Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation

Area of Science:

• Nonlinear dynamics
• Pattern formation in physical systems
• Mathematical physics

Background:

• The Kuramoto-Sivashinsky equation with dispersive terms shows significant pattern formation changes.
• Dispersive effects can dramatically alter dynamics in nonlinear partial differential equations.

Purpose of the Study:

• Investigate pattern formation in the dispersive Swift-Hohenberg equation (DSHE).
• Analyze the nature and behavior of defects, termed 'seams', in DSHE stripe patterns.
• Relate DSHE phenomena to the anisotropic complex Ginzburg-Landau equation (ACGLE).

Main Methods:

• Analytical derivation of the amplitude equation for DSHE near threshold.
• Identification of seams as spiral waves in the ACGLE.
• Perturbative analysis in the strong dispersion limit.
• Numerical integration of both DSHE and ACGLE.

Main Results:

• DSHE exhibits stripe patterns with spatially extended defects called seams.
• Seams in DSHE are analogous to spiral waves in the ACGLE.
• Formulas for spiral wave core velocity and spacing were derived.
• A relationship between stripe pattern amplitude, wavelength, and propagation velocity was found under strong dispersion.

Conclusions:

• The DSHE provides a model system for studying defects in pattern formation.
• The connection to ACGLE allows for analytical insights into seam dynamics.
• Analytical and numerical results confirm the behavior of seams and spiral waves.