Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 35Q55, 35Q60, 35Q70; Secondary: 70S05, 78A35.

 Citation:

•  [1] T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrodinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.doi: 10.1017/S030821050000353X. [2] J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential, Math. Res. Lett., 7 (2000), 329-342.doi: 10.4310/MRL.2000.v7.n3.a7. [3] W. Appel and M. Kiessling, Mass and spin renormalization in lorentz electrodynamics, Ann. Phys., 289 (2001), 24-83.doi: 10.1006/aphy.2000.6119. [4] A. Babin and A. Figotin, Wavepacket preservation under nonlinear evolution, Commun. Math. Phys., 278 (2008), 329-384.doi: 10.1007/s00220-007-0406-0. [5] A. Babin and A. Figotin, Nonlinear dynamics of a system of particle-like wavepackets, in Instability in Models Connected with Fluid Flows (Ed. C. Bardos and A. Fursikov), International Mathematical Series, Vol. 6, Springer, 2008.doi: 10.1007/978-0-387-75217-4_3. [6] A. Babin and A. Figotin, Wave-corpuscle mechanics for electric charges, J. Stat. Phys., 138 (2010), 912-954.doi: 10.1007/s10955-009-9877-z. [7] A. Babin and A. Figotin, Some mathematical problems in a neoclassical theory of electric charges, Discrete and Continuous Dynamical Systems A, 27 (2010), 1283-1326.doi: 10.3934/dcds.2010.27.1283. [8] A. Babin and A. Figotin, Electrodynamics of balanced charges, Found. Phys., 41 (2011), 242-260.doi: 10.1007/s10701-010-9502-7. [9] A. Babin and A. Figotin, Relativistic dynamics of accelerating particles derived from field equations, Found. Phys., 42 (2012), 996-1014.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, Comm. Math. Phys., 322 (2013), 453-499.doi: 10.1007/s00220-013-1732-z. [11] D. Bambusi and L. Galgani, Some rigorous results on the Pauli-Fierz model of classical electrodynamics, Ann. Inst. H. Poincaré, Phys. Théor, 58 (1993), 155-171. [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, Arch. Rational Mech. Anal., 82 (1983), 313-345.doi: 10.1007/BF00250555. [14] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375.doi: 10.1007/BF00250556. [15] M. Del Pino and J. Dolbeault, The optimal Euclidean $L_p$-Sobolev logarithmic inequality, J. Funct. Anal., 197 (2003), 151-161.doi: 10.1016/S0022-1236(02)00070-8. [16] I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Annals of Physics, 100 (1976), 62-93. [17] I. Bialynicki-Birula and J. Mycielski, Gaussons: Solitons of the logarithmic Schrödinger equation, Physica Scripta, 20 (1979), 539-544.doi: 10.1088/0031-8949/20/3-4/033. [18] T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.doi: 10.1016/0362-546X(83)90022-6. [19] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10 AMS, Providence RI, 2003. [20] T. Cazenave and A. Haraux, Équations d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math., 5 (1980), 21-51. [21] T. Cazenave and P.-L.Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. [22] J. Frohlich, T.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys., 225 (2002), 223-274.doi: 10.1007/s002200100579. [23] H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 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, Ann. Henri Poincare, 7 (2006), 621-660.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, Math. Nachr., 242 (2002), 91-118.doi: 10.1002/1522-2616(200207)242:1<91::AID-MANA91>3.0.CO;2-Z. [27] J. Jackson, Classical Electrodynamics, 3rd Edition, Wiley, 1999. [28] T. Kato, Nonlinear Schrödinger equations, in Schrödinger Operators (H. Holden and A Jensen, eds.), Lecture Notes in Physics 345, Springer Verlag, 1989.doi: 10.1007/3-540-51783-9_22. [29] M. Kiessling, Electromagnetic Field Theory without Divergence Problems 1. The Born Legacy, J. Stat. Physics, 116 (2004), 1057-1122.doi: 10.1023/B:JOSS.0000037250.72634.2a. [30] A. Komech, Quantum Mechanics: Genesis and Achievements, Springer, Dordrecht, 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, Comm. Math. Phys., 203 (1999), 1-19.doi: 10.1007/s002200050023. [32] C. Lanczos, The Variational Principles of Mechanics, 4th ed., Dover, 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, Rev. Math. Phys., 21, 459-510 (2009).doi: 10.1142/S0129055X09003669. [35] V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Reidel, Boston, 1981. [36] C. Møller, The Theory of Relativity, 2nd edition, Oxford, 1982. [37] P. Morse and H. Feshbach, Methods of Theoretical Physics, Vol. I, McGraw-Hill, 1953. [38] A. Nayfeh, Perturbation methods, Wiley, 1973. [39] P. Pearle, Classical electron models, in Electromagnetism Paths to Research (D. Teplitz ed.), Plenum, New York, 1982, pp. 211-295. [40] F. Rohrlich, Classical Charged Particles, Addison-Wesley, 3d ed., 2007.doi: 10.1142/6220. [41] J. Schwinger, Electromagnetic mass revisited, Foundations of Physics, 13 (1983), 373-383.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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