• Previous Article
    Analytical study of resonance regions for second kind commensurate fractional systems
  • DCDS-B Home
  • This Issue
  • Next Article
    A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations
doi: 10.3934/dcdsb.2020292

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

* 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.

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

[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. DüllG. 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. GermainN. 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. HaasL. Garcia and J. Goedert, Quantum ion acoustic waves, Phys. Plasmas., 10 (2003), 3858-3866.   Google Scholar

[12]

J. K. HunterM. IfrimD. 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. KirrmannG. 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. DüllG. 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. GermainN. 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. HaasL. Garcia and J. Goedert, Quantum ion acoustic waves, Phys. Plasmas., 10 (2003), 3858-3866.   Google Scholar

[12]

J. K. HunterM. IfrimD. 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. KirrmannG. 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

[1]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

[2]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2020393

[3]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003

[4]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[5]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[6]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[7]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[8]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284

[9]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[10]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[11]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[12]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[13]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[14]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125

[15]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[16]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[17]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[18]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[19]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[20]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033

2019 Impact Factor: 1.27

Article outline

[Back to Top]