} $ 4.27e-2 1.34e-4 4.80e-9 5.02e-12
$ \varepsilon =\frac{1}{2^5} $ 5.01e-2 2.49e-4 4.94e-9 5.22e-12
$ \varepsilon =\frac{1}{2^{7}} $ 5.06e-2 2.60e-4 7.75e-9 5.25e-12
$ \varepsilon =\frac{1}{2^9} $ 5.06e-2 2.61e-4 8.17e-9 5.17e-12
$ \varepsilon =\frac{1}{2^{11}} $ 5.06e-2 2.61e-4 8.20e-9 5.25e-12
$ n_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 3.11e-2 7.41e-4 6.27e-8 3.30e-12
$ \varepsilon =\frac{1}{2^3} $ 1.07e-1 8.36e-4 4.98e-8 1.83e-12
$ \varepsilon =\frac{1}{2^5} $ 1.07e-1 6.86e-4 1.67e-8 2.38e-12
$ \varepsilon =\frac{1}{2^{7}} $ 1.07e-1 8.50e-4 1.66e-8 2.19e-12
$ \varepsilon =\frac{1}{2^9} $ 1.07e-1 8.61e-4 1.75e-8 2.38e-12
$ \varepsilon =\frac{1}{2^{11}} $ 1.07e-1 8.62e-4 1.75e-8 3.62e-12
$ e_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 1.41e-2 6.83e-5 3.78e-9 4.77e-12
$ \varepsilon =\frac{1}{2^3} $ 4.27e-2 1.34e-4 4.80e-9 5.02e-12
$ \varepsilon =\frac{1}{2^5} $ 5.01e-2 2.49e-4 4.94e-9 5.22e-12
$ \varepsilon =\frac{1}{2^{7}} $ 5.06e-2 2.60e-4 7.75e-9 5.25e-12
$ \varepsilon =\frac{1}{2^9} $ 5.06e-2 2.61e-4 8.17e-9 5.17e-12
$ \varepsilon =\frac{1}{2^{11}} $ 5.06e-2 2.61e-4 8.20e-9 5.25e-12
$ n_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 3.11e-2 7.41e-4 6.27e-8 3.30e-12
$ \varepsilon =\frac{1}{2^3} $ 1.07e-1 8.36e-4 4.98e-8 1.83e-12
$ \varepsilon =\frac{1}{2^5} $ 1.07e-1 6.86e-4 1.67e-8 2.38e-12
$ \varepsilon =\frac{1}{2^{7}} $ 1.07e-1 8.50e-4 1.66e-8 2.19e-12
$ \varepsilon =\frac{1}{2^9} $ 1.07e-1 8.61e-4 1.75e-8 2.38e-12
$ \varepsilon =\frac{1}{2^{11}} $ 1.07e-1 8.62e-4 1.75e-8 3.62e-12
Table 2.  Temporal errors of the scheme at $ T = 1 $ for Example 4.1 with different $ \varepsilon $, $ h = 1/2^4 $
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.85e-3 3.80e-4 7.31e-5 1.38e-5 3.55e-6 8.03e-7
rate - 2.28 2.38 2.41 1.96 2.14
$ \varepsilon =\frac{1}{2^3} $ 6.81e-4 1.69e-4 4.22e-5 1.05e-5 2.63e-6 6.52e-7
rate - 2.01 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 7.56e-4 1.85e-4 4.61e-5 1.15e-5 2.87e-6 7.13e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 7.63e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.14e-3 3.26e-4 7.08e-5 1.56e-5 3.86e-6 9.46e-7
rate - 1.80 2.20 2.18 2.01 2.03
$ \varepsilon =\frac{1}{2^3} $ 2.16e-3 5.15e-4 1.28e-4 3.20e-5 7.98e-6 1.98e-6
rate - 2.07 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 2.33e-3 5.71e-4 1.42e-4 3.54e-5 8.84e-6 2.19e-6
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 2.33e-3 5.74e-4 1.43e-4 3.56e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 2.33e-3 5.74e-4 1.43e-4 3.57e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 2.33e-3 5.74e-4 1.43e-4 3.57e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.85e-3 3.80e-4 7.31e-5 1.38e-5 3.55e-6 8.03e-7
rate - 2.28 2.38 2.41 1.96 2.14
$ \varepsilon =\frac{1}{2^3} $ 6.81e-4 1.69e-4 4.22e-5 1.05e-5 2.63e-6 6.52e-7
rate - 2.01 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 7.56e-4 1.85e-4 4.61e-5 1.15e-5 2.87e-6 7.13e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 7.63e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
rate - 2.03 2.01 2.00 2.00 2.01
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.14e-3 3.26e-4 7.08e-5 1.56e-5 3.86e-6 9.46e-7
rate - 1.80 2.20 2.18 2.01 2.03
$ \varepsilon =\frac{1}{2^3} $ 2.16e-3 5.15e-4 1.28e-4 3.20e-5 7.98e-6 1.98e-6
rate - 2.07 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 2.33e-3 5.71e-4 1.42e-4 3.54e-5 8.84e-6 2.19e-6
rate - 2.03 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 2.33e-3 5.74e-4 1.43e-4 3.56e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 2.33e-3 5.74e-4 1.43e-4 3.57e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 2.33e-3 5.74e-4 1.43e-4 3.57e-5 8.89e-6 2.21e-6
rate - 2.02 2.01 2.00 2.00 2.01
Table 3.  Temporal errors of the scheme at $ T = 1 $ for Example $ 4.4 $ with different $ \varepsilon $, $ h = 1/2^4 $
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 3.23e-2 8.06e-3 2.01e-3 5.02e-4 1.25e-4 3.11e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^3} $ 2.01e-2 5.03e-3 1.26e-3 3.14e-4 7.83e-5 1.94e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 1.92e-2 4.80e-3 1.20e-3 3.00e-4 7.49e-5 1.86e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-4 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-4 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-04 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.01e-2 2.51e-3 6.26e-4 1.56e-4 3.90e-5 9.71e-6
rate - 2.01 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^3} $ 6.06e-3 1.52e-3 3.80e-4 9.49e-5 2.37e-5 5.92e-6
rate - 2.00 2.00 2.00 2.00 2.00
$ \varepsilon =\frac{1}{2^5} $ 6.27e-3 1.57e-3 3.93e-4 9.82e-5 2.45e-5 6.11e-6
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.46e-5 6.10e-6
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.45e-5 6.04e-6
rate - 2.00 2.00 2.00 2.00 2.02
$ \varepsilon =\frac{1}{2^{11}} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.45e-5 6.03e-6
rate - 2.00 2.00 2.00 2.01 2.02
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 3.23e-2 8.06e-3 2.01e-3 5.02e-4 1.25e-4 3.11e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^3} $ 2.01e-2 5.03e-3 1.26e-3 3.14e-4 7.83e-5 1.94e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^5} $ 1.92e-2 4.80e-3 1.20e-3 3.00e-4 7.49e-5 1.86e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-4 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-4 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{11}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-04 7.47e-5 1.85e-5
rate - 2.00 2.00 2.00 2.00 2.01
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.01e-2 2.51e-3 6.26e-4 1.56e-4 3.90e-5 9.71e-6
rate - 2.01 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^3} $ 6.06e-3 1.52e-3 3.80e-4 9.49e-5 2.37e-5 5.92e-6
rate - 2.00 2.00 2.00 2.00 2.00
$ \varepsilon =\frac{1}{2^5} $ 6.27e-3 1.57e-3 3.93e-4 9.82e-5 2.45e-5 6.11e-6
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^{7}} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.46e-5 6.10e-6
rate - 2.00 2.00 2.00 2.00 2.01
$ \varepsilon =\frac{1}{2^9} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.45e-5 6.04e-6
rate - 2.00 2.00 2.00 2.00 2.02
$ \varepsilon =\frac{1}{2^{11}} $ 6.28e-3 1.57e-3 3.94e-4 9.84e-5 2.45e-5 6.03e-6
rate - 2.00 2.00 2.00 2.01 2.02
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