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274 Polynomial Approximation of Differential Equations
Now the error En,m := |U ′ (0) − p ′ n(0)|, depends also on the discretization parameter
m. One can prove that limn,m→+∞ En,m = 0. We give in table 12.2.2 some computed
values. As the reader will note, with nearly the same number of collocation nodes as
the previous algorithm, one may obtain better results.
n En,3 En,4 En,5
8 .087622 .115119 .126074
12 .057295 .029799 .018844
16 .031526 .004030 .006925
Table 12.2.2 - Errors for the approximation
of problem (12.2.1) by the scheme (12.2.4).
12.3 The nonlinear Schrödinger equation
Many phenomena in plasma physics, quantum physics and optics are described by the
nonlinear Schrödinger equation, which is introduced for instance in trullinger, za-
kharov and pokrovsky(1986). After dimensional scaling the equation takes the form
(12.3.1) i ∂U
∂t (x,t) + ζ ∂2 U
∂x 2 (x,t) + ǫ |U(x,t)|2 U(x,t) = 0 x ∈ I, t ∈]0,T],
where i = √ −1 is the complex unity, ζ > 0 is proportional to the so called dispersion
parameter, ǫ ∈ R and T > 0. The unknown U : I × [0,T] → C is a function
assuming complex values. Generally we have I ≡ R and U is required to decay to zero
at infinity. For simplicity, we take I =] − 1,1[ and impose the homogeneous boundary