Numerical methods for bifurcation problems

Abstract

Most physical systems are governed by evolution equations of the general form

$$ \frac{{\partial \Psi }}{{\partial t}} = L\Psi + W(\Psi ) $$

((1))

where L is the Laplacian operator and W represents some combination of multiplicative and nonlinear terms. Some examples are:

$$ \frac{{\partial U}}{{\partial t}} = - (U\cdot\nabla )U - \nabla P + \nu {\nabla ^2}U(Navier - Stokes) $$

((2))

$$ \frac{{\partial A}}{{\partial t}} = \mu A - {\left| A \right|^2}A + {\nabla ^2}A(Ginzburg - Landau) $$

((3))

$$ - i\frac{{\partial \Psi }}{{\partial t}} = \left[ {\frac{1}{2}{\nabla ^2} + \mu - V(x)a{{\left| \Psi \right|}^2}} \right]\Psi $$

((4))

(Nonlinear Schrödinger) as well as many other systems, such as the usual Schrödinger equation, reaction-diffusion equations, and the complex Ginzburg-Landau equation. Although the physical system evolves according to the time-dependent equations (1), valuable insight may be gained by studying the closely related equations

$$ 0 = L\Psi + W(\Psi ) $$

((5))

and

$$ \lambda \psi = L\psi + DW(\Psi )\psi $$

((6))

where DW (Ψ) is the linearization or Jacobian of W evaluated at Ψ. (5) describes the steady states of (1) while (6) describes the eigenmodes of (1) about a steady state Ψ.