Solitons in Nonlocal Media

3.2 Role of the Boundary Conditions on the Nonlinear Index Perturbation
conversely, the continuity of bp in υ = η leads to A = B: solving the system I find
A = B = − 1
πp sin (πpζ). Putting this result in eq. (3.18) provides bp(υ, ζ, η) =
− 1
πp sin(πpζ) e−πp|υ−η| . From eq. (3.15), the sought Green function G is
G(ξ, |υ − η|, ζ) = −
∞
m=1
1
πm sin(πmζ) e−(πm|υ−η|) sin(πmξ) (3.19)
G depends on |υ − η| because of translational symmetry along the υ axis. Further-
more, when ξ = ζ and υ = η, i.e. when the response is calculated in the same point
of the forcing term, I have G(ξ = ζ,0) = − ∞
m=1 1
πm sin2 (πmζ), that diverges for
ζ = ha (h = 1, 2, . . .), as espected.
3.2.2.2 Perturbation Profile
As stated above, eq. (3.19) is a diverging harmonic series in ξ = ζ, υ = η, and must be
inserted into eq. (3.11) to find ∆ρ; to compute the total perturbation from an intensity
profile |A(ζ, η)| 2 , I take the series out of the integral, obtaining:
∆ρ(ξ, υ) =
=
∞
−∞
∞
m=1
1
dη
0
∞ 1
πm sin(πmξ)sin(πmζ)e−πm|υ−η| |A(ζ, η)| 2 dζ =
m=1
1
πm sin(πmξ)
∞
−∞
dη
1
0
sin(πmζ)e −πm|υ−η| |A(ζ, η)| 2 dζ
(3.20)
Eq. (3.20) is the Fourier series along the axis ξ for the perturbation profile, that is
∆ρ(ξ, υ) =
∞
m=1
where the harmonic coefficients Vm(υ) are given by
Vm(υ) =
∞
−∞
dη
1
0
1
πm Vm(υ)sin(πmξ) (3.21)
sin(πmζ)e −πm|υ−η| |A(ζ, η)| 2 dζ. (3.22)
If the intensity profile is in the form |A(ξ, υ)| 2 = fξ(ξ)fυ(υ) I derive that Vm(υ) =
V ξ mV υ m(υ), being
V ξ 1
m =
0 ∞
V υ m(υ) =
−∞
fξ(ζ)sin(πmζ)dζ (3.23)
fυ(η)e −πm|υ−η| dη (3.24)
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