a reduced model for internal waves interacting with submarine ...

homogeneuous (flat) regions at the extremes and keep the dynamics away from
these regions. Hence both profile and initial disturbance will be defined and kept
away fromξ=0=2l.
From system (4.1), a hierarchy of one-dimensional models can be derived by
considering the different regimes (linear, weakly nonlinear or strongly nonlinear)
as well as the flat or corrugated bottom cases. The Weakly Nonlinear Corrugated
Bottom Model (WNCM) was already obtained in (2.30). In curvilinear coordinates
for a periodic domain it reads
⎧
η t − 1 [ ] (1−αη)u1
ξ ⎪⎨ M(ξ)
= 0,
⎪⎩
u 1t + α
M(ξ) u 1 u 1ξ − 1
M(ξ) η ξ= √ β ρ 2
ρ 1
1
M(ξ) T [0,2l]
Settingα=0 we obtained the Linear Corrugated Bottom Model (LCM)
⎧
η t − 1
⎪⎨ M(ξ) u 1ξ= 0,
⎪⎩
u 1t − 1
M(ξ) η ξ= √ β ρ 2
ρ 1
1
M(ξ) T [0,2l]
[
u1
]
ξt . (4.2)
[
u1
]
ξt . (4.3)
The flat bottom versions are obtained by simply taking M(ξ)=1 for allξ∈
Π[0, 2l]. To fix a notation, let us use the abbreviations in Table 4.1 to refer to each
model.
4.2 Method of lines
To find the solution for the initial value problem of systems SNCM, WNCM,
SNFM, WNFM, LCM is a nontrivial task. That is why we resort to numerical
methods to find approximate solutions.
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