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

the IVP reads
η t − u 1 ξ = 0,
u 1t −η ξ
= √ β ρ 2
ρ 1
T [0,2l]
[
u1
]
ξt ,
η(ξ, 0)=η 0 (ξ),
(4.10)
⎧⎪⎨⎪⎩
u 1 (ξ, 0)=u 10 (ξ).
The symbol of the operatorT [0,2l] [·] ξ (that is the composition of one spatial derivative
with the Hilbert transform) in the new coordinates is
− kπ l coth ( kπ
l
)
h 2
,
L
see Appendix C. Applying Fourier Transform (see Appendix C, Eq. (C.2) for its
definition) to problem (4.10) we have
⎧
ˆη t = ikû1,
⎪⎨
û 1t
(
1+ √ β ρ 2
ρ 1
kπ
l coth ( kπ
l
ˆη(k, 0)=̂η 0 (k),
))
h 2
= ikˆη, for k0
L
(4.11)
⎪⎩ u 1 (k, 0)=û10(k).
Substitute û1 from the first equation into the second:
ˆη t t =−
k 2
1+ √ β ρ 2
ρ 1
kπ
The general solution for this ODE is
coth( kπ
l l
) ˆη=−ω 2 (k)ˆη.
h 2
L
ˆη(k, t)=c 1 exp ( iω(k)t ) + c 2 exp ( −iω(k)t ) , k0,
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