192
S. V. Vasylyuk et al.
In this case in the zero approximation by the external field (or at the field switched
off), both these equations for each branch “±” coincide and reduce themselves to
the form:
1
2μ
1 + τ p v ⊥
∂ 2 Φ ± (ξ ± )
∂ξ 2
±
+
2g 11
ξ ± + x
±
c
+ g 10
ξ ± + x
±
c
Φ 3
± (ξ ± )
+
⎡
⎣ Ω +
⎛
⎝
x
+
y
⎞
⎠ +
1
μ
+ τ p v ⊥
1
2
+
1
μ
⎤
⎦ Φ ± (ξ ± ) = 0. (14.49)
In this case, we designate g 00 ≡ g 11 ≡ g p ; (14.50) g 00 ≡ g 11 ≡ g p ” (14.51).
That is, under conditions of the absence of the external field or in zero approximation by it, parameters g i j , as follows from their definitions (14.20) ÷ (14.22),
(14.25), do not depend on the variable x = ξ ± + x
±
c (id est, they are constants) and
coincide with pairs according to definitions (14.50), (14.51).
In the approximation considered (|β( p)| → 0 , | p| → 0 {Π } → 0) the dependence of μ on p is rather weak and can be neglected. In this case, the functions Φ
(0)
± (ξ ± ) do not obtain additional, besides x
±
c (τ ), dependence on τ , which
is forbidden by the system (14.39) [7].
For simplifications of computations, make 2g + g ⊥ ≡ g, and also
Ω +
1
μ
± τ p v ⊥
1
2
+
1
μ
≡ ε ± ; (14.52) 1 ± τ v ⊥ ≡ s ± .
(14.53)
Make (14.49) to the more compact form [26–29]:
s ±
2μ
(0)
Φ
± (ξ ± ) + g
(0)
Φ
3
± (ξ ± ) + ε ±
(0)
Φ
±
(ξ ± ) = 0.
(14.54)
This equation is well known [7] as the soliton equation.
Using the results of the first part of this work, let us consider now in the approximation [8], zero by field (this approximation in the physical sense corresponds to
the external field switched off), and the conductivity, caused by the transfer of the
injected charge. From the following formulae of the first part [7]
Ω +
1
μ
± τ v ⊥
1
2
+
1
μ
≡ ε ±
(1.52)
1 ± τ v ⊥ ≡ s ±
.
(1.53)
“From (2.8) for each mode of the I ± , we can, at last, write [8] current, caused by
the transfer of the injected charge
I + =
e
M
N
(1 + τ ν ⊥ ) sin( p) −
g
2 sin
4
(φ) sin( p)
8(1 + τ ν ⊥ ) cos 2 ( p)
(2.9)