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Symmetry in Maxwell's Equations

Symmetry in Maxwell's Equations

Once the fields have been calculated using Maxwell's four equations, the Lorentz force equation gives the force that the fields exert on a charged particle moving with a certain velocity. The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. Maxwell's equations and the Lorentz force law together encompass all the laws of electricity and magnetism. The symmetry that Maxwell introduced into his mathematical framework may not be...

Standing Waves in a Cavity

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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:

Modes of Standing Waves: II

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The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions are derived from the physical understanding of how the standing waves are sustained, that is, how the vibrating particles of the medium behave at the boundaries imposed on them.
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Modes of Standing Waves - I

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A close look at earthquakes provides evidence for the conditions appropriate for resonance, standing waves, and constructive and destructive interference. A building may vibrate for several seconds with a driving frequency matching the building's natural frequency of vibration; this produces a resonance that results in one building collapsing while the neighboring buildings do not. Often, buildings of a certain height are devastated, while other taller buildings remain intact. This phenomenon...

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Updated: Jun 25, 2026

Fabrication And Characterization Of Photonic Crystal Slow Light Waveguides And Cavities

Fabrication And Characterization Of Photonic Crystal Slow Light Waveguides And Cavities

Published on: November 30, 2012

Spontaneous symmetry breaking in coupled parametrically driven waveguides.

Nir Dror¹, Boris A Malomed

¹Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|March 5, 2009

Summary

We analyzed spontaneous symmetry breaking in nonlinear Schrödinger equations, revealing new scenarios for soliton behavior and bistability in coupled systems. This research advances understanding of stable symmetric and asymmetric solitons in nonlinear optics.

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Area of Science:

Nonlinear Optics and Photonics
Soliton Dynamics
Complex Systems

Background:

Introduces a novel system of linearly coupled parametrically driven damped nonlinear Schrödinger equations.
Models lasers based on nonlinear dual-core waveguides and parallel ferromagnetic films with parametric amplification.
Addresses a gap in systematic studies of spontaneous symmetry breaking (SSB) in dissipative nonlinear coupled systems.

Purpose of the Study:

To systematically analyze spontaneous symmetry breaking (SSB) of fundamental and multiple solitons in a parametrically amplified dual-core nonlinear waveguide system.
To investigate the stability of symmetric and asymmetric solitons and their bifurcations.
To explore the potential for bistability and novel soliton behaviors relevant to applications.

Main Methods:

Analysis of spontaneous symmetry breaking (SSB) in a system of linearly coupled parametrically driven damped nonlinear Schrödinger equations.
Direct simulations to identify the stability of asymmetric solitons.
Computation of stability eigenvalues for symmetric and antisymmetric solitons.

Main Results:

Identified three distinct SSB scenarios for fundamental solitons.
Discovered a vast bistability region, differing from standard dual-core-fiber models.
Observed restabilization of symmetric solitons post-SSB bifurcation and stability of both symmetric and asymmetric solitons in the coupled system, unlike the decoupled model where all solitons were unstable.

Conclusions:

The study provides comprehensive stability maps for symmetric solitons and reveals the complex dynamics of asymmetric solitons, with some families being entirely unstable.
Investigated SSB bifurcation of two-soliton bound states, showing asymmetric double-peak states can decouple and remain stable.
The findings offer insights into soliton control and bistability in nonlinear optical systems, with potential applications in laser technology and optical signal processing.