96
V. Liapidevskii and N. Gavrilov
By choosing the corresponding values 𝛽 ± , Eq. (7) describe steady-state flows for all
mentioned above models BM, ULM, BLM and HM.
Solitary Waves
Consider the special class of solutions for ULM, BLM, SM, which satisfy for |x| →
∞ the following conditions
h → h 0 , 𝜁 → 𝜁 0 , u → u 0 , w → u 0 ,
h
′
→ 0, h
′′
→ 0, 𝜁
′
→ 0, 𝜁
′′
→ 0
(8)
(here and below, the prime denotes differentiation with respect to the variable x).
Solutions of the problem (7)–(8) describe solitary waves moving with the velocity
u 0 in the laboratory system of coordinates.
Solitary Waves in SM
For SM the solitary waves have been investigated in [7, 8]. The soliton-like solution
for SM can be found in quadratures. To see that we rewrite (7) for SM in dimensionless variables
h ∗ = h∕H 1 , u ∗ = u∕
√
̄
bH 1 , p ∗ = p∕ ̄
bH 1 ,
t ∗ = t
√
̄
b∕H 1 , x ∗ = x∕H 1 , Fr s = u 0 ∕
√
̄
bH 1 .
(“star” is omitted below).
The governing equations for (7)–(8) take the form
hu = h 0 Fr s , (1 − h)v = (1 − h 0 )Fr s ,
1
2
v
2
+ p =
1
2
Fr
2
s
+ p 0 ,
1
2
u
2
+ h + p +
1
3
hu
2 h
′′
−
1
6
u
2
(h
′
)
2
=
1
2
Fr
2
s + h 0 + p 0 .
(9)
Equation (9) can be reduced to ODE [9]
(h
′
)
2
= G(h) =
3(h − h 0 ) 2 (Fr 2
s
− h + h 2 )
Fr 2
s
h
2
0
(1 − h)
.
(10)
It follows from (10) that the solitary waves exist for