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Related Concept Videos

Electromagnetic Wave Equation01:24

Electromagnetic Wave Equation

Maxwell's equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose trajectory can thus be determined using suitable boundary conditions. The objective of electromagnetism is thus theoretically complete.
However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations: What...
The de Broglie Wavelength02:32

The de Broglie Wavelength

In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...
Plane Electromagnetic Waves I01:30

Plane Electromagnetic Waves I

The existence of combined electric and magnetic fields that propagate through space as electromagnetic (EM) waves is the most significant prediction of Maxwell's equations. As Maxwell's equations hold in free space, the predicted electromagnetic waves do not require a medium for their propagation. An EM wave comprises an electric field, defined as the force per charge on a stationary charge, and a magnetic field, which is the force per charge on a moving charge.
The EM field is assumed to be a...
Standing Waves in a Cavity01:28

Standing Waves in a Cavity

A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the nodal planes, it creates a cavity where standing waves are formed. The cavity between the two planes is analogous to a stretched string held at the points x = 0 and x = L. Here, the distance 'L' between the two planes must be an integer multiple of half of the wavelength. The wavelengths that satisfy this condition are given by:
Standing Electromagnetic Waves01:15

Standing Electromagnetic Waves

Electromagnetic waves can be reflected; the surface of a conductor or a dielectric can act as a reflector. As electric and magnetic fields obey the superposition principle, so do electromagnetic waves. The superposition of an incident wave and a reflected electromagnetic wave produces a standing wave analogous to the standing waves created on a stretched string.
Suppose a sheet of a perfect conductor is placed in the yz-plane, and a linearly polarized electromagnetic wave traveling in the...
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.

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Updated: May 16, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
11:03

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Nonlinear Dirac equation solitary waves in external fields.

Franz G Mertens1, Niurka R Quintero, Fred Cooper

  • 1Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany. franzgmertens@gmail.com

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 11, 2012
PubMed
Summary
This summary is machine-generated.

We studied nonlinear Dirac equations with self-interaction and external fields. Solitary waves in this system exhibit particle-like motion, but instabilities arise from unexpected sources.

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Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

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Last Updated: May 16, 2026

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Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Area of Science:

  • Theoretical Physics
  • Quantum Field Theory
  • Nonlinear Dynamics

Background:

  • Nonlinear Dirac equations (NLDEs) are crucial for modeling various physical phenomena.
  • Understanding the behavior of solitary waves in NLDEs under external fields is essential.

Purpose of the Study:

  • To investigate exact solutions and solitary-wave dynamics of 1+1 dimensional NLDEs with scalar-scalar self-interaction.
  • To analyze the influence of external electromagnetic fields on solitary-wave behavior using a variational approximation.

Main Methods:

  • Exact solutions were found for specific external fields.
  • A variational approximation with collective coordinates was employed to study solitary-wave dynamics.
  • Numerical simulations were used to validate the variational approximation.

Main Results:

  • Solitary-wave centers in the approximation follow relativistic particle trajectories.
  • For time-independent fields, energy is conserved, but momentum becomes time-dependent.
  • Instabilities do not follow the dP/dq̇ < 0 criterion, suggesting different sources.

Conclusions:

  • The variational approximation accurately describes stable solitary-wave behavior for small forcing terms.
  • The time evolution of collective coordinates (position, momentum) indicates the onset of solitary-wave instability.
  • Instabilities in this NLDE system may originate from mechanisms distinct from those in the nonlinear Schrödinger equation.