Basic Points of the Approximate Approach
Equation (1) with constant coefficients α, β, and μ > 0, has a family of soliton
solutions:
Φ s x, t
ð Þ= Φ +
D
2
th λ x − vt + Δ
ð
Þ− th λ x − vt − Δ
ð
Þ
½
,
ð2Þ
which depend on an arbitrary pedestal Φ x, t
ð Þ and on one more parameter, namely,
dimensional variable λΔ = r; D = Φ m th 2r, λ =
1
2
ffiffiffiffiffiffiffiffiffiffi
v m − v 0
β
q
th 2r, Δ = r
ffiffiffiffiffiffiffiffiffiffi
β
v m − v 0
q
cth2r,
v − v 0
ð
Þ
v m − v 0
ð
Þ = th
2 2r, Φ m =
α
μ − 2Φ, v m =
α
2
6μ +
Φ
3 α − μΦ
À
Á
, and v 0 = Φ α − μΦ
À
Á
. Soliton
amplitude A = max Φ s − Φ
À
Á
and width L at HWFM (
1
2 A level) is also expressed
through parameter r = λΔ: A = Φ m 1 − ch
− 1 2r
ð
Þ , chλL = 2 + ch2r.
Besides soliton solutions, Eq. (1) has a one-parameter family of kink solutions
Φ
+
k x, t
ð Þ= Φ +
Φ m
2
1±thλ m x − v m t
ð
Þ
½
,
ð3Þ
where λ m =
1
2 Φ m
ffiffiffi ffi
μ
6β
q
.
Soliton solution (2) may be written in the exact form as a combination of kink
solutions
with
re-normalized
(due
to
kink
interaction)
parameters:Φ m → D, λ m → λ, v m → v:
Φ s x, t
ð Þ= Φ
+
k + Φ
−
k − ðΦ + DÞ
ð 4Þ
Parameter r is equal to the ratio of the distance between the kinks (2Δ) and the
characteristic scale of field jumps in the kinks λ
− 1
m
À
Á
. At r ≪ 1, the soliton amplitudes are small due to the strong overlapping of the kinks. Their velocity is close to
minimal v 0 and the size does not depend on 2Δ; it is determined only by the λ
− 1
m
scale. In general, such solitons are close to the KdV solitons and have no obvious
signs of a compound structure. At r ≫ 1, the soliton amplitude and velocity are
close to the maximum possible values of Φ m , v m . The soliton shape tends to a
rectangle and size L, which almost coincides with the distance between the kinks
2Δ, hence the composite structure of the solitary wave becomes apparent. The
structure of expression (4) is typical for the solutions obtained by the method of
matched asymptotic expansions. The general solution consists of the sum of
solutions in internal regions with fast field variation Φ
+
k + Φ
−
k (soliton field jumps
≈ thλ m x − v m t
ð
Þ
½
) after deduction of their total asymptotic behavior that is a solution
in the external regions with slow field variation (in this case, they are constant
values ðΦ + DÞ). Expression (4) gives a correct description of the general structure
of the field. This representation will be used below for describing
quasi-nonstationary processes. Construction of the solutions by the method of
280
I. A. Soustova et al.
Equation (1) with constant coefficients α, β, and μ > 0, has a family of soliton
solutions:
Φ s x, t
ð Þ= Φ +
D
2
th λ x − vt + Δ
ð
Þ− th λ x − vt − Δ
ð
Þ
½
,
ð2Þ
which depend on an arbitrary pedestal Φ x, t
ð Þ and on one more parameter, namely,
dimensional variable λΔ = r; D = Φ m th 2r, λ =
1
2
ffiffiffiffiffiffiffiffiffiffi
v m − v 0
β
q
th 2r, Δ = r
ffiffiffiffiffiffiffiffiffiffi
β
v m − v 0
q
cth2r,
v − v 0
ð
Þ
v m − v 0
ð
Þ = th
2 2r, Φ m =
α
μ − 2Φ, v m =
α
2
6μ +
Φ
3 α − μΦ
À
Á
, and v 0 = Φ α − μΦ
À
Á
. Soliton
amplitude A = max Φ s − Φ
À
Á
and width L at HWFM (
1
2 A level) is also expressed
through parameter r = λΔ: A = Φ m 1 − ch
− 1 2r
ð
Þ , chλL = 2 + ch2r.
Besides soliton solutions, Eq. (1) has a one-parameter family of kink solutions
Φ
+
k x, t
ð Þ= Φ +
Φ m
2
1±thλ m x − v m t
ð
Þ
½
,
ð3Þ
where λ m =
1
2 Φ m
ffiffiffi ffi
μ
6β
q
.
Soliton solution (2) may be written in the exact form as a combination of kink
solutions
with
re-normalized
(due
to
kink
interaction)
parameters:Φ m → D, λ m → λ, v m → v:
Φ s x, t
ð Þ= Φ
+
k + Φ
−
k − ðΦ + DÞ
ð 4Þ
Parameter r is equal to the ratio of the distance between the kinks (2Δ) and the
characteristic scale of field jumps in the kinks λ
− 1
m
À
Á
. At r ≪ 1, the soliton amplitudes are small due to the strong overlapping of the kinks. Their velocity is close to
minimal v 0 and the size does not depend on 2Δ; it is determined only by the λ
− 1
m
scale. In general, such solitons are close to the KdV solitons and have no obvious
signs of a compound structure. At r ≫ 1, the soliton amplitude and velocity are
close to the maximum possible values of Φ m , v m . The soliton shape tends to a
rectangle and size L, which almost coincides with the distance between the kinks
2Δ, hence the composite structure of the solitary wave becomes apparent. The
structure of expression (4) is typical for the solutions obtained by the method of
matched asymptotic expansions. The general solution consists of the sum of
solutions in internal regions with fast field variation Φ
+
k + Φ
−
k (soliton field jumps
≈ thλ m x − v m t
ð
Þ
½
) after deduction of their total asymptotic behavior that is a solution
in the external regions with slow field variation (in this case, they are constant
values ðΦ + DÞ). Expression (4) gives a correct description of the general structure
of the field. This representation will be used below for describing
quasi-nonstationary processes. Construction of the solutions by the method of
280
I. A. Soustova et al.