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

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Exercises 15

−1

w (inverse tan function), and

solve the equation for w to ﬁnd all bounded, real periodic solutions

for w and therefore U. Express the solution in terms of Jacobi elliptic

functions. (Hint: See Chapter 4.)

(d) Use the above to ﬁnd all bounded wave solutions U that tend to zero

at −∞ and 2π at +∞. These are called kink solutions; they turn out

to be solitons.

1.4 Consider the KdV equation

+ 6uu

+ u

xxx

= 0.

(a) Make the transformation x → k

x, t → k

t, u → k

u, k  0, and ﬁnd

l, m, n so that the KdV equation is invariant under the transformation.

(b) Make the transformation u = t

−2/3

f (v), v = xt

−1/3

, f = 2(log F)



Find an equation for the similarity solution f and an equation for F;

then obtain a rational solution to the KdV equation. (See Section 3.2

for a discussion of self-similar/similarity solutions.)

1.5 Find the bounded traveling wave solution to the generalized KdV

+ (n + 1)(n + 2)u

+ u

xxx

= 0

where n = 1, 2,....

1.6 Consider the modiﬁed KdV (mKdV) equation

− 6u

+ u

xxx

= 0.

(a) Make the transformation x → a

x, t → a

t, u → a

u and ﬁnd l, m,

n, so that the equation is invariant under the transformation.

(b) Introduce u(x, t) = (3t)

−1/3

f (ξ), ξ = x(3t)

−1/3

to reduce the mKdV

equation to the following ODE



= ξ f + 2 f

+ α

where α is constant. The equation for f is called the second Painlev

equation, cf. Ablowitz and Segur (1981).

1.7 Show that a solitary wave solution of the Boussinesq equation,

− u

+ 3(u

)

− u

xxxx

= 0,

is u(x, t) = a sech

[b(x − ct) + d], for suitable relations between the

constants a, b, c, and d. Verify that the Boussinesq solitary wave can

propagate in either direction.

1.8 Find the bounded traveling wave solution to the equation

= u

+ u

xxxx

16 Introduction

Hint: Set ξ = x − ct, integrate twice with respect to ξ, and use u

= q(u)

to solve the resulting equation.

1.9 Consider the sine–Gordon equation

− u

= sin u.

(a) Using the transformation χ = γ(x − vt), τ = γ(t − vx) write the

equation in terms of the new coordinates χ, τ; ﬁnd γ in terms of v,

−1 < v < 1 so that the equation is invariant under the transformation.

(b) Consider the transformation ξ = (x + t)/2,η = (x − t)/2. Find the

equation in terms of the new coordinates ξ, η. Show that this equation

has a self-similar solution of the form u(ξ, η) = f (z), z = ξη. Then

ﬁnd an equation for w = exp(if). The equation for w is related to the

third Painlev

e equation, cf. Ablowitz and Segur (1981). (See Section

3.2 for a discussion of self-similar/similarity solutions.)

1.10 Seek a similarity solution of the Klein–Gordon equation,

− u

= u

in the form u(x, t) = t

f (xt

) for suitable values of m and n. Show that

f (z), z = x/t satisﬁes the equation (z

− 1) f



+ 4zf



+ 2 f = f

. (See

Section 3.2 for a discussion of self-similar/similarity solutions.)

Linear and nonlinear wave equations

In Chapter 1 we saw how the KdV equation can be derived from the FPU

problem. We also mentioned that the KdV equation was originally derived for

weakly nonlinear water waves in the limit of long or shallow water waves.

Researchers have subsequently found that the KdV equation is “universal” in

the sense that it arises whenever we have a weakly dispersive and a weakly

quadratic nonlinear system. Thus the KdV equation has also been derived from

other physical models, such as internal waves, ocean waves, plasma physics,

waves in elastic media, etc. In later chapters we will analyze water waves in

depth, but ﬁrst we will discuss some basic aspects of waves.

Broadly speaking, the study of wave propagation is the study of how signals

or disturbances or, more generally, information is transmitted (cf. Bleistein,

1984). In this chapter we begin with a study of “dispersive waves” and we

will introduce the notion of phase and group velocity. We will then brieﬂy

discuss: the linear wave equation, the concept of characteristics, shock waves

in scalar ﬁrst-order partial diﬀerential equations (PDEs), traveling waves of the

viscous Burgers equation, classiﬁcation of second-order quasilinear PDEs, and

the concept of the well-posedness of PDEs.

2.1 Fourier transform method

Consider a PDE in evolution form, ﬁrst order in time, and in one spatial

dimension,

= F[u, u

, u

,...],

where F is, say, a polynomial function of its arguments. We will consider the

initial value problem on |x| < ∞ and assume u → 0suﬃciently rapidly as

|x|→∞with u(x, 0) = u

(x) given. To begin with, suppose we consider the

linear homogeneous case, i.e.,

18 Linear and nonlinear wave equations



j=0

(x, t)u

where u

≡ ∂

u/∂x

and a

(x, t) are prescribed coeﬃcients. When the a

are

constants,



j=0

, (2.1)

and u

(x) decays fast enough, then we can use the method of Fourier transforms

to solve this equation. Before we do that, however, let us recall some basic facts

about Fourier transforms (cf. Ablowitz and Fokas, 2003).

The function u(x, t) can be expressed using the (spatial) Fourier transform as

u(x, t) =

2π



b(k, t)e

ikx

dk, (2.2)

where it is assumed that u is smooth and |u|→0as|x|→∞suﬃciently

rapidly; also, unless otherwise speciﬁed,



represents an integral from −∞ to

+∞ in this chapter. For our purposes it suﬃces to require that u ∈ L

meaning that



|u|dx and



|u|

dx are both ﬁnite. Substituting (2.2)into(2.1),

assuming the interchange of derivatives and integral, leads to



ikx

− b



j=0

(ik)

dk = 0.

It follows that

= b



j=0

(ik)

or that

= −iω(k)b,

with

−iω(k) =



j=0

(ik)

. (2.3)

We call ω(k), which we assume is real, the dispersion relation corresponding

to (2.1). For example, if N = 3, then ω(k) = ia

− ka

− ik

+ k

and ω(k)

is real if a

, a

are real and a

, a

are pure imaginary. We can solve this ODE

to get

b(k, t) = b

(k)e

−iω(k)t

2.2 Terminology: Dispersive and non-dispersive equations 19

where b

(k) ≡ b(k, 0). From this one obtains

u(x, t) =

2π



(k)e

i[kx−ω(k)t]

dk.

Hence b

(k) plays the role of a weight function, which depends on the initial

conditions according to the inverse Fourier transform:

(k) =



u(x, 0)e

−ikx

dx.

Strictly speaking, we now have an “algorithm” in terms of integrals for

solving our problem for u(x, t).

Note that, in retrospect, we can also obtain this relation by substituting

u(x, t) =

2π



(k)e

i[kx−ω(k)t]

into the PDE.

There is, in fact, an alternative method for obtaining the dispersion relation.

It is based on the observation that in this case one can substitute u

= e

i[kx−ω(k)t]

into the PDE and replace the time and spatial derivatives by

∂

→−iω, ∂

→ ik.

Then (2.3) follows from (2.1) directly.

2.2 Terminology: Dispersive and non-dispersive equations

Let us deﬁne and then explain the terminology that is frequently used in con-

junction with these wave problems and Fourier transforms: k is usually called

the wavenumber, ω is the frequency, k and ω real, and θ ≡ kx − ω(k)t is the

phase in the exponent or simply the phase.

The temporal period (or period for short) is denoted by

T ≡

2π

The meaning of the period is that whenever t → t + nT , where n is an inte-

ger, then the phase remains the same modulo 2π and therefore e

iθ

remains

unchanged. Similarly, we call λ = 2π/k the wavelength and note that when-

ever x → x + nλ the phase remains the same modulo 2π and therefore e

iθ

remains unchanged. Furthermore, we call

c(k) ≡

ω(k)

20 Linear and nonlinear wave equations

the phase velocity, since θ = k(x − c(k)t). There is also the notion of group

velocity v

(k) ≡ ω



(k); i.e., the speed of a slowly varying group of waves. We

will discuss its importance later.

An equation in one space and one time dimension is said to be dispersive

when ω(k) is real-valued and ω



(k)  0. The meaning of dispersion will be fur-

ther elucidated when we discuss the long time asymptotics of these equations.

Consider the ﬁrst-order linear equation

− a

= 0, (2.4)

where a

 0, real and constant. We see that

−iω = ika

which gives the linear dispersion relation

ω(k) = −a

In this case ω



(k) ≡ 0, which means the PDE is non-dispersive.

Now let us look at the ﬁrst-order constant coeﬃcient equation (2.1). In this

case the dispersion relation (2.3) shows that if all the a

are real, then ω is real

if and only if all the even powers of k (or even values of the index j) vanish;

i.e., a

= 0for j = 2, 4, 6,.... In that case it follows from (2.3) that ω(k)isan

odd function of k, in which case the dispersion relation takes the general form

ω(k) = −



j=0

(−1)

2 j+1

A further example is the linearized KdV equation given by

+ u

xxx

= 0, (2.5)

i.e., (1.5) without the nonlinear term. By substituting u

= e

i[kx−ω(k)t]

one

obtains its dispersion relation,

ω(k) = −k

Thus ω is real and ω



= −6k  0, which means that this is a dispersive

equation.

We can use Fourier transforms to solve this equation. As indicated above,

using the Fourier transform, u(x, t) can be expressed as

u(x, t) =

2π



b(k, t)e

ikx

dk,

2.2 Terminology: Dispersive and non-dispersive equations 21

and for the linearized KdV equation (2.5) one ﬁnds that b

= ik

b hence

u(x, t) =

2π



(k)e

i(kx+k

dk.

However, by itself this is not a particularly insightful solution, since one cannot

evaluate this integral explicitly. Often having an integral formula alone does

not give useful qualitative information. Similarly, we usually cannot explicitly

evaluate the integrals corresponding to most linear dispersive equations. This is

exactly the place where asymptotics of integrals plays an extremely important

role: it will allow us to approximate the integrals with simple, understandable

expressions. This will be a recurring theme in this book: namely, analytical

methods can yield solutions that are inconvenient or uninformative and asymp-

totics can be used to obtain valuable information about these problems. Later

we will brieﬂy discuss the long-time asymptotic analysis of the linearized KdV

equation.

In so far as we are dealing with dispersive equations, we have seen that

requiring ω(k) to be real implies that ω(k) is odd. If in addition, u(x, t)isreal,

this knowledge can be encoded into b

(k) as follows. We have that

∗

(x, t) =

2π



∗

(k)e

−i[kx−ω(k)t]

dk,

where u

∗

denotes the complex conjugate of u. Calling k



= −k yields

∗

(x, t) = −

2π



−∞

∞

∗

(−k



−i[−k



x−ω(−k



)t]



2π



∞

−∞

∗

(−k



−i[−k



x−ω(−k



)t]



Then if we require that u is real-valued then u(x, t) = u

∗

(x, t). In addition,

ω(−k



) = −ω(k). Combined, one obtains that

2π



(k)e

i[kx−ω(k)t]

dk =

2π



∗

(−k



i[k



x−ω(k



)t]



This identity is satisﬁed for all (x, t) if and only if

∗

(k) = b

(−k).

Note that we cannot apply the Fourier transform method for the nonlinear

problem, sometimes called the inviscid Burgers equation,

+ uu

= 0,

because of the nonlinear product. We study the solution of this equation by

using the method of characteristics; this is brieﬂy discussed later in this chapter.

22 Linear and nonlinear wave equations

We also note that for some problems, u

= exp(i(kx − ω(k)t)) yields an ω(k)

that is not purely real. For example, for the so-called heat or diﬀusion equation

= u

we ﬁnd ω(k) = −ik

. The Fourier transform method works as before giving

u(x, t) =

2π



∞

−∞

(k)e

ikx−k

dk.

We see that the solution decays, i.e., the solution diﬀuses with increasing t.

2.3 Parseval’s theorem

The L

-norm of a function f (x) is deﬁned by

f 

≡



|f (x)|

dx.

Let

f (k) be the Fourier transform of f (x). Parseval’s theorem [see e.g.,

Ablowitz and Fokas (2003)] states that

f (x)

2π



f (k)

In many cases the L

-norm of the solutions of PDEs has the meaning of energy

(i.e., is proportional to the energy in physical units). The physical meaning

associated with Parseval’s relation is that the energy in physical space is equal

to the energy in frequency (or sometimes called spectral or Fourier) space.

Moreover, we know that when ω is real-valued then



|u(x, t)|

dx =

2π



(k)|

dk = constant.

Hence, it follows from Parseval’s theorem that energy is conserved in lin-

ear dispersive equations. While it is possible to prove this result using direct

integration methods, we get it “for free” in linear PDEs using Parseval’s

theorem.

2.4 Conservation laws

We saw above how Parseval’s theorem is used to prove energy conservation.

However, there may be other conserved quantities. These often play a very

useful role in the analysis of problems, as we will see later.

2.5 Multidimensional dispersive equations 23

A conservation law (or relation) has the general form

∂

∂t

T (x, t) +

∂

∂x

F(x, t) = 0;

we call T the density of the conserved quantity, and F the ﬂux. Let us integrate

this relation from x = −∞ to x = ∞:

∂

∂t



Tdx+ F(x, t)

x→+∞

x→−∞

= 0.

The second term is zero, since we assume that F decays at inﬁnity, which

leads to

∂

∂t



Tdx= 0,

i.e.,



Tdx= constant.

For example, let us study the conservation laws for the linearized KdV

equation:

+ u

xxx

= 0.

This equation is already in the form of a conservation law, with T

= u and

= u

; that is,



udx is conserved. This, however, is only one of many

conservation laws corresponding to (2.5). Another example is energy conser-

vation. Using Parseval’s theorem, we have already proven that linear dispersive

equations with constant coeﬃcients satisfy energy conservation. Since (2.5)is

solvable by Fourier transforms we know from Parseval’s theorem that energy

is conserved. An alternative way of seeing this is by multiplying (2.5)byu and

integrating with respect to x. It can be checked that this leads to

∂

∂t





∂

∂x



−



= 0,

from which it follows that



|u|

dx = constant.

2.5 Multidimensional dispersive equations

So far we have focused on dispersive equations in one spatial dimension. The

method of Fourier transforms can be generalized to solve constant coeﬃcient

multidimensional equations, where a typical solution takes the form:

24 Linear and nonlinear wave equations

Figure 2.1 Phase contours in two dimensions (n = 2).

u(x, t) =

2π



(k)e

i[k·x−ω(k)t]

and x = (x

, x

,...,x

), k = (k

, k

,...,k

), cf. Whitham (1974). As before, ω

is the frequency (assumed real) and T = 2π/ω is the period. The wavenumber

becomes the vector k = ∇θ, where θ ≡ k · x − ω(k)t, and the wavelength

λ = 2π

k/|k|, where

k = k/|k| and |·|is the (Euclidean) modulus of the vector.

Thus θ(x + λ) = θ(x) + 2π. The condition that θ be constant describes the

phase contours (see Figure 2.1). We can also deﬁne the phase speed as c(k) =

ω(k)

k/|k|, from which it follows that c · k = ω.

In the multidimensional linear case, an equation is said to be dispersive if ω

is real and (instead of ω



 0) its Hessian is non-singular, that is

det

∂

ω(k)

∂k

 0.

If we have a nonlinear equation, we call it a nonlinear dispersive wave equation

if its linear part is dispersive and the equation is energy-preserving.

2.6 Characteristics for ﬁrst-order equations

First let us use the Fourier transform method to solve (2.4), u

− a

= 0. As

before, we get ω(k) = −a

and

u(x, t) =

2π



(k)e

ik(x+a

dk.

Calling

f (x) ≡ u(x, 0) =

2π



(k)e

ikx

dk,