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Modulation approximation for the quantum Euler-Poisson equation
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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[8] |
P. Germain, Space-time resonance, preprint, arXiv: 1102.1695.
doi: 10.5802/jedp.65. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
J. Shatah,
Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.
doi: 10.1002/cpa.3160380516. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[8] |
P. Germain, Space-time resonance, preprint, arXiv: 1102.1695.
doi: 10.5802/jedp.65. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
J. Shatah,
Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.
doi: 10.1002/cpa.3160380516. |
[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. |
[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. |
[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. |
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