Geometric evolution equations for order parameters
نویسنده
چکیده
Energy-decreasing continuum flows of geometric order parameters are considered. The dynamics of pattern formation for an arbitrary geometric quantity is derived using a generalization of Darcy’s law based on the geometric nature of the order parameter. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.
منابع مشابه
STABILITY ANALYSIS FROM FOURTH ORDER NONLINEAR EVOLUTION EQUATIONS FOR TWO CAPILLARY GRAVITY WAVE PACKETS IN THE PRESENCE OF WIND OWING OVER WATER.
Asymptotically exact and nonlocal fourth order nonlinear evolution equations are derived for two coupled fourth order nonlinear evolution equations have been derived in deep water for two capillary-gravity wave packets propagating in the same direction in the presence of wind flowing over water.We have used a general method, based on Zakharov integral equation.On the basis of these evolution eq...
متن کاملEFFECT OF COUNTERPROPAGATING CAPILLARY GRAVITY WAVE PACKETS ON THIRD ORDER NONLINEAR EVOLUTION EQUATIONS IN THE PRESENCE OF WIND FLOWING OVER WATER
Asymptotically exact and nonlocal third order nonlinear evolution equations are derivedfor two counterpropagating surface capillary gravity wave packets in deep water in thepresence of wind flowing over water.From these evolution equations stability analysis ismade for a uniform standing surface capillary gravity wave trains for longitudinal perturbation. Instability condition is obtained and g...
متن کاملNumerical Solution of The First-Order Evolution Equations by Radial Basis Function
In this work, we consider the nonlinear first-order evolution equations: $u_t=f(x,t,u,u_x,u_{xx})$ for $0 to initial condition $u(x,0)=g(x)$, where $u$ is a function of $x$ and $t$ and $f$ is a known analytic function. The purpose of this paper is to introduce the method of RBF to existing method in solving nonlinear first-ord...
متن کاملInteraction of Particles with Non-central Potential: Gradient Flows and Singular Solutions for Evolution of Geometric Continuum Quantities
Evolutionary PDEs for geometric order parameters that admit propagating singular solutions are introduced and discussed. These singular solutions arise as a result of the competition between nonlinear and nonlocal processes in various familiar vector spaces. Several examples are given. The motivating example is the directed self assembly of a large number of particles for technological purposes...
متن کاملIntegrable geometric evolution equations for curves
The vortex filament flow and planar filament flow are examples of evolution equations which commute with Euclidean isometries and are also integrable, in that they induce completely integrable PDE for curvature—the focusing nonlinear Schödinger equation and the mKdV equations, respectively. In this note we outline an approach for classifying integrable geometric evolution equations for planar c...
متن کامل