Solitons in Nonlocal Media

3.2 Role of the Boundary Conditions on the Nonlinear Index Perturbation
with K and µ < Kπ2
a 2 given constants. Eq. (3.27) governs reorientation in liquid
crystals in an isotropic configuration (30) and is largely used as a ruling equation for
refractive index in nonlinear nonlocal media (34; 83). I consider the same geometry
investigated in section 3.2.2. Using the same normalizations I get
κ∇ 2 ξυ ∆ρ + µ∆ρ + |A(ξ, υ)|2 = 0 (3.28)
In the next section I compute the corresponding Green function, necessary to eval-
uate the nonlinear perturbation through eq. (3.11).
3.2.3.1 Green Function for the Screened Poisson Equation
To find the Green function G I have to solve
κ∇ 2 G + µG = δ(ξ − ζ, υ − η) (3.29)
with boundary conditions as in section 3.2.2.1. Developing G in a Fourier series re-
spect to ξ as previously done for the Poisson equation, I can write ∞
m=1 bm(υ, ζ, η)sin(πmξ).
Substitution of the latter into eq. (3.31) leads to
∞
m=1
κ ∂2bm ∂υ2 + µbm − κbm (πm) 2
sin(πmξ) = δ(ξ − ζ, υ − η) (3.30)
Every coefficient bm is determined by ∂2bm ∂υ2 − (πm) 2 − µ
κ
bm = 2
κ sin(πmζ)δ(υ −η),
which is equal to eq. (3.17) with the transformation (πm) 2
→ (πm) 2 − µ
1.
κ From eq.
(3.19) it is easy to compute
3.2.3.
G(ξ, |υ − η|, ζ) = −
∞
m=1
1
(πm) 2 − µ
κ
1 To have a real square root ∀m, the relationship µ < Kπ 2
sin(πmζ) e −
(πm) 2 − µ
κ |υ−η| sin(πmξ) (3.31)
47
a 2 must hold true, as anticipated in section