Solitons in Nonlocal Media

3.3 Soliton Trajectory
to n = 1 is always zero for the definition of 〈x〉. Finally, if η(y) is even, all the odd
terms in eq. (3.62) are zero, being y 2j+1 = 0 ∀j ∈ N, independently from Veq, i.e. in
every medium.
3.3.2 Power series for the Equivalent Force
In general, Wn = ∞
l=0 cl n (〈x〉 − x0) l can be written using a Taylor expansion around
〈x〉 = x0, where x0 is the initial beam position, Substituting in eq. (3.62) I get
F m X (〈x〉) =
∞
n=0 l=0
∞
c l n (〈x〉 − x0) l 〈y n 〉 η
(3.63)
Without loss of generality I can set x0 = 0. If the problem is invariant under the
transformation x → −x (reflection with respect to the plane x = 0), the force on the
beam must be odd; this implies c2l n = 0 (l = 0, 1, 2, . . .). Inserting the last in eq. (3.63)
I get F m X (〈x〉) = ∞ ∞ n=0 l=0 c2l+1 n 〈x〉 2l+1 〈yn 〉 η . The beam undergoes an equivalent
potential
V m X (〈x〉,s) = −
〈x〉
0
F m X (x ′ , s)dx ′ = −
∞
∞
n=0 l=0
1
(2l + 2) c2l+1
n 〈x〉 2l+2 〈y n 〉 η (3.64)
For small displacements from x = 0 in (3.64), x powers larger than 2 can be ne-
glected; hence, the potential V m X
takes the form
V m X (〈x〉,s) = − 1
∞
c
2
n=0
1 n 〈y n
〉 η 〈x〉 2
(3.65)
Eq. (3.65) tells me that for small amplitude motions the beam is subjected to
a classical harmonic oscillator potential, with an equivalent spring constant K =
∞
n=0 c1 n 〈y n 〉 η dependent on s. The quantity K depends on beam profile momenta
〈y〉 η , that remain unchanged in soliton propagation: in the latter case K varies with s
only through the coefficients c l n.
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