Let us commence by brie°y reviewing the concept of variational derivatives. Let u(x; t) be a function of (x; t) 2 [0; L]£[0;1), and G(u; ux) be a real-valued function of u; ux. We often call G \local energy." We then de¯ne a \global energy" by

J(u) = Z L 0

G(u; ux)dx: (2.1)

Below we consider three cases: when u is a real-valued scalar function, a complex-valued scalar function, and vector of real-and complex-valued functions. Suppose u is a real-valued scalar function. The variation of the global

energy J(u) is de¯ned with the Ga^teaux derivative:

±J(u; ´) = lim "!0