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doi: 10.3934/dcdsb.2020292

Modulation approximation for the quantum Euler-Poisson equation

Dongfen Bian 1, , Huimin Liu 2,, and  Xueke Pu 3,

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
2. 

Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China
3. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Huimin Liu

Received  April 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author D. Bian is supported by NSFC under the Contract 11871005. The second author H. Liu is supported by NSFC under the Contract 12001338, the Youth Fund of Shanxi University of Finance and Economics of China under Z06180 and the Science and Technology Innovation Project of Shanxi Province of China under 2020L0256. The last author X. Pu is supported by NSFC under the contract 11871172 and the Natural Science Foundation of Guangdong Province of China under 2019A1515012000

The nonlinear Schrödinger (NLS) equation is used to describe the envelopes of slowly modulated spatially and temporally oscillating wave packet-like solutions, which can be derived as a formal approximation equation of the quantum Euler-Poisson equation. In this paper, we rigorously justify such an approximation by taking a modified energy functional and a space-time resonance method to overcome the difficulties induced by the quadratic terms, resonance and quasilinearity.

Keywords: Modulation approximation, nonlinear Schrödinger equation, Euler-Poisson equation, modified energy, space-time resonance.
Mathematics Subject Classification: Primary: 35M10; Secondary: 35Q35.
Citation: Dongfen Bian, Huimin Liu, Xueke Pu. Modulation approximation for the quantum Euler-Poisson equation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020292
References:
[1]

R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E., in Beijing Lectures in Harmonic Analysis, Princeton University Press, (1986), 3–45. Google Scholar

[2]

W. Craig, Nonstrictly hyperbolic nonlinear systems, Math. Ann., 277 (1987), 213-232.  doi: 10.1007/BF01457361.  Google Scholar

[3]

P. Cummings and C. E. Wayne, Modified energy functionals and the NLS approximation, Discrete Contin. Dyn. Syst., 37 (2017), 1295-1321.  doi: 10.3934/dcds.2017054.  Google Scholar

[4]

W.-P. Düll, Justification of the Nonlinear Schrödinger approximation for a quasilinear wave equation, preprint, arXiv: 1602.08016. Google Scholar

[5]

W.-P. Düll, Justification of the nonlinear Schrödinger approximation for a quasilinear Klein-Gordon equation, Comm. Math. Phys., 355 (2017), 1189-1207.  doi: 10.1007/s00220-017-2966-y.  Google Scholar

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W.-P. Düll and M. Heß, Existence of long time solutions and validity of the Nonlinear Schrödinger approximation for a quasilinear dispersive equation, J. Differ. Equ., 264 (2018), 2598-2632.  doi: 10.1016/j.jde.2017.10.031.  Google Scholar

[7]

W.-P. Düll, G. Schneider and C. E. Wayne, Justification of the Nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth, Arch. Ration. Mech. Anal., 220 (2016), 543-602.  doi: 10.1007/s00205-015-0937-z.  Google Scholar

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P. Germain, Space-time resonance, preprint, arXiv: 1102.1695. doi: 10.5802/jedp.65.  Google Scholar

[9]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimention 3, Ann. Math., 175 (2012), 691-754.  doi: 10.4007/annals.2012.175.2.6.  Google Scholar

[10]

Y. Guo and X. Pu, KdV limit of the Euler-Poisson system, Arch. Ration. Mech. Anal., 211 (2014), 673-710.  doi: 10.1007/s00205-013-0683-z.  Google Scholar

[11]

F. Haas, L. Garcia and J. Goedert, Quantum ion acoustic waves, Phys. Plasmas., 10 (2003), 3858-3866.   Google Scholar

[12]

J. K. Hunter, M. Ifrim, D. Tataru and T. K. Wong, Long time solutions for a Burgers-Hilbert equation via a modified energy method, Proc. Am. Math. Soc., 143 (2015), 3407-3412.  doi: 10.1090/proc/12215.  Google Scholar

[13]

J. Jackson, Classical Electrodynamics, Wiley, 1999. Google Scholar

[14]

L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium, Math. USSR-Sb., 60 (1988), 457-483.  doi: 10.1070/SM1988v060n02ABEH003181.  Google Scholar

[15]

P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A., 122 (1992), 85-91.  doi: 10.1017/S0308210500020989.  Google Scholar

[16]

D. Lannes, Space time resonances, Seminaire Bourbaki, 2011/2012 (2013), 1043-1058.   Google Scholar

[17]

D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in phase space analysis with applications to PDEs, Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer, New York, 84 (2013), 181–213. doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[18]

H. Liu and X. Pu, Long wavelength limit for the quantum Euler-Poisson equation, SIAM J. Math. Anal., 48 (2016), 2345-2381.  doi: 10.1137/15M1046587.  Google Scholar

[19]

H. Liu and X. Pu, Justification of the NLS approximation for the Euler-Poisson equation, Comm. Math. Phys., 371 (2019), 357-398.  doi: 10.1007/s00220-019-03576-4.  Google Scholar

[20]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834-878.  doi: 10.1137/120875648.  Google Scholar

[21]

G. Schneider, Justification of the NLS approximation for the KdV equation using the Miura transformation, Adv. Math. Phys., 2011 (2011), Art. ID 854719, 4 pp. doi: 10.1155/2011/854719.  Google Scholar

[22]

G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem I. The case of zero surface tension, Comm. Pure Appl. Math., 53 (2000), 1475-1535.  doi: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V.  Google Scholar

[23]

G. Schneider and C. E. Wayne, Justification of the NLS approximation for a quasilinear water wave model, J. Differ. Equ., 251 (2011), 238-269.  doi: 10.1016/j.jde.2011.04.011.  Google Scholar

[24]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.  doi: 10.1002/cpa.3160380516.  Google Scholar

[25]

K. Shimizu and Y. Ichikawa, Automodulation of ion oscillation modes in plasma, J. Phys. Soc. Jpn., 33 (1972), 789-792.  doi: 10.1143/JPSJ.33.789.  Google Scholar

[26]

N. Totz, A justification of the modulation approximation to the 3D full water wave problem, Comm. Math. Phys., 335 (2015), 369-443.  doi: 10.1007/s00220-014-2259-7.  Google Scholar

[27]

N. Totz and S. Wu, A rigorous justification of the modulation approximation to the 2D full water wave problem, Comm. Math. Phys., 310 (2012), 817-883.  doi: 10.1007/s00220-012-1422-2.  Google Scholar

show all references

References:
[1]

R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E., in Beijing Lectures in Harmonic Analysis, Princeton University Press, (1986), 3–45. Google Scholar

[2]

W. Craig, Nonstrictly hyperbolic nonlinear systems, Math. Ann., 277 (1987), 213-232.  doi: 10.1007/BF01457361.  Google Scholar

[3]

P. Cummings and C. E. Wayne, Modified energy functionals and the NLS approximation, Discrete Contin. Dyn. Syst., 37 (2017), 1295-1321.  doi: 10.3934/dcds.2017054.  Google Scholar

[4]

W.-P. Düll, Justification of the Nonlinear Schrödinger approximation for a quasilinear wave equation, preprint, arXiv: 1602.08016. Google Scholar

[5]

W.-P. Düll, Justification of the nonlinear Schrödinger approximation for a quasilinear Klein-Gordon equation, Comm. Math. Phys., 355 (2017), 1189-1207.  doi: 10.1007/s00220-017-2966-y.  Google Scholar

[6]

W.-P. Düll and M. Heß, Existence of long time solutions and validity of the Nonlinear Schrödinger approximation for a quasilinear dispersive equation, J. Differ. Equ., 264 (2018), 2598-2632.  doi: 10.1016/j.jde.2017.10.031.  Google Scholar

[7]

W.-P. Düll, G. Schneider and C. E. Wayne, Justification of the Nonlinear Schrödinger equation for the evolution of gravity driven 2D surface water waves in a canal of finite depth, Arch. Ration. Mech. Anal., 220 (2016), 543-602.  doi: 10.1007/s00205-015-0937-z.  Google Scholar

[8]

P. Germain, Space-time resonance, preprint, arXiv: 1102.1695. doi: 10.5802/jedp.65.  Google Scholar

[9]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimention 3, Ann. Math., 175 (2012), 691-754.  doi: 10.4007/annals.2012.175.2.6.  Google Scholar

[10]

Y. Guo and X. Pu, KdV limit of the Euler-Poisson system, Arch. Ration. Mech. Anal., 211 (2014), 673-710.  doi: 10.1007/s00205-013-0683-z.  Google Scholar

[11]

F. Haas, L. Garcia and J. Goedert, Quantum ion acoustic waves, Phys. Plasmas., 10 (2003), 3858-3866.   Google Scholar

[12]

J. K. Hunter, M. Ifrim, D. Tataru and T. K. Wong, Long time solutions for a Burgers-Hilbert equation via a modified energy method, Proc. Am. Math. Soc., 143 (2015), 3407-3412.  doi: 10.1090/proc/12215.  Google Scholar

[13]

J. Jackson, Classical Electrodynamics, Wiley, 1999. Google Scholar

[14]

L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium, Math. USSR-Sb., 60 (1988), 457-483.  doi: 10.1070/SM1988v060n02ABEH003181.  Google Scholar

[15]

P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A., 122 (1992), 85-91.  doi: 10.1017/S0308210500020989.  Google Scholar

[16]

D. Lannes, Space time resonances, Seminaire Bourbaki, 2011/2012 (2013), 1043-1058.   Google Scholar

[17]

D. Lannes, F. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in phase space analysis with applications to PDEs, Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer, New York, 84 (2013), 181–213. doi: 10.1007/978-1-4614-6348-1_10.  Google Scholar

[18]

H. Liu and X. Pu, Long wavelength limit for the quantum Euler-Poisson equation, SIAM J. Math. Anal., 48 (2016), 2345-2381.  doi: 10.1137/15M1046587.  Google Scholar

[19]

H. Liu and X. Pu, Justification of the NLS approximation for the Euler-Poisson equation, Comm. Math. Phys., 371 (2019), 357-398.  doi: 10.1007/s00220-019-03576-4.  Google Scholar

[20]

X. Pu, Dispersive limit of the Euler-Poisson system in higher dimensions, SIAM J. Math. Anal., 45 (2013), 834-878.  doi: 10.1137/120875648.  Google Scholar

[21]

G. Schneider, Justification of the NLS approximation for the KdV equation using the Miura transformation, Adv. Math. Phys., 2011 (2011), Art. ID 854719, 4 pp. doi: 10.1155/2011/854719.  Google Scholar

[22]

G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem I. The case of zero surface tension, Comm. Pure Appl. Math., 53 (2000), 1475-1535.  doi: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V.  Google Scholar

[23]

G. Schneider and C. E. Wayne, Justification of the NLS approximation for a quasilinear water wave model, J. Differ. Equ., 251 (2011), 238-269.  doi: 10.1016/j.jde.2011.04.011.  Google Scholar

[24]

J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.  doi: 10.1002/cpa.3160380516.  Google Scholar

[25]

K. Shimizu and Y. Ichikawa, Automodulation of ion oscillation modes in plasma, J. Phys. Soc. Jpn., 33 (1972), 789-792.  doi: 10.1143/JPSJ.33.789.  Google Scholar

[26]

N. Totz, A justification of the modulation approximation to the 3D full water wave problem, Comm. Math. Phys., 335 (2015), 369-443.  doi: 10.1007/s00220-014-2259-7.  Google Scholar

[27]

N. Totz and S. Wu, A rigorous justification of the modulation approximation to the 2D full water wave problem, Comm. Math. Phys., 310 (2012), 817-883.  doi: 10.1007/s00220-012-1422-2.  Google Scholar

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