Studies on Some Aspects of Light Beam Propagation Through ...

32 Solitons in Bulk CUbic-Quintic Media
For a solution of this problem, let us assume a trial solution of the form
r2
B(z, r) = C(z) exp[- ()2 + ib(z)r 2 ],
2w z
(2.9)
where C(z) is the maximum amplitude, b(z) is the curvature parameter,
w{z) is the beam radius. Ideally, the trial function should include a possibility
to model the dynamically varying radial shape function of the beam.
But that will make the variational analysis more complicated.
The reduced Lagrangian is then given by
(L) = foo Lrdr, (2.10)
./0
Now we can find the variation of (L) with respect to the va.rious Gaussian
parameters C(z), C(z)*, w(z) and b(z). We have
and
Thus the variation with C(z) and C(z)* gives
Co(L) = .T CC* 3 + b ICl 2 T w5
BC '/, 2 zW z 2
ICI 2 T 1 w 5
A.
+ __ {_ + 4b2 }- + _1 ICI4T3w3
2k w 4 2 2V2
A.2 6 4 3 pa ICI 2N + 2
:l
(2.12)
+ 3y'3I C I T 1JJ + 2k (N + 1)5/2 Tg(TJ)W , (2.1:3)