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Examples 275
conditions U(±1,t) = 0, t ∈ [0,T]. Finally, an initial condition U(x,t) = U0(x), x ∈ Ī,
is provided. A conservation property is immediately obtained by noting that
(12.3.2)
d
|U|
dt I
2 (x,t) dx =
= i ζ U
I
∂2Ū ∂x2 + ǫ |U|4 − ζ Ū ∂2U ∂x
I
U ∂Ū
∂t
2 − ǫ |U|4
∂U
+ Ū (x,t) dx
∂t
(x,t) dx = 0, ∀t ∈]0,T],
where the upper bar denotes the complex conjugate and we used integration by parts.
This means that the quantity
I |U|2 (x,t)dx is constant with respect to t. Another
quantity which does not vary with respect to t is
ζ
∂U
∂x
2 − ǫ
2 |U|4
(x,t)dx.
We denote by V and W the real and imaginary parts of U respectively, hence
U = V + iW. Thus, we rewrite (12.3.1) as a system
(12.3.3)
with
⎧
⎪⎨
⎪⎩
− ∂W
∂t (x,t) = −ζ ∂2 V
∂x 2 (x,t) − ǫ |U(x,t)|2 V (x,t)
∂V
∂t (x,t) = −ζ ∂2 W
∂x 2 (x,t) − ǫ |U(x,t)|2 W(x,t)
(12.3.4) V (±1,t) = W(±1,t) = 0, t ∈ [0,T].
I
x ∈ I, t ∈]0,T],
The initial conditions are V (x,0) = V0(x), W(x,0) = W0(x), x ∈ Ī, U0 = V0 + iW0.
For the approximation we use the collocation method based on the Legendre Gauss-
Lobatto points for the variable x, and an implicit finite-difference scheme for the variable
t (see section 10.6). The initial polynomials in P 0 n, n ≥ 2 (see (6.4.11)) are such that
p (0)
n := Ĩw,nV0 and q (0)
n := Ĩw,nW0 , where the interpolation operator is defined in
section 3.3 (w ≡ 1). We subdivide the interval [0,T] in m ≥ 1 equal parts of size
h := T/m. The successive polynomials p (k)
n , q (k)
n ∈ P 0 n, 1 ≤ k ≤ m, are determined at
the nodes η (n)
i , 1 ≤ i ≤ n − 1, by the formulas