Ablowitz M.A. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons
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5.5 Solitary wave solutions 125
map is contained in the solution of the non-local system (I–II). Also, solving
equations (I)–(II) for η, q reduces to solving the remaining water wave equa-
tion, Laplace’s equation (5.4), which is a linear problem due to the now fixed
boundary conditions.
The non-local equation can be derived from Laplace’s equation, the kine-
matic condition and the bottom boundary condition via Green’s identity (cf.
Haut and Ablowitz, 2009). The essential ideas are outlined below. We also
mention that the reader will find valuable discussions of non-local systems
with auxiliary parameters in Fokas (2008). Fokas has made numerous impor-
tant contributions in the study of such non-local nonlinear equations. We define
an associated potential ψ(x, y, z) satisfying
Δψ(x, y, z) = 0, in D,ψ
y
(x, y, z = −h) = 0. (5.41)
Then, from Green’s identity,
0 =
D(η)
((
Δψ
)
φ −
(
Δφ
)
ψ
)
dV
=
∂D(η)
(φ(∇ψ · ˆn) − ψ(∇φ · ˆn)) dS
where dV, dS are the volume and surface elements respectively, and ˆn is the
unit normal. It then follows that
ψ(x, y,η)η
t
dxdy =
q
ψ
z
(x,η) −∇
x,y
ψ(x, y,η) ·∇η
dxdy, (5.42)
where we have used η
t
= ∇φ ·
n,
n = (−∇η, 1) on the free surface as well as the
bottom boundary condition and the condition |η| decaying at infinity.
We assume a solution of (5.41) can be written in the form
ψ(x, y, z) =
ˆ
ξ(k, l)
ˆ
ψ
k,l
(x, y, z)dkdl,
ˆ
ψ
k,l
= e
i(kx+ly)
cosh[κ(z + h)].
Inserting ψ
k,l
(x, y, z)into(5.42) yields
R
2
e
i(kx+ly)
cosh(κ(η + h)η
t
dxdy
=
R
2
q
e
i(kx+ly)
κ sinh(κ(η + h) − ie
i(kx+ly)
cosh(κ(η + h)
(
k ·∇
)
η
dxdy.
(5.43)
126 Water waves and KdV-type equations
Then, using
e
i(kx+ly)
κ sinh(κ(η + h) − ie
i(kx+ly)
cosh(κ(η + h)
(
(k, l) ·∇
)
η
= −i∇·
e
ikx
sinh(κ(η + h)
κ
(k, l)
in (5.43) and integrating by parts yields the non-local equation (I).
The second equation is essentially a change of variables. Differentiating
q(x, y, t) = φ(x, y,η(x, y, t), t) leads to
q
x
+ φ
x
+ φ
z
η
x
, q
y
= φ
y
+ φ
z
η
y
.
These equations and
η
t
+ ∇φ ·∇η = φ
y
yield φ
x
, φ
y
, φ
z
in terms of derivatives of η, q. Then from Bernoulli’s equation
with surface tension included, (5.28), we obtain equation (II).
If |η|, |∇q| are small then equations (I) and (II) simplify to the linearized
water wave equations. Calling the Fourier transform ˆη =
dxdye
i(kx+ly)
η,
and similarly for the derivatives of q, we find from equations (I) and (II)
respectively
i ˆη
t
cosh κh +
k ·
=
∇q
κ
sinh κh = 0
(
q
t
+ (g + σκ
2
)ˆη = 0.
Then from these two equations we get
ˆη
tt
= −(gκ + σκ
3
) tanh κh ˆη
which is the linearized water wave equation for η in Fourier space.
We also remark that one can find integral relations by taking k, l → 0in
equations (I) and (II) above. The first three, corresponding to powers k
0
, l
0
,
k, l,aregivenby
∂
∂t
∞
−∞
∞
−∞
dxdy η(x, y, t) = 0 (Mass)
∂
∂t
∞
−∞
∞
−∞
dxdy(xη) =
∞
−∞
∞
−∞
dxdy q
x
(η + h)(COM
x
)
∂
∂t
∞
−∞
∞
−∞
dxdy(yη) =
∞
−∞
∞
−∞
dxdy q
y
(η + h)(COM
y
).
They correspond to conservation of mass and the motion of the center of mass
in x, y respectively. The right-hand sides of (COM
x
, y) are related to the x, y-
momentum respectively. Higher powers of k, l lead to “virial” type identities.
5.5 Solitary wave solutions 127
From the above system (I) and (II), one can also derive both deep- and
shallow-water approximations to the water wave equations. In deep water, non-
linear Schr
¨
odinger equation (NLS) systems result (see Ablowitz et al., 2006;
Ablowitz and Haut, 2009b). In shallow water the Benney–Luke (BL) equa-
tion (Benney and Luke, 1964), properly modified to take into account surface
tension, can be obtained.
To do this it is convenient to make all variables non-dimensional:
x
1
=
x
1
λ
, x
2
= γ
x
2
λ
, t
=
c
0
λ
t , aη
= η,q
=
aλg
c
0
q ,σ
=
σ
gh
2
where c
0
=
gh and λ, a are the characteristic horizontal length and ampli-
tude, and γ is a non-dimensional transverse length parameter. The equations
are written in terms of non-dimensional variables = a/h 1, small-
amplitude, μ = h/λ 1 long waves; γ 1, slow transverse variation.
Dropping
we find
dxdye
i(kx+ly)
iη
t
cosh[˜κμ(1 + εη)] + sinh[˜κμ(1 + εη)]
(k, l) ·
˜
∇q
˜κμ
= 0 (5.44)
where (k, l) ·
˜
∇ = kq
x
+ γ
2
lq
y
,˜κ
2
= k
2
+ γ
2
l
2
.
In (5.44)weuse
cosh[˜κμ(1 + εη)] ∼ 1 +
μ
2
2
˜κ
2
, sinh[˜κμ(1 + εη)] ∼ μ˜κ +
μ
3
6
˜κ
3
+ εμη˜κ.
Then, after inverse Fourier transforming, and with (k, l) → (i∂x, i∂y) we find
from equations (I) and (II)
1 −
μ
2
2
˜
Δ
η
t
+
˜
Δ −
μ
2
6
˜
Δ
2
q + ε
˜
∇η ·
˜
∇q
+ εη
˜
Δq = 0 (Ia)
η = −q
t
−
ε
2
|
˜
∇q|
2
+ ˜σμ
2
˜
Δη (IIa)
where
˜
Δ=∂
2
x
+ δ
2
∂
2
y
,
˜
∇η ·
˜
∇q = η
x
q
x
+ δ
2
η
y
q
y
, |
˜
∇q|
2
=
q
2
x
+ δ
2
q
2
y
, and
˜σ = σ − 1/3.
Eliminating the variable η we find, when ,μ, δ 1, the Benney–Luke (BL)
equation with surface tension included:
q
tt
−
˜
Δq + ˜σμ
2
˜
Δ
2
q + ε(∂
t
|
˜
∇q|
2
+ q
t
˜
Δq) = 0, (5.45)
where ˜σ = σ − 1/3. Alternatively, one can derive the BL system from the
classical water wave equations via asymptotic expansions as we have done
previously for the Boussinesq models. The equations look somewhat differ-
ent due to different representations of the velocity potential (see also the
exercises).
128 Water waves and KdV-type equations
We remark that if we take the maximal balance = μ
2
= δ
2
then using mul-
tiple scales the BL equation can be reduced to the unidirectional KP equation,
which we have shown can be further reduced to the KdV equation if there is
no transverse variation. Namely letting ξ = x −t, T = t/2, w = q
ξ
we find the
KP equation in the form
∂
ξ
(w
T
− ˜σw
ξξξ
+ 3(ww
ξ
)) + w
yy
= 0.
Hence the non-local system (I)–(II) contains the well-known integrable water
wave reductions. We also note that higher-order asymptotic expansions of soli-
tary waves and lumps can be obtained from equations (I) and (II) using the
same non-dimensionalization as we used for the BL equation. These expan-
sions yield solitary waves close to their maximal amplitude (Ablowitz and
Haut, 2009a, 2010)
Exercises
5.1 Derive the KP equation, with surface tension included, from the Benney–
Luke (BL) equation (5.45): i.e., letting w = q
ξ
,ε= μ
2
= δ
2
, find
w
T ξ
+
1/3 −
ˆ
T
w
ξξξξ
+ w
yy
+ (3ww
ξ
)
ξ
= 0(KP)
where T = t,
ˆ
T = T/ρgh
2
, which is the well-known Kadomtsev–
Petviashvili (KP) equation.
5.2 From the non-dimensional KP equation derived in Exercise 5.1,show
that the equation in dimensional form is given by
1
c
0
η
xt
+
3
2h
(ηη
x
)
x
+ γη
xxxx
= −
1
2
η
yy
where γ =
h
2
6
(1 − 3
ˆ
T ).
5.3 Find the dimensional form of a soliton solution to the KdV equation
when
ˆ
T > 1/3, i.e., γ<0.
5.4 Derive the equivalent BL equation for the velocity potential φ(x, y, 0, t)
from the classical water wave equations with surface tension included.
5.5 Rewrite the lump solution given in this section in dimensional form.
5.6 Given the “Boussinesq” model
u
tt
− Δu +Δ
2
u + ||∇u||
2
= 0,
make suitable assumptions about the size of terms, then rescale, and
derive a corresponding KP-type equation.
Exercises 129
5.7 Given a “modified Boussinesq” model
u
tt
− Δu +Δ
2
u + ||∇u||
2+n
= 0
with n a positive integer, make suitable assumptions about the size of
terms, then rescale, and derive a corresponding “modified KP”-type
equation.
5.8 Beginning with (5.22) assume right-going waves and obtain the KdV
equation with a higher-order correction.
6
Nonlinear Schr¨odinger models and water waves
Adifferent asymptotic regime will be investigated in this chapter. Since nonlin-
ear, dispersive, energy-preserving systems generically give rise to the nonlinear
Schr
¨
odinger (NLS) equation, it is important to study this situation. In this
chapter first the NLS equation will be derived from a nonlinear Klein–Gordon
and the KdV equation. Then a discussion of how the NLS equation arises
in the limit of deep water will be undertaken. A number of results associ-
ated with NLS-type equations, including Benney–Roskes/Davey–Stewartson
systems, will also be obtained and analyzed.
6.1 NLS from Klein–Gordon
We start with the so-called “u-4” model, which arises in theoretical physics, or
the nonlinear Klein–Gordon (KG) equation – with two choices of sign for the
cubic nonlinear term:
u
tt
− u
xx
+ u ∓ u
3
= 0. (6.1)
Note, by multiplying the equation with u
t
we obtain the conservation of
energy law
∂
t
u
2
t
+ u
2
x
+ u
2
∓ u
4
/2
− ∂
x
(2u
x
u
t
) = 0
and hence the equation has a conserved density T
1
= u
2
t
+ u
2
x
+ u
2
∓ u
4
/2.
Note the “potential” is proportional to V(u) = u
2
∓ u
4
/2, hence the term “u-4”
model is used. We assume a real solution to (6.1) with a “rapid” phase and
slowly varying functions of x, t:
u(x, t) = u(θ, X, T ; ε); X = εx, T = εt,θ= kx − ωt, |ε|1,
130
6.1 NLS from Klein–Gordon 131
where the dispersion relation for the linear problem in (6.1)isω
2
= 1 + k
2
.
Based on the form of assumed solution the following operators are introduced
in the asymptotic analysis:
∂
t
= −ω∂
θ
+ ε∂
T
∂
x
= k∂
θ
+ ε∂
X
.
The weakly nonlinear asymptotic expansion
u = εu
0
+ ε
2
u
1
+ ε
3
u
2
+ ···,
and the differential operators are substituted into (6.1) leading to
(−ω∂
θ
+ ε∂
T
)
2
− (k∂
θ
+ ε∂
X
)
2
u + u ∓ u
3
= 0
or
(ω
2
−k
2
)∂
2
θ
−ε(2ω∂
θ
∂
T
+ 2k∂
θ
∂
X
) + ε
2
∂
2
T
−∂
2
X
(εu
0
+ ε
2
u
1
+ ···) +
+ (εu
0
+ ε
2
u
1
+ ···) ∓ (εu
0
+ ε
2
u
1
+ ···)
3
= 0.
We will find the equations up to O(ε
3
); the first two are:
3O(ε): (ω
2
− k
2
)∂
2
θ
u
0
+ u
0
= 0
⇒ u
0
= Ae
iθ
+ c.c.
O(ε
2
): (ω
2
− k
2
)∂
2
θ
u
1
+ u
1
= (2ωiA
T
+ 2kiA
X
)e
iθ
+ c.c.
where c.c. denotes the complex conjugate. Note: throughout the discussion we
will use the dispersion relation ω
2
− k
2
= 1. In order to remove secular terms,
we require that the coefficient on the right-hand side of the O(ε
2
) equation
be zero. This leads to an equation for A(X, T ) that, to go to higher order, we
modify by using the asymptotic expansion
2i(ωA
T
+ kA
X
) = εg
1
+ ε
2
g
2
+ ···. (6.2)
The quantity A(X, T ) is often called the slowly varying envelope of the wave.
As usual, we absorb the homogeneous solutions into u
0
so u
1
= 0. Now let us
look at the next-order equation
O(ε
3
): (ω
2
− k
2
)∂
2
θ
u
2
+ u
2
= −(A
TT
− A
XX
)e
iθ
+ c.c.
± (Ae
iθ
+ A
∗
e
−iθ
)
3
+ g
1
e
iθ
+ c.c.
In order to remove secular terms, we eliminate the coefficient of e
iθ
.This
implies
g
1
= (A
TT
− A
XX
) ∓ 3A
2
A
∗
.
132 Nonlinear Schr¨odinger models and water waves
Combining the above relation with the asymptotic expansion in (6.2)wehave
2iω(A
T
+ ω
(k)A
X
) = ε(A
TT
− A
XX
∓ 3|A|
2
A), (6.3)
where we used the dispersion relation to find ω
(k) = k/ω. After removing the
secular terms, we can solve the O(ε
3
) equation for u
2
giving
u
2
= ∓
1
8
A
3
e
3iθ
+ c.c.
Note, in (6.3), we can transform the equation for A and remove the ε (group
velocity) term. To do this, we change the coordinate system so that we are
moving with the group velocity ω
(k) – sometimes this is called the group-
velocity frame:
ξ = X − ω
(k)T ∂
T
= −ω
(k)∂
ξ
+ ε∂
τ
τ = εT = ε
2
t ∂
X
= ∂
ξ
.
Now, we use (6.3) to find
2iω(−ω
A
ξ
+ εA
τ
+ ω
A
ξ
) = ε((ω
)
2
A
ξξ
− A
ξξ
∓ 3|A|
2
A) + O(ε
2
).
We can simplify this equation by using the dispersion relation as follows:
ω
= k/ω and
ω
=
ω − ω
k
ω
2
=
1
ω
−
kω
ω
2
=
1
ω
1 − (ω
)
2
,
to arrive at the nonlinear Schr
¨
odinger equation in canonical form:
iA
τ
+
ω
2
A
ξξ
±
3
2ω
|A|
2
A = 0. (6.4)
It is important to note that, like the derivation of the Korteweg–de Vries
equation in the previous chapter, the small parameter, ε, has disappeared from
the leading-order NLS equation. The NLS equation is a maximally balanced
asymptotic system.
The behavior of (6.4) depends on the plus or minus sign. The NLS equation
witha“+”, here
ω
=
ω
2
− k
2
ω
3
=
1
ω
3
,
so ω
/ω > 0. The NLS with a “+” sign is said to be “focusing” and we will
see gives rise to “bright” solitons. Bright solitons have a localized shape and
decay at infinity. With a “−”, NLS is said to be “defocusing” and we will
see admits “dark” soliton solutions. Dark solitons have a constant amplitude
at infinity. The terminology “bright” and “dark” solitons is typically used in
6.2 NLS from KdV 133
nonlinear optics. We will discuss these solutions later, and nonlinear optics in
a subsequent chapter.
In our derivation of NLS, we transformed our coordinate system to coincide
with the group velocity of the wave. An alternative form, useful in nonlinear
optics and called the retarded frame, is
t
= T −
1
ω
(k)
X ∂
T
= ∂
t
χ = εX ∂
X
= −
1
ω
∂
t
+ ε∂
χ
.
Next, we substitute the above relations into (6.3) to find
2iω
A
t
+ ω
−1
ω
A
t
+ εA
χ
= ε
A
t
t
−
1
(ω
)
2
A
t
t
∓ 3|A|
2
A
.
Again we simplify this equation into a nonlinear Schr
¨
odinger equation
iA
χ
+
ω
2(ω
)
3
A
t
t
±
3
2ωω
|A|
2
A = 0. (6.5)
Note that in the above equation, the “evolution” variable is χ and the “spatial”
variable is t
.
6.2 NLS from KdV
We can also derive the NLS equation from the KdV equation asymptoti-
cally for small amplitudes. To do that, we expand the solution of the KdV
equation
u
t
+ 6uu
x
+ u
xxx
= 0
as
u = εu
0
+ ε
2
u
1
+ ε
3
u
2
+ ···,
where for convenience the assumption of small amplitude is taken through
the expansion of u rather than in the equation itself. Since the nonlinear
term, uu
x
, is quadratic rather than cubic, it will turn out that the reduction
to the NLS equation requires more work than in the KG equation. As we
will see, in this case one needs to remove secularity at one additional power
of ε, which in this case is O(ε
3
). Another difficulty is manifested by the
need to consider a mean term in the solution. Indeed, the equation to leading
order is
u
0,t
+ u
0,xxx
= 0,
134 Nonlinear Schr¨odinger models and water waves
whose real solution is given by
u
0
= A(X, T )e
iθ
+ c.c. + M(X, T ), (6.6)
where again T = εt and X = εx are the slow variables, θ = kx − ωt is the fast
variable, with the dispersion relation ω = −k
3
, and M(T, X) is a real, slowly
varying, mean term, i.e., it corresponds to the coefficient of e
inθ
with n = 0. The
quantity A(X, T ) is the slowly varying envelope of the rapidly varying wave
contribution. As we will see, in this case the mean term turns out to be O(ε),
however, that is not always the case.
1
Note that the addition of the complex
conjugate (“c.c.”) in (6.6) corresponds to the first term and not to the mean
term, which is real-valued.
Substituting ∂
t
= −ω∂
θ
+ ε∂
T
and ∂
x
= k∂
θ
+ ε∂
X
in the KdV equation
leads to
(−ω∂
θ
+ ε∂
T
)u + 6u(k∂
θ
+ ε∂
X
)u + (k∂
θ
+ ε∂
X
)
3
u = 0.
Using the above expansion for the solution, u = εu
0
+ ε
2
u
1
+ ε
3
u
2
+ ···
gives
(−ω∂
θ
+ ε∂
T
)(εu
0
+ ε
2
u
1
+ ε
3
u
2
+ ···)
+ 6(εu
0
+ ε
2
u
1
+ ε
3
u
2
+ ···)(k∂
θ
+ ε∂
X
)(εu
0
+ ε
2
u
1
+ ε
3
u
2
+ ···)
+ (k∂
θ
+ ε∂
X
)
3
(εu
0
+ ε
2
u
1
+ ε
3
u
2
+ ···) = 0.
The leading-order solution of Lu
0
= 0, L defined below, is given by (6.6).
We recall that the dispersion relation for the exponential term is given by
ω(k) = −k
3
.TheO(ε
2
) equation is
Lu
1
= −u
0,T
− 3k
2
u
0,θθX
− 6u
0
ku
0,θ
= (−A
T
e
iθ
+ c.c.) − M
T
+ (3k
2
A
X
e
iθ
+ c.c.)
− 6(Ae
iθ
+ c.c.) + M)(Aike
iθ
+ c.c.),
where we have defined the linear operator (using ω = −k
3
),
Lu = k
3
(u
θ
+ u
θθθ
).
Thus, we rewrite the O(ε) equation as
Lu
1
= (−A
T
− ω
A
X
− 6iMkA)e
iθ
+ c.c. − 6(ikA
2
e
2iθ
+ c.c.) − M
T
,
where we have used the dispersion relation, from which one gets that
ω
(k) = −3k
2
. In order to remove secular terms we note that such terms arise
1
For example, in finite-depth water waves the mean term is O(1) (Benney and Roskes, 1969;
Ablowitz and Segur, 1981).