Soliton Propagation Through Photorefractive Media
131
Applying the minimization criteria:
dh
d Q
i
= 0
dh
d Q i
= 0
(65)
with i = 1, 2.
Using Eqs. (65), (64) can be expressed as
2Q 1
1 − A(Q
2
1 − 2Q
2
2 )
= 0
(66)
so that
Q
2
2 =
Q
2
1 − 1
2
(67)
Substituting (66) in (67), the equation obtained is
Q
2
1 + Q
2
1 −
3A
2
Q
4
1 + AQ
2
1 = 0
(68)
Taking A=1, and multiplying by
2
3
and integrating (68) the solution obtained is
Q 1 = −
4
3
sech
2
√
2
√
3
T
(69)
Substituting for T from Eq. (59), the soliton solutions obtained is
Q 1 = −
4
3
sech
2
√
2
√
3
√ λ(t − k 1 z)
(70)
When (t − k 1 z) → ∞, A 1 (z, t) → 0 and hence this is a bright solitary solution, a
profile of which is shown in Fig. 1. Following the same procedure for Eq. (63), we
get the solution as
Q 2 = −
4
3
sech
2
√
2
√
3
√ λ(t − k 2 z)
(71)
131
Applying the minimization criteria:
dh
d Q
i
= 0
dh
d Q i
= 0
(65)
with i = 1, 2.
Using Eqs. (65), (64) can be expressed as
2Q 1
1 − A(Q
2
1 − 2Q
2
2 )
= 0
(66)
so that
Q
2
2 =
Q
2
1 − 1
2
(67)
Substituting (66) in (67), the equation obtained is
Q
2
1 + Q
2
1 −
3A
2
Q
4
1 + AQ
2
1 = 0
(68)
Taking A=1, and multiplying by
2
3
and integrating (68) the solution obtained is
Q 1 = −
4
3
sech
2
√
2
√
3
T
(69)
Substituting for T from Eq. (59), the soliton solutions obtained is
Q 1 = −
4
3
sech
2
√
2
√
3
√ λ(t − k 1 z)
(70)
When (t − k 1 z) → ∞, A 1 (z, t) → 0 and hence this is a bright solitary solution, a
profile of which is shown in Fig. 1. Following the same procedure for Eq. (63), we
get the solution as
Q 2 = −
4
3
sech
2
√
2
√
3
√ λ(t − k 2 z)
(71)