Solitons in Nonlocal Media

3.4 Soliton Oscillations in a Finite-Size Geometry
2 2
ǫa sin (θ − δ) − sin (θ0 − δ) , with m = k0ne cos δ. In section 3.2.4.2 I demonstrated
that, for typical (experimental) powers, only terms up to P 2 must be considered to reach
a good approximation; hence, I can set ∆n = ǫa sin[2(θ0 − δ)]Ψ + cos[2(θ0 − δ)]Ψ2 and, from eq. (3.42), perturbation angle is Ψ = γPg1 +γ 2 P 2 g2. Thus, considering only
terms up to P 2 , I get:
∆n ∼
= ǫa sin[2(θ0 − δ)] γPg1 + γ 2 P 2
g2 + cos[2(θ0 − δ)]γ 2 P 2 g 2 1
Therefore, the equivalent potential Veq (defined in section 3.3.1) is:
where I defined
V L
eq = − ǫak0
2ne cos δ sin[2(θ0 − δ)]
Veq = V L
eq + V NL
eq
|A| 2 = Ce −[ξ2 /ω 2 x+υ 2 /ω 2 t] with C = 2Z0/ neπωxωta 2
γP
πω 2 t
V NL
eq = − ǫak0
2ne cos δ cos[2(θ0 − δ)] γ2 P 2
πω 2 t
∞
∞
−∞
−∞
υ2
−
ω e 2 t g1dυ + γ2P 2
πω2 t
υ2
−
ω e 2 t g 2 1dυ
The force W0 acting on the soliton is (section 3.3.1):
being W L 0
∂V L
eq
= and W
ξ=〈ξ〉 NL
∂ξ
0
W0 = W L 0 + W NL
0
∂V NL
eq
= ∂ξ
∞
−∞
(3.74)
(3.75)
υ2
−
ω e 2
t g2dυ
(3.76)
(3.77)
the terms stemming from linear and
ξ=〈ξ〉
quadratic parts 1 of ∆n, respectively. Substituting definitions of g 1/2 [see eqs. (3.43)]
in (3.76), the two forces W L 0
and W NL
0 are:
W L 0 = ǫak0
2ne cos δ sin[2(θ0
∞
− δ)] γCP sin[2(θ0 − δ)] V
m=1
ξ mV υ
m cos(πm 〈ξ〉)
+γ 2 C 2 P 2
∞
∞
sin[4(θ0 − δ)] cos(πm 〈ξ〉)
1 With respect to Ψ.
67
m=1
l=1
G m l Hm l
(3.78a)