June  2012, 32(6): 2233-2251. doi: 10.3934/dcds.2012.32.2233

On the fluid dynamical approximation to the nonlinear Klein-Gordon equation

1. 

Department of Applied Mathematics and, Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, Hsinchu 30010, Taiwan

2. 

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 OWA, United Kingdom

Received  February 2011 Revised  December 2011 Published  February 2012

We study the nonrelativistic, semiclassical and nonrelativistic-semiclassical limits of the (modulated) nonlinear Klein-Gordon equations from its hydrodynamical structure via WKB analysis. The nonrelativistic-semiclassical limit is proved rigorously by modulated energy method.
Citation: Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233
References:
[1]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Commun. in Partial Differential Equations, 25 (2000), 737. doi: 10.1080/03605300008821529. Google Scholar

[2]

R. Carles, Geometric optics and instability for semi-classical Schrödinger equations,, Arch. Rational Mech. Anal., 183 (2007), 525. doi: 10.1007/s00205-006-0017-5. Google Scholar

[3]

Q. Chang, Y.-S. Wong and C.-K. Lin, Numerical computation for long-wave short-wave interaction equations in semi-classical limit,, Journal of Computational Physics, 227 (2008), 8489. doi: 10.1016/j.jcp.2008.05.015. Google Scholar

[4]

T. Colin and A. Soyeur, Some singular limits for evolutionary Ginzburg-Landau equations,, Asymptotic Analysis, 13 (1996), 361. Google Scholar

[5]

B. Desjardins, C.-K. Lin and T.-C. Tso, Semiclassical limit of the derivative nonlinear Schrödinger equation,, Math. Models Methods Appl. Sci., 10 (2000), 261. Google Scholar

[6]

B. Desjardins and C.-K. Lin, On the semiclassical limit of the general modified NLS equation,, J. Math. Anal. Appl., 260 (2001), 546. Google Scholar

[7]

N. Ercolani and R. Montgometry, On the fluid approximation to a nonlinear Schrödinger equation,, Physics Letters A, 180 (1993), 402. doi: 10.1016/0375-9601(93)90290-G. Google Scholar

[8]

I. Gasser, C.-K. Lin and P. Markowich, A review of dispersive limit of the (non)linear Schrödinger-type equation,, Taiwanese J. of Mathematics., 4 (2000), 501. Google Scholar

[9]

V. L. Ginzburg and L. P. Pitaevskiĭ, On the theory of superfluidity,, Sov. Phys. JETP, 34(7) (1958), 858. Google Scholar

[10]

E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time,, Proc. Amer. Math. Soc., 126 (1998), 523. doi: 10.1090/S0002-9939-98-04164-1. Google Scholar

[11]

S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy,, Comm. Pure Appl. Math., 52 (1999), 613. doi: 10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L. Google Scholar

[12]

J.-H. Lee and C.-K. Lin, The behavior of solutions of NLS equation of derivative type in the semiclassical limit,, Chaos, 13 (2002), 1475. doi: 10.1016/S0960-0779(01)00157-6. Google Scholar

[13]

J.-H. Lee, C.-K. Lin and O. K. Pashaev, Shock waves, chiral solitons and semiclassical limit of one-dimensional anyons,, Chaos, 19 (2004), 109. doi: 10.1016/S0960-0779(03)00084-5. Google Scholar

[14]

H.-L. Li and C.-K. Lin, Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems,, Electronic Journal of Differential Equations, 2003 (). Google Scholar

[15]

H.-L. Li and C.-K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model of semiconductors,, Commun. Math. Phys., 256 (2005), 195. doi: 10.1007/s00220-005-1316-7. Google Scholar

[16]

C.-K. Lin and Y.-S. Wong, Zero-dispersion limit of the short-wave-long-wave interaction equations,, Journal of Differential Equations, 228 (2006), 87. doi: 10.1016/j.jde.2006.03.027. Google Scholar

[17]

C.-K. Lin and K.-C. Wu, Singular limits of the Klein-Gordon equation,, Arch. Rational Mech. Anal., 197 (2010), 689. doi: 10.1007/s00205-010-0324-8. Google Scholar

[18]

F.-H. Lin and J. X. Xin, On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation,, Commun. in Math. Phys., 200 (1999), 249. doi: 10.1007/s002200050529. Google Scholar

[19]

T.-C. Lin and P. Zhang, Incompressible and compressible limit of coupled systems of nonlinear Schrödinger equations,, Commun. Math. Phys., 266 (2006), 547. Google Scholar

[20]

S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations,, Mathematische Annalen, 322 (2002), 603. doi: 10.1007/s002080200008. Google Scholar

[21]

A. Messiah, "Quantum Mechanics,", Vol. 1 & 2, (1999). Google Scholar

[22]

H. M. Pilkuhn, "Relativistic Quantum Mechanics,", Texts and Monographs in Physics, (2003). Google Scholar

[23]

M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations,, Commun. in Partial Differential Equations, 27 (2002), 2311. doi: 10.1081/PDE-120016159. Google Scholar

[24]

J. Shatah and M. Struwe, "Geometric Wave Equations,", Courant Lecture Notes in Mathematics, (1998). Google Scholar

[25]

C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse,", Appl. Math. Sci., 139 (1999). Google Scholar

[26]

W. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Mathematics, 73 (1989). Google Scholar

[27]

K.-C. Wu, Convergence of the Klein-Gordon equation to the wave map equation with magnetic field,, J. Math. Anal. Appl., 365 (2010), 584. doi: 10.1016/j.jmaa.2009.11.022. Google Scholar

show all references

References:
[1]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Commun. in Partial Differential Equations, 25 (2000), 737. doi: 10.1080/03605300008821529. Google Scholar

[2]

R. Carles, Geometric optics and instability for semi-classical Schrödinger equations,, Arch. Rational Mech. Anal., 183 (2007), 525. doi: 10.1007/s00205-006-0017-5. Google Scholar

[3]

Q. Chang, Y.-S. Wong and C.-K. Lin, Numerical computation for long-wave short-wave interaction equations in semi-classical limit,, Journal of Computational Physics, 227 (2008), 8489. doi: 10.1016/j.jcp.2008.05.015. Google Scholar

[4]

T. Colin and A. Soyeur, Some singular limits for evolutionary Ginzburg-Landau equations,, Asymptotic Analysis, 13 (1996), 361. Google Scholar

[5]

B. Desjardins, C.-K. Lin and T.-C. Tso, Semiclassical limit of the derivative nonlinear Schrödinger equation,, Math. Models Methods Appl. Sci., 10 (2000), 261. Google Scholar

[6]

B. Desjardins and C.-K. Lin, On the semiclassical limit of the general modified NLS equation,, J. Math. Anal. Appl., 260 (2001), 546. Google Scholar

[7]

N. Ercolani and R. Montgometry, On the fluid approximation to a nonlinear Schrödinger equation,, Physics Letters A, 180 (1993), 402. doi: 10.1016/0375-9601(93)90290-G. Google Scholar

[8]

I. Gasser, C.-K. Lin and P. Markowich, A review of dispersive limit of the (non)linear Schrödinger-type equation,, Taiwanese J. of Mathematics., 4 (2000), 501. Google Scholar

[9]

V. L. Ginzburg and L. P. Pitaevskiĭ, On the theory of superfluidity,, Sov. Phys. JETP, 34(7) (1958), 858. Google Scholar

[10]

E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time,, Proc. Amer. Math. Soc., 126 (1998), 523. doi: 10.1090/S0002-9939-98-04164-1. Google Scholar

[11]

S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy,, Comm. Pure Appl. Math., 52 (1999), 613. doi: 10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L. Google Scholar

[12]

J.-H. Lee and C.-K. Lin, The behavior of solutions of NLS equation of derivative type in the semiclassical limit,, Chaos, 13 (2002), 1475. doi: 10.1016/S0960-0779(01)00157-6. Google Scholar

[13]

J.-H. Lee, C.-K. Lin and O. K. Pashaev, Shock waves, chiral solitons and semiclassical limit of one-dimensional anyons,, Chaos, 19 (2004), 109. doi: 10.1016/S0960-0779(03)00084-5. Google Scholar

[14]

H.-L. Li and C.-K. Lin, Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems,, Electronic Journal of Differential Equations, 2003 (). Google Scholar

[15]

H.-L. Li and C.-K. Lin, Zero Debye length asymptotic of the quantum hydrodynamic model of semiconductors,, Commun. Math. Phys., 256 (2005), 195. doi: 10.1007/s00220-005-1316-7. Google Scholar

[16]

C.-K. Lin and Y.-S. Wong, Zero-dispersion limit of the short-wave-long-wave interaction equations,, Journal of Differential Equations, 228 (2006), 87. doi: 10.1016/j.jde.2006.03.027. Google Scholar

[17]

C.-K. Lin and K.-C. Wu, Singular limits of the Klein-Gordon equation,, Arch. Rational Mech. Anal., 197 (2010), 689. doi: 10.1007/s00205-010-0324-8. Google Scholar

[18]

F.-H. Lin and J. X. Xin, On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation,, Commun. in Math. Phys., 200 (1999), 249. doi: 10.1007/s002200050529. Google Scholar

[19]

T.-C. Lin and P. Zhang, Incompressible and compressible limit of coupled systems of nonlinear Schrödinger equations,, Commun. Math. Phys., 266 (2006), 547. Google Scholar

[20]

S. Machihara, K. Nakanishi and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations,, Mathematische Annalen, 322 (2002), 603. doi: 10.1007/s002080200008. Google Scholar

[21]

A. Messiah, "Quantum Mechanics,", Vol. 1 & 2, (1999). Google Scholar

[22]

H. M. Pilkuhn, "Relativistic Quantum Mechanics,", Texts and Monographs in Physics, (2003). Google Scholar

[23]

M. Puel, Convergence of the Schrödinger-Poisson system to the incompressible Euler equations,, Commun. in Partial Differential Equations, 27 (2002), 2311. doi: 10.1081/PDE-120016159. Google Scholar

[24]

J. Shatah and M. Struwe, "Geometric Wave Equations,", Courant Lecture Notes in Mathematics, (1998). Google Scholar

[25]

C. Sulem and P.-L. Sulem, "The Nonlinear Schrödinger Equation. Self-focusing and Wave Collapse,", Appl. Math. Sci., 139 (1999). Google Scholar

[26]

W. Strauss, "Nonlinear Wave Equations,", CBMS Regional Conference Series in Mathematics, 73 (1989). Google Scholar

[27]

K.-C. Wu, Convergence of the Klein-Gordon equation to the wave map equation with magnetic field,, J. Math. Anal. Appl., 365 (2010), 584. doi: 10.1016/j.jmaa.2009.11.022. Google Scholar

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