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

3. For system (2.28) a similar unidirectional reduction can be obtained leading
to
η t +η x − 3 2 αηη x− ρ 2
ρ 1
√ β
2 H[η xt]=0,
which is a regularized Benjamin-Ono (BO) equation over an infinite bottom
layer.
Hence Eq. (2.37) is a generalization of the BO equation for intermediate
depth and the presence of a topography. This equation is valid only when
backscattering is negligible.
2.5 Solitary wave solutions
The ILW equation we referred to in Section 2.4 and derived in [8, 14, 17] is of the
form
η t +η ξ + c 1 ηη ξ + c 2 T [η ξξ ]=0 (2.40)
where c 1 =− 3 2 α and c 2= ρ 2
√ β
ρ 1 2
solitary wave solutions [14, 8]
. Eq. (2.40) admits a family in the parameterθof
η(x)=
a cos 2 θ
, x=ξ− ct, (2.41)
cos 2 θ+sinh 2 (x/λ)
where
a= 4c 2θ tanθ
h 2 c 1
, λ= h 2
θ , c=1− 2c 2
h 2
θ cot(2θ),
with 0≤θ≤π/2. Alternatively, we consider a Regularized Intermediate Long
Wave equation
η t +η ξ + c 1 ηη ξ − c 2 T [η ξt ]=0 (2.42)
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