Nonlinear Schrödinger equation with a short-range compensating field

Sergey A. Rashkovskiy; Sergey A. Rashkovskiy

doi:10.3934/cam.2025021

Communications in Analysis and Mechanics

2025, Volume 17, Issue 2: 520-549. doi: 10.3934/cam.2025021

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Research article

Nonlinear Schrödinger equation with a short-range compensating field

Sergey A. Rashkovskiy ^,

Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Vernadskogo Ave., 101/1, Moscow, 119526, Russia

Received: 14 February 2025 Revised: 21 April 2025 Accepted: 06 May 2025 Published: 15 May 2025
81Sxx, 81P10, 81Q80

Within the self-consistent Maxwell-Pauli theory, a nonlinear Schrödinger equation with a short-range compensating field was derived. Stationary and nonstationary solutions of the obtained nonlinear Schrödinger equation for the hydrogen atom were investigated. It is shown that spontaneous emission and the associated rearrangement of the internal structure of the atom, which is traditionally called a spontaneous transition, have a simple and natural description within the classical field theory without any quantization and additional hypotheses. The solution of the nonlinear Schrödinger equation shows that, depending on the frequency of spontaneous emission, the compensating field behaves differently. At relatively low frequencies of spontaneous emission, there is no radiation (waves) of the short-range compensating field, and this field does not carry away energy. In this case, the damping rate of the spontaneous emission coincides with that obtained in quantum electrodynamics (QED). At relatively high frequencies of spontaneous emission, radiation (waves) of the compensating field arises, which, along with electromagnetic radiation, carry away some of the energy. In this case, the damping rate of the spontaneous emission is greater (up 1.5 times) than that predicted by QED.
- classical field theory,
- self-consistent Maxwell-Pauli theory,
- nonlinear Pauli equation,
- nonlinear Schrödinger equation,
- spontaneous emission
Citation: Sergey A. Rashkovskiy. Nonlinear Schrödinger equation with a short-range compensating field[J]. Communications in Analysis and Mechanics, 2025, 17(2): 520-549. doi: 10.3934/cam.2025021

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Abstract

Within the self-consistent Maxwell-Pauli theory, a nonlinear Schrödinger equation with a short-range compensating field was derived. Stationary and nonstationary solutions of the obtained nonlinear Schrödinger equation for the hydrogen atom were investigated. It is shown that spontaneous emission and the associated rearrangement of the internal structure of the atom, which is traditionally called a spontaneous transition, have a simple and natural description within the classical field theory without any quantization and additional hypotheses. The solution of the nonlinear Schrödinger equation shows that, depending on the frequency of spontaneous emission, the compensating field behaves differently. At relatively low frequencies of spontaneous emission, there is no radiation (waves) of the short-range compensating field, and this field does not carry away energy. In this case, the damping rate of the spontaneous emission coincides with that obtained in quantum electrodynamics (QED). At relatively high frequencies of spontaneous emission, radiation (waves) of the compensating field arises, which, along with electromagnetic radiation, carry away some of the energy. In this case, the damping rate of the spontaneous emission is greater (up 1.5 times) than that predicted by QED.

References

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