Related Experiment Video
Updated: Jul 19, 2026
12:14
The Generation of Higher-order Laguerre-Gauss Optical Beams for High-precision Interferometry
Published on: August 12, 2013
Scale disparities in the complex Swift-Hohenberg equation for lasers.
Carlos Martel1, Miguel Hoyuelos
1Departamento de Fundamentos Matemáticos, ETSI Aeronáuticos, Universidad Politécnica de Madrid, 28040 Madrid, Spain.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 10, 2006
Summary
The complex Swift-Hohenberg equation for class C lasers reveals overlooked scale disparities between dispersion and diffusion effects. This study presents a new deduction and analysis, highlighting these crucial differences in physical system modeling.
Area of Science:
- Theoretical Physics
- Nonlinear Dynamics
- Laser Physics
Background:
- The complex Swift-Hohenberg (CSH) equation is a fundamental model for various physical systems, including class C lasers.
- Derivations of the CSH equation from Maxwell-Bloch equations often assume terms are of the same order, potentially overlooking scale disparities.
Purpose of the Study:
- To carefully deduce the asymptotically nonuniform CSH equation without relying on scaling assumptions.
- To investigate the implications of scale disparities between dispersion and diffusion effects in physical systems.
- To analyze the simplest solutions of the derived CSH equation and present numerical simulations.
Main Methods:
- A simpler, scaling-free procedure was employed to deduce the complex Swift-Hohenberg equation.
- Stability analysis was performed on the simplest solutions of the derived equation.
- Numerical simulations were conducted to visualize the scale disparities.
Main Results:
- The derived CSH equation inherently contains asymptotic order terms indicating unequal contributions of dispersion and diffusion.
- These scale disparities, often neglected in literature, significantly impact the equation's behavior.
- Numerical simulations clearly demonstrated the presence and effects of these scale differences.
Conclusions:
- The asymptotically nonuniform CSH equation provides a more accurate description of physical systems like class C lasers.
- Recognizing and analyzing scale disparities between dispersion and diffusion is crucial for qualitative and quantitative accuracy in modeling.
- This work emphasizes the need for careful derivation and analysis of generic order parameter equations.
Related Concept Videos
02:32The 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...
01:25Ostwald’s Dilution Law
Consider a binary electrolyte AB with a concentration ‘c’ that reversibly dissociates into its constituent ions. The degree of this dissociation is represented by ⍺. This means that the equilibrium concentration of each ionic species can be expressed as ⍺c. As well as this, the fraction of the electrolyte that remains undissociated at equilibrium is given by (1−⍺). The corresponding equilibrium concentration for this undissociated portion is then calculated as (1−⍺)c. For such solutions,...
01:28Debye–Huckel–Onsager Conductance Equation
The Debye-Hückel-Onsager equation is a cornerstone of physical chemistry, providing a method to determine the molar conductance (Λm) and molar conductance at infinite dilution (Λ°m) for uni-univalent electrolytes.Uni-univalent electrolytes are electrolytes that dissociate in solution to produce one cation with a +1 charge and one anion with a –1 charge per formula unit.This equation addresses two crucial phenomena: the asymmetry effect and the electrophoretic effect. According to this equation,...
01:26Linear Approximation in Frequency Domain
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
01:26Scaling
In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...
01:07Propagation of Waves
When a wave propagates from one medium to another, part of it may get reflected in the first medium, and part of it may get transmitted to the second medium. In such a case, the interface of the two mediums can be considered as a boundary that is neither fixed nor free.
Consider a scenario where a wave propagates from a string of low linear mass density to a string of high linear mass density. In such a case, the reflected wave is out of phase with respect to the incident wave, however the...
Consider a scenario where a wave propagates from a string of low linear mass density to a string of high linear mass density. In such a case, the reflected wave is out of phase with respect to the incident wave, however the...