26
R. Grimshaw and C. Yuan
obtained by [16] for internal undular bores. In section “Solitary Wave Train” we
present the modulation equations for a solitary wave train, and describe briefly how
these may be used when waves propagate in a region where 𝛼 = 𝛼(𝜏) varies. Then
in section “Numerical Simulations of a Process Model” we discuss how a solitary
wave train behaves when there is a change of polarity, that is, there is a critical point
where 𝛼 changes sign. We conclude in section “Discussion”.
Solitary Wave Train
The equations for the asymptotic description of a solitary wave train were initially
developed by [7]. They can also be obtained from the Whitham equations for a modulated periodic wave train by taking the solitary wave limit, see [5, 16]. That approach
is summarised in the Appendix, and here we just present the outcome.
U = a sech
2
(𝛾𝜃; m)} + d , 𝜃 = k(X − V𝜏) , V − 𝛼d =
𝛼a
3
= 4𝛾
2 k
2
,
(17)
with three parameters, the amplitude a, the pedestal d and the wavenumber k, to be
determined. The pedestal d satisfies
d 𝜏 + 𝛼dd X = 0 ,
(18)
and so can be regarded as a known quantity. The equation for conservation of waves
(30) provides one equation for k
k 𝜏 + (kV) X = 0 .
(19)
and the third equation is (37), that is
{
a
2
k𝛾
} 𝜏 + V{
a
2
k𝛾
} X +
a
2
k𝛾
𝛼d X = 0 ,
(20)
The pair (19, 20) form a nonlinear hyperbolic system for a solitary wave train, and
can be solved explicitly. The wavenumber k can be eliminated from (19) and (20) to
yield
A 𝜏 + (𝛼d +
𝛼a
3
)A X + A 𝛼d X = 0 , A = {
a
3
𝛼
}
1∕2
.
(21)
This is now an equation for the amplitude a alone, and is readily solved using characteristics. Then, with a, d and hence V also known, the wavenumber k can be found
from (19) which is a linear hyperbolic equation for k.
Formally the modulation equations for a single solitary wave can be found by
considering modulations in 𝜏 alone. Then (18) shows that d is a constant which can
be set to zero without loss of generality. Next, it follows from (19, 21) that k, A are