36
R. Grimshaw and C. Yuan
It is important to note that the theory and modelling is developed for the transformed equation (13) which has only one variable coefficient, namely 𝛼, rather than
for the original equation (7) which has five variable coefficients, namely c, Q, 𝜇.𝜆, 𝜎.
The back transformation from (13) to (7) involves only amplitude factors and changing the time and space scales, see the sequence of equations (9, 10, 11, 12). Hence
the essential dynamics is not changed when expressed in the physical variables.
Although, importantly the amplitudes may be significantly altered during the back
transformation, the polarity of the waves is not changed. We have focussed on two
principal scenarios. In the first there is no polarity change, that is, the coefficient 𝛼
does not change sign. Then as the undular bore propagates in the variable medium,
the leading waves separate and form a solitary wave train, with subsequent adiabatic behaviour. In the second case there is a polarity change, that is the coefficient
𝛼 changes at a critical point. The passage of the undular bore through the critical
point does involve some non-adiabatic behaviour, which leads after the critical point
to the formation of a rarefaction wave of the same polarity as the original waves in
the undular bore, and the formation of a solitary wave train of the opposite polarity
riding on the pedestal formed by the rarefaction wave. Application of simulations of
the transformed equation (13) to actual oceanic situations seem to be quite rare, but
we note the recent study by [23].
Acknowledgements RG was supported by the Leverhulme Trust through the award of a Leverhulme Emeritus Fellowship.
Appendix
Here we summarise the derivation of the equations (18, 19, 20) presented in [16].
When the coefficient 𝛼 in (12) is a constant the KdV equation supports a periodic
travelling wave, U(X − V𝜏), the well-known cnoidal wave,
U = a {b(m) + cn
2
(𝛾𝜃; m)} + d , 𝜃 = k(X − V𝜏) ,
(28)
V − 𝛼d =
𝛼a
3
{ 2 − m
m
−
3E(m)
mK(m)
}
= 4𝛾
2 k
2
{
2 − m −
3E(m)
K(m)
}
.
(29)
Here cn(x; m) is the Jacobian elliptic function of modulus m, 0 < m < 1, and K(m)
and E(m) are the elliptic integrals of the first and second kind, The expression (28)
has period 2𝜋 in 𝜃 so that 𝛾 = K(m)∕𝜋, while the spatial period is 2𝜋∕k. The (troughto-crest) amplitude is a and the mean value over one period is d. It is a threeparameter family with parameters k, m, d say. As the modulus m → 1, this becomes
a solitary wave, since then b → 0 and cn(x) → sech(x), while 𝛾 → ∞, k → 0 with
𝛾k = Γ fixed. As m → 0, b → −1∕2, 𝛾 → 1∕2, cn(x) → cos (x), and it reduces to a
sinusoidal wave (a∕2) cos (𝜃) of small amplitude a ∼ m and wavenumber k.