Ablowitz M.A. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons

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5.4 Shallow-water theory 115

We will interpret the above equation in terms of the dispersion relation for

water waves [see (5.8)], noting κ

= k

+ l

with l = 0, so κ = k:

= gk tanh(kh) = gk



kh −

(kh)

+ ···



In shallow-water waves, kh  1 so if we retain only the leading-order term in

the Taylor series expansion above, we have

ω = ±k



gh = ±kc

This is the dispersion relation for the linear wave equation and c

is the wave

speed. Retaining the next term in the Taylor series of tanh(kh)gives

ω ∼±k

gh(1 −

(kh)

)

∼±k



gh(1 −

(kh)

)

= ±(kc

−

Now, we wish to see what linear PDE gives rise to the above dispersion

relation. Taking the positive root, replacing ω with i∂

and k with −i∂

, we ﬁnd

i∂

η = c

(−i∂

)η −

(−i∂

)

η ⇒

+ η

xxx

= 0.

This is exactly the dimensional linear KdV equation (5.26)! To get the

nonlinear term we used the multiple-scales method.

5.4.4 Adding surface tension

The model equations (5.4) through (5.7) do not take into account the eﬀects

of surface tension. Actually only Bernoulli’s equation (5.10c)isaﬀected by

surface tension. The modiﬁcation is due to an additional pressure term from

surface tension eﬀects involving the curvature at the surface. We will not go

through the derivation here: rather we will just state the result. First, recall

Bernoulli’s equation used so far

|∇φ|

+ gη = 0, (5.27)

116 Water waves and KdV-type equations

on z = η. Adding the surface tension term gives (cf. Lamb 1945 or Ablowitz

and Segur 1981)

|∇φ|

+ gη =

∇·

⎛

⎜

⎝

∇η



1 + |∇η|

⎞

⎟

⎠



1 + η



+ η



1 + η



− 2η





1 + η

+ η



(5.28)

on z = η where T is the surface tension coeﬃcient. Retaining dimensions and

keeping linear terms that will aﬀect the previous result to O(ε), we have:

|∇φ|

+ gη −

(η

+ η

) = 0 z = η. (5.29)

Using (5.29) and the same asymptotic procedure as before, the corresponding

leading-order asymptotic equation for the free surface is found to be

+ η

+ γη

xxx

ηη

= 0. (5.30)

The only diﬀerence between this equation and (5.26) is the coeﬃcient γ of the

third-derivative term. This term incorporates the surface tension

γ =

−

2ρg



1 −

ρgh



(1 − 3

T )

T =

ρgh

Note (5.30) was also derived by Korteweg and deVries [see Chapter 1, equa-

tion (1.4)], and we can see that, depending on the relative sizes of the

parameters (in particular

T < 1/3or

T > 1/3),γ will be positive or nega-

tive. This aﬀects the types of solutions and behavior allowed by the equation

and is discussed later. In particular, when we include transverse waves, we will

discover the Kadomtsev–Petviashvili (KP) equation.

5.4.5 Including transverse waves: The KP equations

Our investigation so far has been for waves in shallow water without transverse

modulations. We will now relax our assumptions and include weak transverse

variation.

First, let us look at the dispersion relation for water waves. Using the

same ideas as previously [see the discussion leading to (5.8)] the reader can

5.4 Shallow-water theory 117

verify that the dispersion relation in multidimensions (two space, one time) is

given by



gκ +



tanh(κh), (5.31)

where κ

= k

+ l

; here k, l are the wavenumbers that correspond to the x- and

y-directions, respectively. Recall that the wavelengths of a wave solution in the

form αe

i(kx+ly−ωt)

are λ

= 2π/k and λ

= 2π/l. We further assume that (recall

our earlier deﬁnitions of scales)

δ =

 1.

The above relation says that the wavelength in the y-direction is much larger

than the wavelength in the x-direction. This is what is referred to as weak

transverse variation.

We now derive the linear PDE associated with the dispersion relation (5.31)

with weak transverse variation. Since we are still in the shallow-water regime,

we have |hk|∝|h/λ

|1 and we assume that κh = kh



1 + l

 1. Then

we expand the hyperbolic tangent term in a Taylor series to ﬁnd

≈



gκ +



κh −

(κh)



+ ···

= ghκ



1 +

gρ



1 −

(κh)



+ ···

ω =



gh k



1 +





1 +

gρ

+ O(l

)





1 −

(hk)

+ O(l

)



Use of the binomial expansion and assuming |l/κ|1 and the maximal

balance relation l

∼ O(k

), we end up with (c



gh),

ωk ≈ c



+ k



2gρ

−



+ O(k

)



Now we can write down the linear PDE associated with this dispersion relation

using ω → i∂

, k →−i∂

+ η

(1 − 3

T )η

xxxx

= 0,

where we recall

T = T /ρgh

. This is the linear Kadomtsev–Petviashvili (KP)

equation. To derive the full nonlinear version, we must resort to multiple scales

118 Water waves and KdV-type equations

(Ablowitz and Segur, 1979, 1981). Though we will not do that here, if we take

a hint from our work on the KdV equation, we can expect that the nonlinear

part of our equation will not change when we add in slow transverse (y) depen-

dence. This is true and the full KP equation, ﬁrst derived in 1970 (Kadomtsev

and Petviashvili, 1970), in dimensional form is

∂



+ η

ηη

+ γη

xxx



= 0, or

+ η

ηη

+ γη

xxx

= −



−∞



(5.32)

where γ =

(1 −3

T ). Usually in water waves surface tension is small and we

have

T < 1/3 which gives rise to the so-called KPII equation

+ η

(ηη

)

xxxx

= −

The equation termed the KPI equation arises when

T = T /ρgh

> 1/3;

i.e., when surface tension eﬀects are large. Alternatively, we can rescale the

equation into non-dimensional form

∂

+ 6uu

+ u

xxx

) + 3σu

= 0, (5.33)

where σ has the following meaning

• σ =+1 =⇒ KPII: typical water waves, small surface tension,

• σ = −1 =⇒ KPI: water waves, large surface tension.

We note that if η

= 0in(5.32) and we rescale, the resulting KP equation can

be reduced to the KdV equation in standard form

± 6uu

+ u

xxx

= 0. (5.34)

The “+” corresponds to

T < 1/3 whereas the “–” arises when

T > 1/3.

5.5 Solitary wave solutions

As discussed in Chapter 1, the KdV and KP equations admit special, exact

solutions known as solitary waves. We also mentioned in Chapter 1 that a

solitary wave was noted by John Scott Russell in 1834 when he observed a

wave detach itself from the front of a boat brought to rest. This wave evolved

into a localized rounded hump of water that Russell termed the Great Wave of

Translation. He followed this solitary wave on horseback as it moved along the

Union Canal between Edinburgh and Glasgow. He noted that it hardly changed

5.5 Solitary wave solutions 119

its shape or lost speed for over two miles; see Russell (1844); Ablowitz and

Segur (1981); Remoissenet (1999) and www.ma.hw.uk/solitons. Today, scien-

tists often use the term soliton instead of solitary wave for localized solutions

of many equations despite the fact that the original deﬁnition of a soliton

reﬂected the fact that two solitary waves interacted elastically.

5.5.1 A soliton in dimensional form for KdV

Recall from above that the KdV equation in standard, non-dimensional

form, (5.34), can be written

± 6uu

+ u

xxx

= 0, and ± when γ ≷ 0,

where

γ =

(1 − 3

T ).

A soliton solution admitted by the non-dimensional equation (5.34) is given by

u(x, t) = ±2β

sech

(β(x ∓ 4β

t − x

)). (5.35)

Notice that the speed of the wave, c = 4β

, is twice the amplitude of the wave.

Also, the “+” corresponds to γ>0 that is physically a positive elevation wave

traveling on the surface like the one ﬁrst observed by Russell. The “–” case

results from γ<0 and corresponds to a dip in the surface of the water. In fact,

an experiment done only recently with high surface tension produced just such

a “depression” wave (Falcon et al., 2002).

The solitary wave or soliton solution equation (5.35) is in non-dimensional

form. We can convert this solution of the non-dimensional equation (5.34)

directly to a solution of the dimensional equation (5.26). We will only con-

sider here γ>0; we leave it as an exercise to ﬁnd the dimensional soliton

when γ<0. First consider (5.24)intheform



+ 3U



where we denote all variables with a prime. We will rescale the variables

appropriately. Assume the following transformation of coordinates



= l

ξ, T



= l

T, U



= l

Then (5.24) becomes

ξξξ

+ 3

= 0.

120 Water waves and KdV-type equations

If we multiply the whole equation by l

then we get

ξξξ

+ 3UU

= 0.

To get (5.24) into standard form, (5.34), we set

and

One solution is (l

, l

) = (1, 6, 2/3). Now our soliton solution takes the form



(ξ, T) =

4β

sech



β(ξ −

T − x

)



Next use, ξ



= x



− t



and T



= εt



to ﬁnd



, t



) =

4β

sech



β(x



− (1 +

ε)t



− x



The dimensional solution employs the following change of variables

η = aU



, x = λ



, t =



Substituting this into our solution, we have

η(x, t) =

4β

sech



−



1 +



t −



Now we use the relations a/h = ε, μ = h/λ

, and μ

= ε to write down the

dimensional solution to the KdV equation in the form

εβ

sech



√

ε(x − (1 +

ε)c

t − x

)



Note that both β and ε are non-dimensional numbers, and  is related to the

size of a typical wave amplitude. Recall that a = |η

max

| and c

= gh.The

excess speed beyond the long wave speed is

2β

ε. Russell observed this as

well! Later, in 1871, Boussinesq found this relationship from the more general

point of view of weakly nonlinear waves moving in two directions. Boussinesq

also found KdV-type equations (Boussinesq, 1877). Korteweg and de Vries

found this result in 1895 concentrating on unidirectional water waves. They

also found a class of periodic solutions in terms of elliptic functions and called

them “cnoidal” functions (see also the discussion in Chapter 1).

5.5 Solitary wave solutions 121

y = 0

= 0

Figure 5.2 Lump solution of KPI, on the left. To better illustrate, we also plot

slices along the x-andy-directions on the right.

5.5.2 Solitons in the KP equation

Recall the KPI equation (with “large” surface tension) in non-dimensional

form, (5.33), is

∂

+ 6uu

+ u

xxx

) − 3u

= 0.

This equation admits the following non-singular traveling “lump” soliton

solution (cf. Ablowitz and Clarkson, 1991) (see Figure 5.2)

u(x, y, t) = 2∂

⎡

⎢

⎣

log

⎛

⎜

⎝

(ˆx −2k

ˆy)

+ 4k

ˆy

⎞

⎟

⎠

⎤

⎥

⎦

ˆx = x − 12



+ k



t − x

ˆy = y −12k

t − y

Note that this lump only moves in the positive x-direction. Though the one-

dimensional solitons in the previous section have been observed, this lump

solution has not yet been seen in the laboratory. Presumably, this is because of

the diﬃculties of working with large surface tension ﬂuids. But, as mentioned

above, only recently has the one-dimensional KdV “depression” solitary wave

been observed for large surface tension (Falcon et al., 2002).

If there exists a solution to (5.33) such that u, u

, u

xxx

→ 0asx →−∞

then we have

+ 6uu

+ u

xxx

= −3σ



−∞



In particular, if u, u

, u

xxx

→ 0 and u

→ 0asx →∞,wealsohave



∞

−∞

(x, y, t) dx = 0.

122 Water waves and KdV-type equations

But if we give a general initial condition u(x, y, t = 0) = u

(x, y), it need not

satisfy this constraint at the initial instant. For the linearized version of the

KP equation, one can show (Ablowitz and Villarroel, 1991) that if the initial

condition satisﬁes



∞

−∞

(x, y) dx  0,

nevertheless the solution obeys



∞

−∞

u(x, y, t) dx = 0, for all t  0

and the function ∂u/∂t is discontinuous at t = 0. Ablowitz and Wang (1997)

show why KP models of water waves lead to such constraints asymptotically

and how a two-dimensional Boussinesq equation has an “initial value layer”

that “smooths” the discontinuity in



(x, y, t) dx.

5.5.3 KdV and related models

Suppose we consider the KdV equation in the form

+ u

+ ε(u

xxx

+ αuu

) = 0 (5.36)

where α is constant. First look at the linear problem and assume the wave

solution u ∼ exp(i(kx − ωt). The dispersion relation is then

ω = k − εk

which, for |k|1, implies ω ∼−εk

. In particular, note that ω →−∞, which

can present some numerical diﬃculties. One way around this issue is to use the

relation u

= −u

+ O(ε) to alter the equation slightly (Benjamin et al., 1972)

+ u

+ ε(−u

xxt

+ αuu

) = 0. (5.37)

Note that (5.36)–(5.37) are asymptotically equivalent to O(). But for large k,

the dispersion relation for the linear problem is now

ω =

1 + εk

∼

εk

for |k  1. Numerically, (5.37) has some advantages over (5.36). But, we still

have a “small” parameter, ε, in both equations so they are not asymptotically

reduced. Alternatively, if we use the following transformation in (5.36), as

indicated by multiple-scale asymptotics,

ξ = x − t, T = εt,

∂

= ∂

,∂

= −∂

+ ε∂

5.5 Solitary wave solutions 123

then we get the equation

+ u

ξξξ

+ αuu

= 0, (5.38)

which has no direct dependence on the small parameter ε. Equation (5.38)

is also useful for numerical computation since T = O(1) corresponds to t =

O(1/ε).

Finally, recall the Fermi–Pasta–Ulam (FPU) model of nonlinear coupled

springs,

m¨y

= k(y

i+1

+ y

i−1

− 2y

) + α((y

i+1

− y

)

− (y

− y

i−1

)

) (5.39)

discussed in Chapter 1. In the continuum limit, use y

i+1

= y(x) + hy



(x) +



(x)/2 + ···, x = ih, to ﬁnd

− h

xxxx

+ 2αh

+ ···,

where we have used a dot to denote

. Making the following substitutions

t = τ/hω, ω

= k/m, ε = 2αh, and δ

ε = h

/12, where the last two equalities

arise from the stipulation of maximal balance, we have

ττ

− y

= ε(δ

xxxx

+ y

). (5.40)

This is a Boussinesq-type equation and is also known to be integrable

(Ablowitz and Segur, 1981; Ablowitz and Clarkson, 1991). The term integrable

has several interpretations. We will use the notion that if we can exactly lin-

earize (as opposed to solving the equation via a perturbation expansion) the

equation then we consider the equation to be integrable (see Chapters 8 and 9).

Linearization or direct methods when applied to an integrable equation allow

us to ﬁnd wide classes of solutions.

Performing asymptotic analysis, in particular multiple scales on (5.40) with

the expansion

y ∼ y

+ εy

+ ···

= φ(X; T), X = x − τ, T = ετ,

we ﬁnd that, after removing secular terms (see also Chapter 1),

2φ

+ φ

+ δ

XXXX

= 0.

Or, if we make the substitution φ

= u then we end up with the integrable KdV

equation

XXX

= 0.

124 Water waves and KdV-type equations

Another lattice equation, called the Toda lattice that is known to be

integrable, is given by

m¨y

= e

k(y

i+1

−y

)

− e

k(y

−y

i−1

)

If we expand the exponential and keep quadratic nonlinear terms, we get

the FPU model (5.39), which in turn asymptotically yields the KdV equa-

tion. This further shows that the KdV equation (5.34) arises widely in applied

mathematics and the FPU problem itself is very “close” to being integrable.

5.5.4 Non-local system and the Benney–Luke equations

In a related development, a recent reformulation (see Ablowitz et al., 2006)of

the fully nonlinear water wave equations with surface tension leads to two

equations for two unknowns: η and q = q(x, y, t) = φ(x, y,η(x, y, t)). The

equations are given by the following “simple looking” system



∞

−∞



∞

−∞

dxdye

i(kx+ly)



iη

cosh[κ(η + h)]

(k, l) ·∇q sinh

κ(η + h)



= 0 (I)

|∇q|

+ gη −

(η

+ ∇q ·∇η)

2(1 + |∇η|

)

= σ∇·

⎛

⎜

⎝

∇η



1 + |∇η|

⎞

⎟

⎠

(II)

where (k, l)·∇q = kq

+lq

,κ

= k

,σ= T/ρ where T is the surface tension

and ρ is the density; here η and derivatives of q are assumed to decay rapidly

at inﬁnity. The ﬁrst of the above two equations is non-local and is written in a

“spectral” form.

The non-local equation satisﬁes the Laplace equation, the kinematic condi-

tion and the bottom boundary condition. The second equation is Bernoulli’s

equation rewritten in terms of the new variable q. These variables were intro-

duced by Zakharov (1968) in his study of the Hamiltonian formulation of the

water wave problem. Subsequently they were used by Craig and co-workers

(Craig and Sulem, 1993; Craig and Groves, 1994) in their discussion of the

Dirichlet–Neumann (DN) map methods in water waves. In this regard we

note:

= φ

−∇φ ·∇η = ∇φ ·



where



n = (−∇η, 1) is the normal on y −η = 0. Thus ﬁnding q(x, y, t), η(x, y, t)

yields η

, which in turn leads to the normal derivative ∂φ/∂n; hence the DN