62
N. Makarenko et al.
𝛼 1 (𝜂) =
𝜋 + 𝜂
√
𝜋F 1
, 𝛼 2 (𝜂) =
𝜋 − r𝜂
√
𝜋F 2
since we have at the leading order in 𝜎 the relations 𝜆 j = 1∕
√
𝜋F j (j = 1, 2) resulted
under the condition (15). Denominator Q in (17) has the following form
2Q(𝜂; F 1 , F 2 ) =
=
(
𝜋F
2
1 − 2
√ 𝜋F 1 𝜂 cot 𝛼 1 (𝜂) + 𝜂
2
cot
2
𝛼 1 (𝜂)
) (
𝜂 + 𝜋
sin
2
𝛼 1 (𝜂)
−
√
𝜋F 1 cot 𝛼 1 (𝜂)
)
+
+
( 𝜋F
2
2
r 2 − 2
√
𝜋F 2
r
𝜂 cot 𝛼 2 (𝜂) + 𝜂
2
cot
2
𝛼 2 (𝜂)
) (
𝜋 − r𝜂
sin
2
𝛼 2 (𝜂)
−
√
𝜋F 2 cot 𝛼 2 (𝜂)
)
.
Small-amplitude waves can be modelled by simplified weakly nonlinear version of
the Eq. (17) which is written as
(
d𝜂
dx
) 2
= 𝜂
2 D 0 + D 1 𝜂 + D 2 𝜂 2
Q(0; F 1 , F 2 )
(18)
where the coefficients D 0 and D 1 are
D 0 = D(0; F 1 , F 2 ) =
√
𝜋F 1 cot
√
𝜋
F 1
+
√
𝜋F 2 cot
√ 𝜋
F 2
− 1,
D 1 = D
′
𝜂 (0; F 1 , F 2 ) = − cot
2
√ 𝜋
F 1
+ r cot
2
√
𝜋
F 2
+
2
3
(r − 1),
and the explicit form of coefficient D 2 is not important here.
Solitary Waves
Solitary-wave solutions of Eq. (17) are given in the implicit form by the following
formula
x = ±
𝜂
∫
a
√
Q(s; F 1 , F 2 )
D(s; F 1 , F 2 )
ds
s
(19)
where parameter a determines non-dimensional amplitude of the wave. Parametric
range of solitary waves is formed by the domain on the (F 1 , F 2 )-plane where radical
function Q∕D is ensured to be non-negative. It is easy to check that Q(0; F 1 , F 2 ) > 0,