Ablowitz M.A. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons
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5.3 Non-dimensionalization 105
The leading term in the shallow-water dispersion relationship is ω
2
ghκ
2
.
Alternatively, as discussed in earlier chapters, we have that to leading order
the amplitude η satisfies the wave equation,
η
tt
− c
2
0
η
xx
= 0,
with a propagation speed c
2
0
= gh. We will come back to this point when we
discuss the KdV equation later in this chapter.
5.3 Non-dimensionalization
To analyze the full equations, we will first non-dimensionalize them. By doing
this, it will be easier to compare the “size” of each term. This is convenient
since we will then be working with “pure” or dimensionless numbers. For
example, a velocity can be large if measured in units of, say, microns per sec-
ond but in units of light-years per hour it would be small. Indeed small or large
in terms of coefficients in an equation are relative concepts and are meaningful
only in a comparative sense. By studying non-dimensional equations we can
more easily decide which terms are negligible.
For shallow-water waves it is convenient to use the following non-
dimensionalization:
x = λ
x
x
, y = λ
y
y
, z = hz
,
t =
λ
x
c
0
t
,η= aη
,φ=
λ
x
ga
c
0
φ
,
where c
0
=
gh is the shallow-water wave speed, λ
x
,λ
y
are typical wave-
lengths of the initial data (in the x or y direction), and a is the maximum or
typical amplitude of the initial data. The primed variables are dimensionless.
For Laplace’s equation:
1
λ
2
x
φ
x
x
+
1
λ
2
y
φ
y
y
+
1
h
2
φ
z
z
= 0.
For the sake of notational simplicity, we will drop the primed notation so that
φ
zz
+
h
λ
x
2
φ
xx
+
h
λ
y
2
φ
yy
= 0, −1 < z <
aη
h
.
The no-flow condition becomes
∂φ
∂z
= 0, z = −1.
106 Water waves and KdV-type equations
Bernoulli’s equation becomes
φ
t
+
a
2h
⎡
⎢
⎢
⎢
⎢
⎢
⎣
φ
2
x
+
λ
x
λ
y
2
φ
2
y
+
λ
x
h
2
φ
2
z
⎤
⎥
⎥
⎥
⎥
⎥
⎦
+ η = 0, z =
aη
h
.
And the kinematic condition becomes
η
t
+
a
h
⎡
⎢
⎢
⎢
⎢
⎢
⎣
φ
x
η
x
+
λ
x
λ
y
2
φ
y
η
y
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
λ
x
h
2
φ
z
, z =
aη
h
.
Now we define the dimensionless parameters ≡ a/h, δ ≡ λ
x
/λ
y
, and
μ ≡ h/λ
x
: is a measure of the nonlinearity, or amplitude, of the wave; μ
is a measure of the depth relative to the characteristic wavelength, sometimes
called the dispersion parameter; and δ measures the size of the transverse vari-
ations. In summary, the dimensionless equations for water waves propagating
over a flat bottom are
• Euler ideal flow, Laplace’s equation
φ
zz
+ μ
2
φ
xx
+ μ
2
δ
2
φ
yy
= 0, −1 < z <η.
• No flow through the bottom
∂φ
∂z
= 0, z = −1.
• Bernoulli’s or the pressure equation
φ
t
+
2
φ
2
x
+ δ
2
φ
2
y
+
1
μ
2
φ
2
z
+ η = 0, z = η.
• Kinematic condition:
μ
2
η
t
+
φ
x
η
x
+ δ
2
φ
y
η
y
= φ
z
, z = η.
We note that the linear, or small-amplitude, case arises when 1, μ ∼O(1).
Then the above equations reduce, by dropping terms of order , to the linear
equations that are equivalent to those we analyzed earlier, i.e.,
φ
t
+ η = 0, z = 0,
μ
2
η
t
= φ
z
, z = 0,
but now in non-dimensional form.
5.4 Shallow-water theory
We will make some simplifying assumptions about the sizes of , μ, and δ in
order to obtain some interesting limiting equations:
5.4 Shallow-water theory 107
• We will consider shallow water waves (sometimes called long water waves).
This regime corresponds to small depth relative to the water wavelength. In
our system of parameters, this corresponds to μ = h/λ
x
1.
• We will also assume that the wavelength in the transverse direction is much
larger than the propagation wavelength, so δ = λ
x
/λ
y
1.
• As before in the linear regime, we assume small-amplitude waves |ε| =
a/h 1. Recall that a similar assumption was made the Fermi–Pasta–Ulam
(FPU) problem involving coupled nonlinear springs studied in Chapter 1.
• We will make the assumption of “maximal balance” (the small terms, i.e.,
nonlinearity and dispersion, are of the same order) as was made in the con-
tinuum approximation of the FPU problem to the Boussinesq model in order
to derive the KdV equation. We will do the same thing now by assuming
maximal balance with ε = μ
2
. This reflects a balance of weak nonlinearity
and weak dispersion.
5.4.1 Neglecting transverse variations
First, we will consider the special case of no transverse waves and later we
will incorporate them into our model. Let us rewrite the fluid equations with
the simplifying assumptions ε = μ
2
– this is our “maximal balance” assump-
tion. Let us also consider the one-dimensional case, i.e., we remove the terms
involving derivatives with respect to y. The four equations then become
εφ
xx
+ φ
zz
= 0, −1 < z <εη (5.10a)
φ
z
= 0, z = −1 (5.10b)
φ
t
+
ε
2
(φ
2
x
+
1
ε
φ
2
z
) + η = 0, z = εη (5.10c)
ε(η
t
+εφ
x
η
x
) = φ
z
, z = εη. (5.10d)
These are coupled, nonlinear partial differential equations in φ and η with
a free boundary, and are very difficult to solve exactly. We will use perturba-
tion theory to obtain equations that are more tractable. We will asymptotically
expand φ as
φ = φ
0
+ εφ
1
+ ε
2
φ
2
+ ···.
Substituting this expansion into (5.10a)gives
φ
0zz
+ ε(φ
0xx
+ φ
1zz
) + ε
2
(φ
1xx
+ φ
2zz
) + ···= 0.
108 Water waves and KdV-type equations
Equating terms with like powers of ε, we find that φ
0zz
= 0. This implies that
φ
0
= A + B(z + 1), where A, B are functions of x, t. But the boundary condi-
tion (5.10b) forces B = 0. Hence the leading-order solution for the velocity
potential is independent of z,
φ
0
= A(x, t).
Proceeding to the next order in ε, we see that φ
1zz
= −A
xx
. Again using the
boundary condition (5.10b),
φ
1
= −A
xx
(z + 1)
2
/2.
As before, we absorb any homogeneous solutions that arise in higher-order
terms into the leading-order term (that is φ
0
). Similarly we find φ
2
= A
xxxx
(z +
1)
4
/4!. Thus, we have the approximation
φ = A −
ε
2
A
xx
(z + 1)
2
+
ε
2
4!
A
xxxx
(z + 1)
4
+ ···, (5.11)
which is valid on the interval −1 < z <εη, in particular, right up to the free
boundary εη. This expansion can be carried out to any order in ε,butthefirst
three terms are sufficient for our purpose.
Substituting (5.11) into Bernoulli’s equation (5.10c) along the free boundary
z = εη gives
A
t
−
ε
2
A
xxt
(1 + εη)
2
+
ε
2
4!
A
xxxxt
(1 + εη)
4
+ ···
+
ε
2
A
x
−
ε
2
A
xxx
(1 + εη)
2
+ ···
2
+
1
2
(−εA
xx
(1 + εη) + ···)
2
+ η = 0.
Retaining only the first two terms leads to
η = −A
t
+
ε
2
A
xxt
− A
2
x
+ ···. (5.12)
Now let us do the same thing for the kinematic equation (5.10d). Substituting
in our expansion for φ and retaining the two lowest-order terms gives
εη
t
+ ε
2
η
x
A
x
= −εA
xx
(1 + εη) +
ε
2
3!
A
xxxx
+ ···. (5.13)
We wish to decouple the system of equations involving A and η. Since we
have an expression for η in terms of A, we substitute (5.12)into(5.13) and
retain only the two lowest-order terms:
5.4 Shallow-water theory 109
ε
−A
tt
+
ε
2
(A
xxtt
− 2A
x
A
xt
) + ε
2
(−A
xt
A
x
)
= −εA
xx
(1 − εA
t
) +
ε
2
3!
A
xxxx
+ ···.
Dividing through by ε and keeping order terms gives
A
tt
− A
xx
= ε
A
xxtt
2
−
A
xxxx
6
− 2A
x
A
xt
− A
xx
A
t
. (5.14)
Within this approximation, this equation is asymptotically the same as the one
Boussinesq derived in 1871, cf. Ablowitz and Segur (1981). Another form,
equally valid up to order ε, is derived by noting that on the left-hand side
of (5.14)wehaveA
tt
= A
xx
+ O(ε). Then, in particular, we can take two x
derivatives to obtain A
xxtt
= A
xxxx
+ O(ε). Now we replace the A
xxtt
/2termin
the above Boussinesq model to get
A
tt
− A
xx
= ε
A
xxxx
3
− 2A
x
A
xt
− A
xx
A
t
, (5.15)
which is still valid up to O(ε).
Now we will make a remark about the linearized model. We have seen that
there are various ways to write the equation; for example,
A
tt
− A
xx
= ε
1
2
A
xxtt
−
1
6
A
xxxx
(5.16)
A
tt
− A
xx
=
ε
3
A
xxxx
. (5.17)
But only one of these two gives rise to a well-posed problem. We can see this
from their dispersion relations: assume a wave solution A(x, t) = exp(i(kx−ωt))
and substitute it into (5.16) (or recalling the correspondences k →−i∂
x
and
ω → i∂
t
), then
−ω
2
+ k
2
= ε
ω
2
k
2
2
−
k
4
6
=⇒ ω
2
=
k
2
+ εk
4
/6
1 + εk
2
/2
.
Note ω
2
> 0 and for very large k, ω
2
∼ k
2
/3 . If we substitute this back into
our wave solution, then we see that A(x, t) = exp(i(kx ± kt/
√
3)) is bounded
for all time. But, if we do the same for (5.17), we get
−ω
2
+ k
2
= ε
k
4
3
.
The large k limit of ω in this case is ω
±
∼±i
√
εk
2
/
√
3. Substituting the
negative root of ω into the wave solution we end up with
A(x, t) ∼ exp(ikx)exp
0
ε
3
k
2
t
.
110 Water waves and KdV-type equations
This solution blows up arbitrarily fast as k →∞so we do not have a conver-
gent Fourier integral solution for “reasonable” (non-analytic) initial data (see
also the earlier discussion of well-posedness). If we wish to do numerical cal-
culations then (5.16) is preferred as it is well-posed, as opposed to (5.17). The
ill-posedness of (5.17) is due to the long-wave approximation. This is a com-
mon difficulty associated with long-wave expansions; see also the discussion
in Benjamin et al. (1972).
Nevertheless since there is still a small parameter, ε, in the equation, we can
and should do further asymptotics on these equations. This will also remove
the ill-posedness and is done in the next section.
First we make another remark about different Boussinesq models. We have
derived an approximate equation for the leading term in the expansion of the
velocity potential φ (5.14)or(5.15). Combined with (5.11) it determines the
velocity potential, accurate to O(). We can also determine an equation for
the wave amplitude η. First we take an x derivative of (5.12) and then use the
substitution u = A
x
in (5.12)–(5.13) to find
Bernoulli: η
x
= −u
t
+
ε
2
(u
xxt
− 2uu
x
) + ···, (5.18)
Kinematic: η
t
= −u
x
+ ε
−η
x
u − ηu
x
+
u
xxx
6
+ ···. (5.19)
This is called the coupled Boussinesq model for η and u.
On the other hand, by differentiating (5.19) with respect to t, and by
differentiating (5.18) with respect to x, we get
η
tt
− η
xx
= ε
(
−η
xt
u − η
x
u
t
− η
t
u
x
− ηu
xt
+
u
xxxt
6
−
u
xxxt
2
+
(u
2
)
xx
2
+ O(ε).
Now we use the relations u
x
= −η
t
+ O(ε) and u
t
= −η
x
+ O(ε)fromthe
kinematic and Bernoulli equations, respectively, and make the replacement u =
−
x
−∞
η
t
dx
. Hence we find
η
tt
− η
xx
= ε
1
3
η
xxxx
+ η
xt
x
−∞
η
t
dx
+ η
2
x
+ η
2
t
+ ηη
xx
+
1
2
∂
2
∂x
2
x
−∞
η
t
dx
2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(5.20)
5.4 Shallow-water theory 111
= ε
η
xxxx
3
+
∂
∂x
η
t
x
−∞
η
t
dx
+ ηη
x
+
1
2
∂
2
∂x
2
x
−∞
η
t
dx
2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
= ε
⎡
⎢
⎢
⎢
⎢
⎢
⎣
η
xxxx
3
+
∂
2
∂x
2
⎛
⎜
⎜
⎜
⎜
⎜
⎝
η
2
2
+
x
−∞
η
t
dx
2
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎤
⎥
⎥
⎥
⎥
⎥
⎦
. (5.21)
The last equation (5.21) is the Boussinesq model for the wave amplitude.
If we omit the non-local term (which can be done via a suitable transfor-
mation) (5.21) is reduced to the Boussinesq model discussed in Chapter 1
[see (1.2), where η = y
x
].
5.4.2 Multiple-scale derivation of the KdV equation
Now let us return to the Boussinesq model for the velocity potential equa-
tion (5.15) to do further asymptotics
A
tt
− A
xx
= ε
A
xxxx
3
− 2A
x
A
xt
− A
xx
A
t
. (5.22)
Assume an asymptotic expansion for A:
A = A
0
+ εA
1
+ ···;
substituting this expansion into (5.22)gives
A
0tt
+ εA
1tt
− A
0xx
− εA
1xx
+ O(ε
2
) = ε
A
0xxxx
3
−
−2A
0x
A
0xt
− A
0xx
A
0t
+ O(ε)
.
Equating the leading-order terms yields the wave equation: A
0tt
−A
0xx
= 0. We
solved this equation earlier and found A
0
(x, t) = F(x − t) + G(x + t) where F
and G are determined by the initial conditions. Anticipating secular terms that
will need to be removed in the next-order equation, we employ multiple scales;
i.e., A
0
= A
0
(ξ, ζ, T ):
A
0
= F(ξ, T ) + G(ζ, T ),
where
ξ = x − t,ζ= x + t, T = εt.
These new variables imply
∂
t
= −∂
ξ
+ ∂
ζ
+ ε∂
T
and ∂
x
= ∂
ξ
+ ∂
ζ
.
112 Water waves and KdV-type equations
Substituting the expressions for the differential operators into (5.22) leads to
((−∂
ξ
+ ∂
ζ
+ ε∂
T
)
2
− (∂
ξ
+ ∂
ζ
)
2
)A =
ε
(∂
ξ
+ ∂
ζ
)
4
3
A − 2(∂
ξ
+ ∂
ζ
)A(∂
ξ
+ ∂
ζ
)(−∂
ξ
+ ∂
ζ
+ ε∂
T
)A
− (∂
ξ
+ ∂
ζ
)
2
A(−∂
ξ
+ ∂
ζ
+ ε∂
T
)A
. (5.23)
First we will assume unidirectional waves and only work with the right-
moving wave. So A
0
= F(ξ, T ) for now. Substituting
A = A
0
+ εA
1
+ ···
into (5.23) and keeping all the O(ε) terms we get
−4A
1ξζ
= 2F
ξT
+
1
3
F
ξξξξ
+ 3F
ξξ
F
ξ
.
This equation can be integrated directly to give
A
1
∼−
1
4
2F
T
+
1
3
F
ξξξ
+
3
2
F
2
ξ
ζ + ···,
remembering that we absorb homogeneous terms in the A
0
solution. In order
to remove secular terms, in particular the ζ term, we require that
2F
T
+
1
3
F
ξξξ
+
3
2
F
2
ξ
= 0;
or, if we take a derivative with respect to ξ and make the substitution U = F
ξ
then we have the Korteweg–de Vries equation
2U
T
+
1
3
U
ξξξ
+ 3UU
ξ
= 0.
Next we discuss the more general case of waves moving to the left and right;
hence A
0
= F(ξ, T ) + G(ζ, T ), and we get the following expression for A
1
− 4A
1ξζ
= 2(F
ξT
−G
ζT
) +
1
3
(F
ξξξξ
++G
ζζζζ
)
+ 3(F
ξ
F
ξξ
+
1
3
F
ξξ
G
ζ
−
1
3
G
ζζ
F
ξ
−G
ζ
G
ζζ
).
When we integrate this expression, secular terms arise from the pieces that are
functions of ξ or ζ alone, not both. Removal of the secular terms implies the
following two equations
2F
ξT
+
1
3
F
ξξξξ
+ 3F
ξξ
F
ξ
= 0
−2G
ζT
+
1
3
G
ζζζζ
− 3G
ζζ
G
ζ
= 0,
5.4 Shallow-water theory 113
which are two uncoupled KdV equations. Simplifying things a bit, we can
rewrite the above two equations in terms of U = F
ξ
and V = G
ζ
:
2U
T
+
1
3
U
ξξξ
+ 3UU
ξ
= 0 (5.24)
2V
T
−
1
3
V
ζζζ
+ 3VV
ζ
= 0. (5.25)
The solution A
1
can be obtained by integrating the remaining terms. Since we
are only interested in the leading-order asymptotic solution we need not solve
for A
1
.
Hence we have the following conclusion: asymptotic analysis of the fluid
equations under shallow-water conditions has given rise to two Korteweg–de
Vries (KdV) equations, (5.24) and (5.25), for the right- and left-going waves.
Given rapidly decaying initial conditions, we can solve these two PDEs for
U and V by the inverse scattering transform (IST). The KdV equation has
been studied intensely and its IST solution is described in Chapter 9. Keeping
only the leading-order terms, we have an approximate solution for the velocity
potential
φ(x, z, t) ∼ A
0
(x, t) = F(x − t,εt) + G(x + t,εt),
=
x−t
−∞
U(ξ
,εt) dξ
+
x+t
−∞
V(ζ
,εt) dζ
,
or for the velocity
u = φ
x
= F
x
+ G
x
+ ···= U + V + ···.
Using the Bernoulli equation (5.12), we have an approximate solution for
the wave amplitude of the free boundary
η(x, t) ∼−A
0t
(x, t) ∼ F
ξ
(ξ, T) −G
ζ
(ζ,T ) = U(x − t,εt) − V(x + t,εt).
Thus the wave amplitude η has right- and left-going waves that satisfy the KdV
equation. Alternatively, we can derive the KdV equation directly by performing
an asymptotic expansion in the η equation (5.21) though we will not do that
here. One can also proceed to obtain higher-order terms and corrections to
the KdV equation, though there is much more in the way of details that are
required. We will not discuss this here.
5.4.3 Dimensional equations
In the previous section, we made our fluid equations non-dimensional for
the purpose of determining what terms are “small” in our approximation
114 Water waves and KdV-type equations
of shallow-water waves. Now that we have approximate equations (5.24)
and (5.25), valid to leading order, we wish to understand the results in
dimensional units. We will now transform back to a dimensional equation
for (5.24).
Recall that we made two changes of variable. We first changed coordinates
x, t,η into primed coordinates x
, t
,η
(though we immediately dropped the
primes). Then the independent variable substitutions ξ = x
− t
and T = εt
were made. First, we remove the ξ and T dependence. Notice that
∂
∂t
U(ξ, T ) = −U
ξ
+ εU
T
∂
∂x
U(ξ, T ) = U
ξ
.
The above two expressions imply that ∂
T
= (∂
t
+ ∂
x
)/ε. Making the above
substitutions into (5.24), we get
2(U
t
+ U
x
) + ε
1
3
U
x
x
x
+ 3UU
x
= 0.
If we neglect left-traveling waves, V, then we have η
∼ U. Recalling our
original non-dimensionalization x = λ
x
x
, t =
λ
x
c
0
t
, = a/h and η = aη
,we
use ∂
x
= λ
x
∂
x
and ∂
t
=
λ
x
c
0
∂
t
to transform the previous expression in U to
2
λ
x
c
0
η
t
a
+ λ
x
η
x
a
+ ε
λ
3
x
3a
η
xxx
+
3λ
x
a
2
ηη
x
= 0 ⇒
2
η
t
c
0
+ η
x
+
a
h
λ
2
x
3
η
xxx
+
3
a
ηη
x
= 0.
From maximal balance, we used ε = a/h = (h/λ
x
)
2
so we can simplify the
non-dimensional KdV equation (5.24) to the dimensional form
1
c
0
η
t
+ η
x
+
h
2
6
η
xxx
+
3
2h
ηη
x
= 0, (5.26)
where c
0
=
gh, g is the gravitational constant of acceleration and h is the
water depth. This equation was derived by Korteweg and de Vries (1895).
Let us consider the linear part of (5.26)
1
c
0
η
t
+ η
x
+
h
2
6
η
xxx
= 0.