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

c 0 chosen, there will be a right- or left-travelling wave.
Then, dropping the asterisks, Eq. (2.26) becomes
⎧
η ⎪⎨ t − [ (1−αη)u 1
]x = 0,
⎪⎩ u 1t +αu 1 u 1x −η x = √ β ρ 2 1
ρ 1 M(ξ) T[ M(˜ξ) [ ] ] (2.30)
(1−αη)u 1 xt + O(β).
Note that
η t = u 1x + O(α); η x = u 1t + O ( α, √ β ) . (2.31)
As in [8], we look for a solution, up to a first order correction inαand √ β, in
the form
η=A 1 u 1 +αA 2 u 1 2 + √ β A 3
1
M(ξ) T[ M(˜ξ)u 1t
]
. (2.32)
Substituting in the system of Eqs. (2.30) up to orderα, √ β, two equations for
u 1 are obtained:
0=A 1 u 1t + 2αA 2 u 1 u 1t + 2αA 1 u 1 u 1x − u 1x + √ β A 3
1
M(ξ) T[ M(˜ξ)u 1tt
]
and
0=u 1t +αu 1 u 1x −
[
A 1 u 1x + 2α A 2 u 1 u 1x + √ β A 3
1
M(ξ) T[ M(˜ξ)u 1xt
] ] −
− √ β ρ 2
ρ 1
1
M(ξ) T[ M(˜ξ)u 1xt
]
+ O
(
α 2 ,β,α √ β ) .
For compatibility A 1 =±1 and to choose a right-going wave we take A 1 =−1.
Therefore
η x =−u 1x + O ( α, √ β ) , (2.33)
30