A Numerical Method and Its Demo on Finding Multiple Unstable Solutions to Nonlinear Variational Systems
by
Xianjin Chen
Texas A&M Univeristy, College Station
Coauthors: Jianxin Zhou

We will present a stable method to solve nonlinear variational elliptic systems of the form

-Du = Fu(x; u, v)    x ∈ W -Dv = -Fv(x; u, v)    x ∈ W

(1)

or

-Du = Fv(x; u, v)    x ∈ W -Dv = Fu(x; u, v)    x ∈ W

(2)

subject to zero Dirichlet boundary conditions, where F:[`(W)] ×R²→ R is of class C¹ in the variables (u, v) and ∇F = (F_u, F_v). Systems of this kind arise as steady-state equations in the study of various types of physical and biological models such as the FitzHugh-Nagumo model in nerve conduction or pattern formation in biology. In particular, the associated energy functionals are strongly indefinite. We will also discuss the convergence of the method and present a short demo in search of multiple unstable solutions for a variational system model.

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Date received: August 16, 2007