# American Institute of Mathematical Sciences

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. [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. [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. [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. [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. [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. [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. [8] A. Babin and A. Figotin, Electrodynamics of balanced charges,, \emph{Found. Phys.}, 41 (2011), 242. doi: 10.1007/s10701-010-9502-7. [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. [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. [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. [12] A. Barut, Electrodynamics and Classical Theory of Fields and Particles,, Dover, (1980). [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. [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. [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. [16] I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics,, \emph{Annals of Physics, 100 (1976), 62. [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. [18] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation,, \emph{Nonlinear Anal.}, 7 (1983), 1127. doi: 10.1016/0362-546X(83)90022-6. [19] T. Cazenave, Semilinear Schrödinger equations,, Courant Lecture Notes in Mathematics, (2003). [20] T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique,, \emph{Ann. Fac. Sci. Toulouse Math.}, 5 (1980), 21. [21] T. Cazenave and P.-L.Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 85 (1982), 549. [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. [23] H. Goldstein, C. Poole and J. Safko, Classical Mechanics,, 3rd ed., (2000). [24] C. Itzykson and J. Zuber, Quantum Field Theory,, McGraw-Hill, (1980). [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. [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. [27] J. Jackson, Classical Electrodynamics,, 3rd Edition, (1999). [28] T. Kato, Nonlinear Schrödinger equations,, in \emph{Schr\, (1989). doi: 10.1007/3-540-51783-9_22. [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. [30] A. Komech, Quantum Mechanics: Genesis and Achievements,, Springer, (2013). doi: 10.1007/978-94-007-5542-0. [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. [32] C. Lanczos, The Variational Principles of Mechanics,, 4th ed., (1986). [33] L. Landau and E. Lifshitz, The Classical Theory of Fields, Pergamon,, Oxford, (1975). [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. [35] V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics,, Reidel, (1981). [36] C. Møller, The Theory of Relativity,, 2nd edition, (1982). [37] P. Morse and H. Feshbach, Methods of Theoretical Physics,, Vol. I, (1953). [38] A. Nayfeh, Perturbation methods,, Wiley, (1973). [39] P. Pearle, Classical electron models,, in \emph{Electromagnetism Paths to Research (D. Teplitz ed.)}, (1982), 211. [40] F. Rohrlich, Classical Charged Particles,, Addison-Wesley, (2007). doi: 10.1142/6220. [41] J. Schwinger, Electromagnetic mass revisited,, \emph{Foundations of Physics, 13 (1983), 373. doi: 10.1007/BF01906185. [42] H. Spohn, Dynamics of Charged Particles and Their Radiation Field,, Cambridge Univ. Press, (2004). doi: 10.1017/CBO9780511535178. [43] J. Stachel, Einstein from B to Z,, Burkhouser, (2002). [44] C. Sulem and P. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Springer, (1999).

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. [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. [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. [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. [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. [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. [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. [8] A. Babin and A. Figotin, Electrodynamics of balanced charges,, \emph{Found. Phys.}, 41 (2011), 242. doi: 10.1007/s10701-010-9502-7. [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. [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. [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. [12] A. Barut, Electrodynamics and Classical Theory of Fields and Particles,, Dover, (1980). [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. [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. [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. [16] I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics,, \emph{Annals of Physics, 100 (1976), 62. [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. [18] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation,, \emph{Nonlinear Anal.}, 7 (1983), 1127. doi: 10.1016/0362-546X(83)90022-6. [19] T. Cazenave, Semilinear Schrödinger equations,, Courant Lecture Notes in Mathematics, (2003). [20] T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique,, \emph{Ann. Fac. Sci. Toulouse Math.}, 5 (1980), 21. [21] T. Cazenave and P.-L.Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 85 (1982), 549. [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. [23] H. Goldstein, C. Poole and J. Safko, Classical Mechanics,, 3rd ed., (2000). [24] C. Itzykson and J. Zuber, Quantum Field Theory,, McGraw-Hill, (1980). [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. [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. [27] J. Jackson, Classical Electrodynamics,, 3rd Edition, (1999). [28] T. Kato, Nonlinear Schrödinger equations,, in \emph{Schr\, (1989). doi: 10.1007/3-540-51783-9_22. [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. [30] A. Komech, Quantum Mechanics: Genesis and Achievements,, Springer, (2013). doi: 10.1007/978-94-007-5542-0. [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. [32] C. Lanczos, The Variational Principles of Mechanics,, 4th ed., (1986). [33] L. Landau and E. Lifshitz, The Classical Theory of Fields, Pergamon,, Oxford, (1975). [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. [35] V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics,, Reidel, (1981). [36] C. Møller, The Theory of Relativity,, 2nd edition, (1982). [37] P. Morse and H. Feshbach, Methods of Theoretical Physics,, Vol. I, (1953). [38] A. Nayfeh, Perturbation methods,, Wiley, (1973). [39] P. Pearle, Classical electron models,, in \emph{Electromagnetism Paths to Research (D. Teplitz ed.)}, (1982), 211. [40] F. Rohrlich, Classical Charged Particles,, Addison-Wesley, (2007). doi: 10.1142/6220. [41] J. Schwinger, Electromagnetic mass revisited,, \emph{Foundations of Physics, 13 (1983), 373. doi: 10.1007/BF01906185. [42] H. Spohn, Dynamics of Charged Particles and Their Radiation Field,, Cambridge Univ. Press, (2004). doi: 10.1017/CBO9780511535178. [43] J. Stachel, Einstein from B to Z,, Burkhouser, (2002). [44] C. Sulem and P. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Springer, (1999).
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