Fronts and Interfaces
If the amplitude is allowed to vary in space, a diffusional term is added to the respective amplitude equation. For a bifurcation at zero eigenvalue in a general RDS, spatial dependence can be incorporated in the formal expansion of Sect. 1.3.2 by scaling the spatial derivatives as ∇ = O(∈(n-1)/2) when the expansion proceeds to the nth order. Starting from a general RDS (1.18), one obtains then the term D∇2a with the amplitude diffusivity D = U† . DU added to the solvability condition in the respective order, yielding a single (scalar) diffusion equation with a polynomial nonlinearity of degree n.
KeywordsPropagation Speed Interphase Boundary Burger Equation Eikonal Equation Phase Field Model
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