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Register a new userThe Swift-Hohenberg Equation Requires Non-Local Modifications to Model Spatial Pattern Evolution of Physical Problems
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Authors
A.J. Roberts
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I argue that ``good mathematical models of spatio-temporal dynamics in two-dimensions require non-local operators in the nonlinear terms. Consequently, the often used Swift-Hohenberg equation requires modification as it is purely local. My aim here is to provoke more critical examination of the rationale for using the Swift-Hohenberg equations as a reliable model of the spatial pattern evolution in specific physical systems.
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I introduce an innovative methodology for deriving numerical models of systems of partial differential equations which exhibit the evolution of spatial patterns. The new approach directly produces a discretisation for the evolution of the pattern amplitude, has the rigorous support of centre manifold theory at finite grid size $h$, and naturally incorporates physical boundaries. The results presented here for the Swift-Hohenberg equation suggest the approach will form a powerful method in computationally exploring pattern selection in general. With the aid of computer algebra, the techniques may be applied to a wide variety of equations to derive numerical models that accurately and stably capture the dynamics including the influence of possibly forced boundaries.
A theoretical model for studying pattern formation in electroconvection is proposed in the form of a modified Swift-Hohenberg equation. A localized state is found in two dimension, in agreement with the experimentally observed ``worm state. The corresponding one dimensional model is also studied, and a novel stationary localized state due to nonadiabatic effect is found. The existence of the 1D localized state is shown to be responsible for the formation of the two dimensional ``worm state in our model.
We show that all meromorphic solutions of the stationary reduction of the real cubic Swift-Hohenberg equation are elliptic or degenerate elliptic. We then obtain them all explicitly by the subequation method, and one of them appears to be a new elliptic solution.
Axisymmetric and nonaxisymmetric patterns in the cubic-quintic Swift-Hohenberg equation posed on a disk with Neumann boundary conditions are studied via numerical continuation and bifurcation analysis. Axisymmetric localized solutions in the form of spots and rings known from earlier studies persist and snake in the usual fashion until they begin to interact with the boundary. Depending on parameters, including the disk radius, these states may or may not connect to the branch of domain-filling target states. Secondary instabilities of localized axisymmetric states may create multi-arm localized structures that grow and interact with the boundary before broadening into domain filling states. High azimuthal wavenumber wall states referred to as daisy states are also found. Secondary bifurcations from these states include localized daisies, i.e., wall states localized in both radius and angle. Depending on parameters, these states may snake much as in the one-dimensional Swift-Hohenberg equation, or invade the interior of the domain, yielding states referred to as worms, or domain-filling stripes.
We consider the one-dimensional Swift-Hohenberg equation coupled to a conservation law. As a parameter increases the system undergoes a Turing bifurcation. We study the dynamics near this bifurcation. First, we show that stationary, periodic solutions bifurcate from a homogeneous ground state. Second, we construct modulating traveling fronts which model an invasion of the unstable ground state by the periodic solutions. This provides a mechanism of pattern formation for the studied system. The existence proof uses center manifold theory for a reduction to a finite-dimensional problem. This is possible despite the presence of infinitely many imaginary eigenvalues for vanishing bifurcation parameter since the eigenvalues leave the imaginary axis with different velocities if the parameter increases. Furthermore, compared to non-conservative systems, we address new difficulties arising from an additional neutral mode at Fourier wave number $k=0$ by exploiting that the amplitude of the conserved variable is small compared to the other variables.
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