matched asymptotic expansions implies finding local corrections near each kink and
their subsequent matching in the regions between the kinks.
Compound Soliton Interaction
Let us consider the interaction of N compound solitons of the Gardner equation
(parameters of the equation are assumed to be constant) with arbitrary ratio of their
sizes. The basic expression for describing their evolution is a superposition of 2N
kinks of alternating polarity (3) after deduction of their total asymptotic behavior:
Φ
0
ð Þ
Ns x, t
ð Þ=
1
2
∑
2N
i = 1
− 1
ð
Þ
i + 1 tanh λ m x − V m t − S i ε t, ε x
ð
Þ
½
ð 5Þ
where S i are kink center coordinates. The procedure of matching the first corrections yields equations for S i ðx, tÞ = S i ðηÞ, η = x − 3t:
dS i
dt
= − 4 e
− S i + 1 − S i
ð
Þ
− e
S i − 1 − S i
ð
Þ
h
i
ð6Þ
and corrects the general solution for field distribution (5) [5]:
Φ
ð0Þ
N i
+ Φ
ð1Þ
N i
x, t
ð Þ=
1
4
∑
2N
i = 1
ð − 1Þ
i + 1 1 −
∂S i
∂x
th
1
2
ðξ − S i Þ
ð 7Þ
Exactly integrable system (6), the Langmuir chain equation, describes the elastic
collision of N solitons without changing their number and parameters. Only phase
shifts of the solitary waves appear. The compound character of the Gardner
equation solitons manifests itself in the asynchronous motion of fronts and decays
of each solitary wave at the stage of their closest convergence. When the kinks
belonging to two different solitons are temporally combined as a result of soliton
collision (Fig. 1), a narrow soliton transforms to an identical soliton of the opposite
polarity moving on the crest of a wide soliton from its front to decay.
The collision process finishes by the inverse transformation of the narrow soliton
to the soliton of initial polarity close to the decay of the wide solitary wave.
Approximate description (5–7) is quite close to the exact one in a wide range of
relative soliton velocities (up to S ̇
i − S i± ̸ V ̇ m ∼ 1 ̸ 2). It is interesting that the exact
N-soliton solution, similarly to the approximate solution, may be represented as a
superposition of quasi-kinks presented by form (7) [5, 6].
It is important that approximate description of N-soliton interactions may also be
constructed for strongly nonlinear models used for describing intense internal
waves (IIW), for example, the Miyta model [7, 8] for a two-layer liquid:
Perturbation Theory for the Compound Soliton …
281
their subsequent matching in the regions between the kinks.
Compound Soliton Interaction
Let us consider the interaction of N compound solitons of the Gardner equation
(parameters of the equation are assumed to be constant) with arbitrary ratio of their
sizes. The basic expression for describing their evolution is a superposition of 2N
kinks of alternating polarity (3) after deduction of their total asymptotic behavior:
Φ
0
ð Þ
Ns x, t
ð Þ=
1
2
∑
2N
i = 1
− 1
ð
Þ
i + 1 tanh λ m x − V m t − S i ε t, ε x
ð
Þ
½
ð 5Þ
where S i are kink center coordinates. The procedure of matching the first corrections yields equations for S i ðx, tÞ = S i ðηÞ, η = x − 3t:
dS i
dt
= − 4 e
− S i + 1 − S i
ð
Þ
− e
S i − 1 − S i
ð
Þ
h
i
ð6Þ
and corrects the general solution for field distribution (5) [5]:
Φ
ð0Þ
N i
+ Φ
ð1Þ
N i
x, t
ð Þ=
1
4
∑
2N
i = 1
ð − 1Þ
i + 1 1 −
∂S i
∂x
th
1
2
ðξ − S i Þ
ð 7Þ
Exactly integrable system (6), the Langmuir chain equation, describes the elastic
collision of N solitons without changing their number and parameters. Only phase
shifts of the solitary waves appear. The compound character of the Gardner
equation solitons manifests itself in the asynchronous motion of fronts and decays
of each solitary wave at the stage of their closest convergence. When the kinks
belonging to two different solitons are temporally combined as a result of soliton
collision (Fig. 1), a narrow soliton transforms to an identical soliton of the opposite
polarity moving on the crest of a wide soliton from its front to decay.
The collision process finishes by the inverse transformation of the narrow soliton
to the soliton of initial polarity close to the decay of the wide solitary wave.
Approximate description (5–7) is quite close to the exact one in a wide range of
relative soliton velocities (up to S ̇
i − S i± ̸ V ̇ m ∼ 1 ̸ 2). It is interesting that the exact
N-soliton solution, similarly to the approximate solution, may be represented as a
superposition of quasi-kinks presented by form (7) [5, 6].
It is important that approximate description of N-soliton interactions may also be
constructed for strongly nonlinear models used for describing intense internal
waves (IIW), for example, the Miyta model [7, 8] for a two-layer liquid:
Perturbation Theory for the Compound Soliton …
281