September  2014, 13(5): 1685-1718. doi: 10.3934/cpaa.2014.13.1685

Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation

1. 

Department of Mathematics, University of California at Irvine, Irvine, CA 92697-3875

Received  December 2013 Revised  January 2014 Published  June 2014

One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to nonlinear Klein-Gordon equation, we derived relativistic Newton's law and Einstein's formula for inertial mass. Here we apply the same approach to nonlinear Schrodinger equation and derive non-relativistic Newton's law for the trajectory of concentration.
Citation: Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685
References:
[1]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134 (2004), 893.  doi: 10.1017/S030821050000353X.  Google Scholar

[2]

J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential,, \emph{Math. Res. Lett.}, 7 (2000), 329.  doi: 10.4310/MRL.2000.v7.n3.a7.  Google Scholar

[3]

W. Appel and M. Kiessling, Mass and spin renormalization in lorentz electrodynamics,, \emph{Ann. Phys.}, 289 (2001), 24.  doi: 10.1006/aphy.2000.6119.  Google Scholar

[4]

A. Babin and A. Figotin, Wavepacket preservation under nonlinear evolution,, \emph{Commun. Math. Phys.}, 278 (2008), 329.  doi: 10.1007/s00220-007-0406-0.  Google Scholar

[5]

A. Babin and A. Figotin, Nonlinear dynamics of a system of particle-like wavepackets,, in \emph{Instability in Models Connected with Fluid Flows (Ed. C. Bardos and A. Fursikov}), (2008).  doi: 10.1007/978-0-387-75217-4_3.  Google Scholar

[6]

A. Babin and A. Figotin, Wave-corpuscle mechanics for electric charges,, \emph{J. Stat. Phys.}, 138 (2010), 912.  doi: 10.1007/s10955-009-9877-z.  Google Scholar

[7]

A. Babin and A. Figotin, Some mathematical problems in a neoclassical theory of electric charges,, \emph{Discrete and Continuous Dynamical Systems A}, 27 (2010), 1283.  doi: 10.3934/dcds.2010.27.1283.  Google Scholar

[8]

A. Babin and A. Figotin, Electrodynamics of balanced charges,, \emph{Found. Phys.}, 41 (2011), 242.  doi: 10.1007/s10701-010-9502-7.  Google Scholar

[9]

A. Babin and A. Figotin, Relativistic dynamics of accelerating particles derived from field equations,, \emph{Found. Phys.}, 42 (2012), 996.  doi: 10.1007/s10701-012-9642-z.  Google Scholar

[10]

A. Babin and A. Figotin, Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation,, \emph{Comm. Math. Phys.}, 322 (2013), 453.  doi: 10.1007/s00220-013-1732-z.  Google Scholar

[11]

D. Bambusi and L. Galgani, Some rigorous results on the Pauli-Fierz model of classical electrodynamics,, \emph{Ann. Inst. H. Poincar\'e, 58 (1993), 155.   Google Scholar

[12]

A. Barut, Electrodynamics and Classical Theory of Fields and Particles,, Dover, (1980).   Google Scholar

[13]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[14]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 347.  doi: 10.1007/BF00250556.  Google Scholar

[15]

M. Del Pino and J. Dolbeault, The optimal Euclidean $L_p$-Sobolev logarithmic inequality,, \emph{J. Funct. Anal.}, 197 (2003), 151.  doi: 10.1016/S0022-1236(02)00070-8.  Google Scholar

[16]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics,, \emph{Annals of Physics, 100 (1976), 62.   Google Scholar

[17]

I. Bialynicki-Birula and J. Mycielski, Gaussons: Solitons of the logarithmic Schrödinger equation,, \emph{Physica Scripta}, 20 (1979), 539.  doi: 10.1088/0031-8949/20/3-4/033.  Google Scholar

[18]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation,, \emph{Nonlinear Anal.}, 7 (1983), 1127.  doi: 10.1016/0362-546X(83)90022-6.  Google Scholar

[19]

T. Cazenave, Semilinear Schrödinger equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[20]

T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique,, \emph{Ann. Fac. Sci. Toulouse Math.}, 5 (1980), 21.   Google Scholar

[21]

T. Cazenave and P.-L.Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 85 (1982), 549.   Google Scholar

[22]

J. Frohlich, T.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation,, \emph{Comm. Math. Phys.}, 225 (2002), 223.  doi: 10.1007/s002200100579.  Google Scholar

[23]

H. Goldstein, C. Poole and J. Safko, Classical Mechanics,, 3rd ed., (2000).   Google Scholar

[24]

C. Itzykson and J. Zuber, Quantum Field Theory,, McGraw-Hill, (1980).   Google Scholar

[25]

B. Jonsson, J. Frohlich, S. Gustafson and I. M. Sigal, Long time motion of NLS solitary waves in a confining potential,, \emph{Ann. Henri Poincare}, 7 (2006), 621.  doi: 10.1007/s00023-006-0263-y.  Google Scholar

[26]

M. Heid, H. Heinz and T. Weth, Nonlinear eigenvalue problems of Schrnodinger type admitting eigenfunctions with given spectral characteristics,, \emph{Math. Nachr.}, 242 (2002), 91.  doi: 10.1002/1522-2616(200207)242:1<91::AID-MANA91>3.0.CO;2-Z.  Google Scholar

[27]

J. Jackson, Classical Electrodynamics,, 3rd Edition, (1999).   Google Scholar

[28]

T. Kato, Nonlinear Schrödinger equations,, in \emph{Schr\, (1989).  doi: 10.1007/3-540-51783-9_22.  Google Scholar

[29]

M. Kiessling, Electromagnetic Field Theory without Divergence Problems 1. The Born Legacy,, \emph{J. Stat. Physics}, 116 (2004), 1057.  doi: 10.1023/B:JOSS.0000037250.72634.2a.  Google Scholar

[30]

A. Komech, Quantum Mechanics: Genesis and Achievements,, Springer, (2013).  doi: 10.1007/978-94-007-5542-0.  Google Scholar

[31]

A. Komech, M. Kunze and H. Spohn, Effective Dynamics for a mechanical particle coupled to a wave field,, \emph{Comm. Math. Phys.}, 203 (1999), 1.  doi: 10.1007/s002200050023.  Google Scholar

[32]

C. Lanczos, The Variational Principles of Mechanics,, 4th ed., (1986).   Google Scholar

[33]

L. Landau and E. Lifshitz, The Classical Theory of Fields, Pergamon,, Oxford, (1975).   Google Scholar

[34]

E. Long and D. Stuart, Effective dynamics for solitons in the nonlinear Klein-Gordon-Maxwell system and the Lorentz force law,, \emph{Rev. Math. Phys.}, (2009), 459.  doi: 10.1142/S0129055X09003669.  Google Scholar

[35]

V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics,, Reidel, (1981).   Google Scholar

[36]

C. Møller, The Theory of Relativity,, 2nd edition, (1982).   Google Scholar

[37]

P. Morse and H. Feshbach, Methods of Theoretical Physics,, Vol. I, (1953).   Google Scholar

[38]

A. Nayfeh, Perturbation methods,, Wiley, (1973).   Google Scholar

[39]

P. Pearle, Classical electron models,, in \emph{Electromagnetism Paths to Research (D. Teplitz ed.)}, (1982), 211.   Google Scholar

[40]

F. Rohrlich, Classical Charged Particles,, Addison-Wesley, (2007).  doi: 10.1142/6220.  Google Scholar

[41]

J. Schwinger, Electromagnetic mass revisited,, \emph{Foundations of Physics, 13 (1983), 373.  doi: 10.1007/BF01906185.  Google Scholar

[42]

H. Spohn, Dynamics of Charged Particles and Their Radiation Field,, Cambridge Univ. Press, (2004).  doi: 10.1017/CBO9780511535178.  Google Scholar

[43]

J. Stachel, Einstein from B to Z,, Burkhouser, (2002).   Google Scholar

[44]

C. Sulem and P. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Springer, (1999).   Google Scholar

show all references

References:
[1]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134 (2004), 893.  doi: 10.1017/S030821050000353X.  Google Scholar

[2]

J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential,, \emph{Math. Res. Lett.}, 7 (2000), 329.  doi: 10.4310/MRL.2000.v7.n3.a7.  Google Scholar

[3]

W. Appel and M. Kiessling, Mass and spin renormalization in lorentz electrodynamics,, \emph{Ann. Phys.}, 289 (2001), 24.  doi: 10.1006/aphy.2000.6119.  Google Scholar

[4]

A. Babin and A. Figotin, Wavepacket preservation under nonlinear evolution,, \emph{Commun. Math. Phys.}, 278 (2008), 329.  doi: 10.1007/s00220-007-0406-0.  Google Scholar

[5]

A. Babin and A. Figotin, Nonlinear dynamics of a system of particle-like wavepackets,, in \emph{Instability in Models Connected with Fluid Flows (Ed. C. Bardos and A. Fursikov}), (2008).  doi: 10.1007/978-0-387-75217-4_3.  Google Scholar

[6]

A. Babin and A. Figotin, Wave-corpuscle mechanics for electric charges,, \emph{J. Stat. Phys.}, 138 (2010), 912.  doi: 10.1007/s10955-009-9877-z.  Google Scholar

[7]

A. Babin and A. Figotin, Some mathematical problems in a neoclassical theory of electric charges,, \emph{Discrete and Continuous Dynamical Systems A}, 27 (2010), 1283.  doi: 10.3934/dcds.2010.27.1283.  Google Scholar

[8]

A. Babin and A. Figotin, Electrodynamics of balanced charges,, \emph{Found. Phys.}, 41 (2011), 242.  doi: 10.1007/s10701-010-9502-7.  Google Scholar

[9]

A. Babin and A. Figotin, Relativistic dynamics of accelerating particles derived from field equations,, \emph{Found. Phys.}, 42 (2012), 996.  doi: 10.1007/s10701-012-9642-z.  Google Scholar

[10]

A. Babin and A. Figotin, Relativistic point dynamics and Einstein formula as a property of localized solutions of a nonlinear Klein-Gordon equation,, \emph{Comm. Math. Phys.}, 322 (2013), 453.  doi: 10.1007/s00220-013-1732-z.  Google Scholar

[11]

D. Bambusi and L. Galgani, Some rigorous results on the Pauli-Fierz model of classical electrodynamics,, \emph{Ann. Inst. H. Poincar\'e, 58 (1993), 155.   Google Scholar

[12]

A. Barut, Electrodynamics and Classical Theory of Fields and Particles,, Dover, (1980).   Google Scholar

[13]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313.  doi: 10.1007/BF00250555.  Google Scholar

[14]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 347.  doi: 10.1007/BF00250556.  Google Scholar

[15]

M. Del Pino and J. Dolbeault, The optimal Euclidean $L_p$-Sobolev logarithmic inequality,, \emph{J. Funct. Anal.}, 197 (2003), 151.  doi: 10.1016/S0022-1236(02)00070-8.  Google Scholar

[16]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics,, \emph{Annals of Physics, 100 (1976), 62.   Google Scholar

[17]

I. Bialynicki-Birula and J. Mycielski, Gaussons: Solitons of the logarithmic Schrödinger equation,, \emph{Physica Scripta}, 20 (1979), 539.  doi: 10.1088/0031-8949/20/3-4/033.  Google Scholar

[18]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation,, \emph{Nonlinear Anal.}, 7 (1983), 1127.  doi: 10.1016/0362-546X(83)90022-6.  Google Scholar

[19]

T. Cazenave, Semilinear Schrödinger equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[20]

T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique,, \emph{Ann. Fac. Sci. Toulouse Math.}, 5 (1980), 21.   Google Scholar

[21]

T. Cazenave and P.-L.Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 85 (1982), 549.   Google Scholar

[22]

J. Frohlich, T.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation,, \emph{Comm. Math. Phys.}, 225 (2002), 223.  doi: 10.1007/s002200100579.  Google Scholar

[23]

H. Goldstein, C. Poole and J. Safko, Classical Mechanics,, 3rd ed., (2000).   Google Scholar

[24]

C. Itzykson and J. Zuber, Quantum Field Theory,, McGraw-Hill, (1980).   Google Scholar

[25]

B. Jonsson, J. Frohlich, S. Gustafson and I. M. Sigal, Long time motion of NLS solitary waves in a confining potential,, \emph{Ann. Henri Poincare}, 7 (2006), 621.  doi: 10.1007/s00023-006-0263-y.  Google Scholar

[26]

M. Heid, H. Heinz and T. Weth, Nonlinear eigenvalue problems of Schrnodinger type admitting eigenfunctions with given spectral characteristics,, \emph{Math. Nachr.}, 242 (2002), 91.  doi: 10.1002/1522-2616(200207)242:1<91::AID-MANA91>3.0.CO;2-Z.  Google Scholar

[27]

J. Jackson, Classical Electrodynamics,, 3rd Edition, (1999).   Google Scholar

[28]

T. Kato, Nonlinear Schrödinger equations,, in \emph{Schr\, (1989).  doi: 10.1007/3-540-51783-9_22.  Google Scholar

[29]

M. Kiessling, Electromagnetic Field Theory without Divergence Problems 1. The Born Legacy,, \emph{J. Stat. Physics}, 116 (2004), 1057.  doi: 10.1023/B:JOSS.0000037250.72634.2a.  Google Scholar

[30]

A. Komech, Quantum Mechanics: Genesis and Achievements,, Springer, (2013).  doi: 10.1007/978-94-007-5542-0.  Google Scholar

[31]

A. Komech, M. Kunze and H. Spohn, Effective Dynamics for a mechanical particle coupled to a wave field,, \emph{Comm. Math. Phys.}, 203 (1999), 1.  doi: 10.1007/s002200050023.  Google Scholar

[32]

C. Lanczos, The Variational Principles of Mechanics,, 4th ed., (1986).   Google Scholar

[33]

L. Landau and E. Lifshitz, The Classical Theory of Fields, Pergamon,, Oxford, (1975).   Google Scholar

[34]

E. Long and D. Stuart, Effective dynamics for solitons in the nonlinear Klein-Gordon-Maxwell system and the Lorentz force law,, \emph{Rev. Math. Phys.}, (2009), 459.  doi: 10.1142/S0129055X09003669.  Google Scholar

[35]

V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics,, Reidel, (1981).   Google Scholar

[36]

C. Møller, The Theory of Relativity,, 2nd edition, (1982).   Google Scholar

[37]

P. Morse and H. Feshbach, Methods of Theoretical Physics,, Vol. I, (1953).   Google Scholar

[38]

A. Nayfeh, Perturbation methods,, Wiley, (1973).   Google Scholar

[39]

P. Pearle, Classical electron models,, in \emph{Electromagnetism Paths to Research (D. Teplitz ed.)}, (1982), 211.   Google Scholar

[40]

F. Rohrlich, Classical Charged Particles,, Addison-Wesley, (2007).  doi: 10.1142/6220.  Google Scholar

[41]

J. Schwinger, Electromagnetic mass revisited,, \emph{Foundations of Physics, 13 (1983), 373.  doi: 10.1007/BF01906185.  Google Scholar

[42]

H. Spohn, Dynamics of Charged Particles and Their Radiation Field,, Cambridge Univ. Press, (2004).  doi: 10.1017/CBO9780511535178.  Google Scholar

[43]

J. Stachel, Einstein from B to Z,, Burkhouser, (2002).   Google Scholar

[44]

C. Sulem and P. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Springer, (1999).   Google Scholar

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