November  2010, 27(4): 1283-1326. doi: 10.3934/dcds.2010.27.1283

Some mathematical problems in a neoclassical theory of electric charges

1. 

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

Received  October 2009 Revised  January 2010 Published  March 2010

We study here a number of mathematical problems related to our recently introduced neoclassical theory for electromagnetic phenomena in which charges are represented by complex valued wave functions as in the Schrödinger wave mechanics. In the non-relativistic case the dynamics of elementary charges is governed by a system of nonlinear Schrödinger equations coupled with the electromagnetic fields, and we prove that if the wave functions of charges are well separated and localized their centers converge to trajectories of the classical point charges governed by Newton's equations with the Lorentz forces. We also found exact solutions in the form of localized accelerating solitons. Our studies of a class of time multiharmonic solutions of the same field equations show that they satisfy Planck-Einstein relation and that the energy levels of the nonlinear eigenvalue problem for the hydrogen atom converge to the well-known energy levels of the linear Schrödinger operator when the free charge size is much larger than the Bohr radius.
Citation: Anatoli Babin, Alexander Figotin. Some mathematical problems in a neoclassical theory of electric charges. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1283-1326. doi: 10.3934/dcds.2010.27.1283
[1]

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

[2]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703

[3]

JinMyong An, JinMyong Kim, KyuSong Chae. Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021221

[4]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431

[5]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1497-1519. doi: 10.3934/cpaa.2021030

[6]

Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102

[7]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021028

[8]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

[9]

Shubin Wang, Guowang Chen. Cauchy problem for the nonlinear Schrödinger-IMBq equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 203-214. doi: 10.3934/dcdsb.2006.6.203

[10]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[11]

Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113

[12]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[13]

Andreas Kirsch. An integral equation approach and the interior transmission problem for Maxwell's equations. Inverse Problems & Imaging, 2007, 1 (1) : 159-179. doi: 10.3934/ipi.2007.1.159

[14]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

[15]

Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations & Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030

[16]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003

[17]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063

[18]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109

[19]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337

[20]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]