Solitons in Nonlocal Media

3.2 Role of the Boundary Conditions on the Nonlinear Index Perturbation
is given by (3.19). So g1 = sin[2(θ0 − δ)] ∞
m=1 1
πm Vm(υ)sin(πmξ) where for Vm(υ)
the equations derived above keep their validity. If the excitation field |A| 2 is not the
product of two functions, both of them dependent only from one transverse variable, I
have to compute g2(ξ, υ) through the general expressions (3.21) and (3.22), with forcing
term |A| 2 θ1 [see eq. (3.43)]. In the opposite case, i.e. when |A(ξ, υ)| 2 = fξ(ξ)fυ(υ), I
have
that
|A(ξ, υ)| 2 θ1(ξ, υ) =
∞
Al(ξ)Bl(υ) (3.45)
l=0
being Al(ξ) = 1 ξ
πlfξ(ξ)V l sin(πlξ) and Bl(υ) = fυ(υ)V υ
l
having defined
g2(ξ, υ) = sin[4(θ0 − δ)]
F m
∞
l (υ) =
−∞
G m l =
1
0
∞
m=0
1
πm sin(πmξ)
(υ). It is easy to demonstrate
∞
l=0
G m l
m
Fl (υ) (3.46)
Bl(η)e −πm|(υ−η)| dη (3.47)
sin(πmζ)Al(ζ)dζ (3.48)
Equation (3.48) is formally identical to eq. (3.23) and can be computed by means
of eq. (C.13), as demonstrated in C.3. Equation (3.47) has no analytical expression,
even in the Gaussian case, so I must evaluate them numerically. Computed g1 and g2
for Gaussian profiles with 〈ξ〉 = 0.5 and 〈ξ〉 = 0.75 are shown in fig. 3.6 and fig. 3.7,
respectively: the two functions possess the same shape with very good approximation,
i.e. g2(ξ, υ) ∼ = cg1(ξ, υ). This means that the reorientation angle in NLC and the solu-
tions of Poisson equation behave nearly the same for ω