Ferrarese G. (editor) Wave Propagation
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84
References
[1]
Jeffrey,
A.
Quasilinear
Hyperbolic
Systems and Waves,
Research
Note
in
Mathematics, 5, Pitman
Publishing,
London, 1976.
[2] Lax, P. D.
Hyperbolic
Systems
of
Conservation
Laws and
the
}~thematical
Theory of Shock Waves,
SlAM
Regional
Conference
Series
in
Applied
Mathematics,
II,
1973.
[3] Glimm,
J.
Solutions
in
the
large
for
nonlinear
hyperbolic
systems
of
equations,
Comm.
Pure
Appl. Math. 13
(1965),
697-715.
[4] Godunov, S.
An
interesting
class
of
quas
ilinear
systems,
Dokl. Akad.
Nauk
SSSR
139
(1961),
521-523.
[5] Godunov, S. Bounds on
the
discrepancy
of
approximate
solutions
constructed
for
the
equations
of
gas dynamics,
J.
Comput.
Math. and Math. Phys . 1
(1961),
623-637.
[6]
Chorin,
A.
Random
choice
solution
of
hyperbolic
systems,
J .
Compo
Phys.
22
(1976),
517-533.
[7]
Jeffrey,
A. The
exceptiona
l
condition
and unboundedness
of
solutions
of
hyperbolic
systems
of
conservation
type,
Proc.
Roy. Soc.
Edinb. 77A
(1977),
1-8.
85
Lecture
8.
Far
Fields,
Solitons
and
Inverse
Scattering
1.
Far
Fields
There
are
many
different
types
of
higher
order
equations
and systems
of
equations
that
characterise
nonlinear
wave
propagation
in
~x
t,
either
with
or
without
dispersion.
A
simplification
frequently
takes
place
in
the
representation
of
the
solutions
to
initial
value
problems
to
such
equations
after
a
suitable
lapse
of
time
or,
equivalently,
suitably
far
from
the
origin,
particularly
when
the
initial
data
is
localised
and so
has
compact
support.
These s
implified
forms
of
solution
are
often
asymptotic
solutions,
and
are
appropriately
called
far
fields.
The
simplest
examples
of
these
are
the
types
of
far
field
behaviour
exhibited
by
the
ordinary
linear
wave
equation
and by a homogeneous
quasilinear
hyperbolic
system
with
n
dependent
variables.
Due to
its
linearity,
the
wave
equation
(c
const.)
(1)
may be
written
either
in
the
form
or
as
(
2.
+ c
2.)
(au _
at
ax
at
au)
c-
ax
o ,
(2)
(
2.
_c
2.)
(au + c au) 0
at
ax
at
ax
Then,
if
u(+)
is
the
solution
of
(3)
au(±)
c-
ax
o .
(4)
it
follows
that
u(t)
is
a
degenerate
solution
of
(2)
and
u(-)
is
a
degenerate
solution
of
(3).
The
general
solution
of
(4)
is
then
f(±)
(x
+
ct)
,
(5)
with
f(
±)
arbitrary
C
l
functions.
Thes e
travelling
wave
solutions
are
such
that
u(+)
propagates
to
the
right
and
u(-)
to
the
left
with
speed
c.
We
thus have
the
s
ituation
that
u(
±)
8
6
are
special
simple
types
of
solution
to
the
wave
equation
(1),
in
the
sense
that
they
only
satisfy
a
first
order
partial
differential
equation.
whereas
the
wave
equation
itself
is
of
second
order.
Such
special
solutions
become
of
considerable
interest
when
the
initial
data
f(±)
is
differentiable
o
(+)
1
with
compact
support.
50
that
f
O-
(x)
! CO' Then.
if
the
support
of
the
initial
data
lies
in
Ixl
< d.
after
an
elapsed
time
dlc
the
interaction
between waves moving
to
the
left
and
right
ceases
and
only
the
solutions
u(-)
and
u(+)
are
observed
to
the
left
and
right
of
the
origin,
respect
ively.
These
are
the
far
fields
of
the
wave
equation
(1).
Since
u(+)
is
transported
along
the
C(+)
characteristics
x-ct·
( and
u(-)
along
the
C(-)
characteristics
X+ct = n. and
neither
family
of
characteristics
intersects
itself.
the
far
fields
of
the
wave
equation
will
propagate
indefinitely
after
the
interaction
has
finished.
The
situation
is
different
in
the
case
of
the
homogeneous
quasilinear
hyperbolic
system
of
hyperbolic
type
l£
+ A
(U)
au
• 0
at
ax
•
in
which U
is
an n x 1
vector
with
elements
~.
u
2
•••••
un and
A(U)
•
[aij(u
l•
u
2
•
••••
un)]
is
an n x n
matrix
with
elements.
depending
on
the
elements
of
U.
(6)
If.
now. we
seek
a
special
solution
of
(6)
in
which
n-l
elements
of
U
are
functions
of
only
the
one
remaining
element.
say
u
l•
we may
set
U·
U(u
l).
Direct
substitution
into
(6) shows
that
(
au
au
_)
j'"
___1 I + 1
A(U)
~U
•
at
ax
~
in
which I
is
the
unit
matrix.
o •
A
non-trivial
solution
of
this
form
only
exists
when
.0,
(8)
showing
that
if
A
is
an
eigenvalue
of
A,
87
(9)
(10)
Since
the
system
(6)
is
hyperbolic
there
are
n
real
eigenvalues
A(l),
A(2),
A(n)
of
A, from which
it
follows
that
when (6)
is
totally
hyperbolic
there
are
n
different
solutions
ul
i
)
satisfying
a
(i)
a
(i)
:~
+ A(i)(U)
:;
- 0
....
for
i -
1,2,
•••
,n.
The
solutions
(10)
are,
of
course,
simple
wave
solutions,
and
for
initial
data
having
compact
support
they
represent
the
far
field
bolutions
after
the
interaction
has
finished.
The
characteristic
curves
e(i)
in
this
case
are
given
by
solving
(11)
for
i ·
a
1,2,
•••
,n.
The
characteristics
comprising
each
family
e(i)
are
again
straight
lines,
but
now
they
are
no
longer
parallel
within
the
family
as
the
gradient
of
a
characteristic
depends on
the
value
of
the
solution
that
is
transported
along
it.
This
leads
to
a breakdown
of
differentiability
when members
of
a
family
of
characteristics
e(i)
intersect,
and to
the
formation
of
a
discontinuous
solution
at
some
finite
elapsed
time
t~i).
Thus
the
simple
waves U =
U(ui
i
»
corresponding
to
the
solutions
ui
i)
of
(10) can
only
form
far
fields
in
the
time
interval
between
the
end
of
the
interaction
period
for
initial
data
with
compact
support
and
the
breakdown
time
for
the
system
t
c
-
min{t~l),
(2)
(n)
t
c
'
•••
, t
c
}.
2.
Reductive
Perturbation
Method
The
far
field
equations
discussed
so
far
are
very
special,
since
the
equations
that
gave
rise
to them
involved
neither
dissipation
nor
dispersion,
and one was,
in
fact,
linear.
In
more
general
situations
both
dissipation
and
dispersion
may be
present,
and
typical
of
the
far
field
equations
that
then
result
are
the
following
nonlinear
evolution
equations:
88
Burgers'
Equation
(dissipative)
av + v av a
2v
at
ax -
\/--2
• (v >
0)
ax
KdV
Equation
(weakly
dispersive)
3
av + v av +
1.1
a v _
O.
(1.1
>
0)
at
ax ax3
Nonlinear
Schr~dinger
Equation
(strongly
dispersive)
av 1 a
2v
2
i - + - - +
alvl
v -
o.
at
2 ax2
An
important
scalar
equation
that
has
either
(13)
or
(14)
as
a
far
(12 )
(13)
(14)
field
equation.
depending
on
the
circumstances.
is
the
Boussinesq
equation
"(15)
in
which
u(x,t)
is
a
one-dimensional
field,
c
is
the
phase
velocity
in
the
long
wave
limit
and
1.1
is
the
dispersion
parameter.
This
occurs
in
the
study
of
water
waves .
A
very
general
quasilinear
system
that
contains
as
special
cases
many
of
the
systems
that
are
of
physical
interest
has
the
form
au
+
A(U)
au
+
at
ax
B +
[~
~
[H
Il
2...
+
K!'
2...)
.)
Il-l
a-I
a
at
a ax
U = 0 •
(p
~
2)
(16)
Here U
is
an n x 1
vector
with
elements
u
l'
u
2'
• • • '
un'
the
matrices
A,
H:, K:
are
all
n x.n
matrices
depending
on U and B
is
an n x 1
vector
depending
on U.
When
wave
propagation
is
involved,
it
is
weakly
dispersive
when B - 0,
and
strongly
dispersive
when B
~
O.
We
now
outline
the
so-called
reductive
perturbation
method due to
Taniuti
and Wei,
referring
for
all
the
details
involved
to
that
paper
or
to
the
review
by
Jeffrey
and
Kakutani
.
Considering
the
weakly
dispersive
case
(B - 0) we
apply
the
Gardiner-
Morikawa
transformation
T
a+1
c
t,
a
l!(p-l)
for
p ~ 2
(17)
to
system
(16)
where ).
is
taken
to
be
a
real
eigenvalue
of
A.
It
is
not
89
necessary
that
all
of
the
eigenvalues
of
A
are
real,
but
when
they
are,
and
the
corresponding
eigenvectors
span
the
space
En
associated
with
A,
the
first
order
system
comprising
the
first
three
terms
of
(16)
will
be
hyperbolic.
Set
(18)
where U
o
is
a
constant
solution
of
the
homogeneous form
of
(16)
(i.e.
B -
0).
Then,
rewriting
the
system
in
terms
of
derivatives
with
respect
to
t and
T.
and
equating
like
powers
of
E,
we
obtain
the
results
(19)
(20)
an,
au
au
(-AI
+ A
o
)
~:
+
-!.
+' {U • (II A)
}-!.
o~
ilT 1 u 0 ilt
+ r
~
[-
x H
a
+ K
a
)
ilPU
l
_
o.
a-I
a-I
aO aO
ilE;P
Here ·A
O'
H:
O'
K:
O
and
(VuA)O
indicate
quantities
appropriate
to
the
solution
u - U
O
'
while
V
u
denotes
the
gradient
operator
with
respect
to
the
elements
of
U.
Then
if
t and r
denote
the
left
and
right
eigenvectors
of
A
O
corresponding
to
the
eigenvalue
>.,
so
that
o and (A
O
- >'1) r o ,
(21)
equation
(19) may be
solved
in
the
form
(22)
with
~
one
of
the
elements
of
U
l
and VI an
arbitrary
vector
function
of
T.
The
compatibility
condition
for
(20) when
solving
for
ilU
2!ilE;
is
t
ilU
l
+ t[V (V A) ]
ilU
l
+ I. i
~
[_m
a
+ K
a
) ilt·: _ 0 • (23)
ilT
l'
u 0
~
s-i
a-I
eo eo
ilE;P
Then
taking
the
boundary
condition
U
~
U
o
as x
~~.
so
that
we may
set
VI :
0,
we
find
that
~
satisfies
the
nonlinear
evolution
equation
9°
ilP~
c
__
1 •
2 ilf;P
o , (24)
where
and
When p • 2
equation
(24) becomes
Burgers'
equation,
and when P • 3
the
KdV
equation.
The
scalar
equation
(24)
thus
governs
the
far
field
behaviour
of
the
homogeneous form
of
system
(16)
that
is
associated
with
the
eigenvalue
A.
There
will
be
such
a
far
field
for
each
real
eigenvalue
A
of
A.
In
concluding
this
section
we
remark
that
although
in
what
follows
we
shall
be
referring
to
properties
of
exact
solutions
of
some
far
field
equations,
it
should
be remembered
that
these
far
field
equations
are
in
the
main
only
asymptotic
approximations
to
the
solution
that
is
of
interest
.
There
are,
in
fact
, a
variety
of
different
methods by which
the
nonlinear
evo
lution
equations
characterising
far
fields
may be
derived,
and
for
an
account
of
three
o
ther
methods we
refer
to
the
paper
by
Jeffrey
and Kawahara ,
to
the
S
cheveningen
Conference
paper
by
Jeffrey
and
to
the
AMS
paper
by Whitham.
Before
moving on
to
discuss
soliton
solutions
to
equations
l i ke (2
4),
we
remark
that
it
is
a
simple
matter
to
show
that
the
coefficient
c
l
in
(24 )
is
i n
fact
proportional
to
(VuA)'r.
This
means
that
when
the
characteristi
c
field
associated
with
A
is
exceptional
c
l
= 0 and
the
nonlinear
term
in
(24)
vanishes.
Further
analysis
is
required
to
derive
the
nonlinear
evolution
equation
that
then
governs
the
far
field,
and
it
has
been shown by
Jeffrey
and
Kakutani
that
a
modified
KdV
type
equation
then
results.
3.
Travelling
Waves
and
Solitons
We
have
seen
that
in
nonlinear
hyperbolic
equations,
waves
propagate
with
a
shape
change
so
that
no
travelling
waves
can
exist
for
such
t
Lons
,
That
is
to
say,
there
is
no
reference
frame
moving
with
a
constant
spee1
s
91
in
which
the
wave
appears
staionary.
However, when
dispersion
or
dissipation
are
present
in
a
nonlinear
evolution
equation,
in
the
sense
that
the
linearised
equation
exhibits
these
effects
as
described
in
Lecture
1,
travelling
waves
become
possible
due to
the
competition
between
the
effects
of
non
linearity
and
dispersion
or
dissipation.
In
~
x
t,
travelling
wave
solutions
have
the
form
v(x
,t)
vet) , x -
st
,
s
const.
,
and,
in
addition
to
the
nonlinear
evolution
equation,
they
must
satisfy
some
appropriate
boundary
conditions
at
infinity.
In
general,
these
will
determine
the
permissible
range
of
values
of
s.
In
the
case
of
Burgers'
equation
and
the
KdV
equation
which
are,
respectively,
examples
of
purely
dissipative
and
purely
dispersive
nonlinear
evolution
equations,
we
find
when
seeking
solutions
for
which
all
the
derivatives
tend
to
zero
as
Ixl
4~,
the
well
known
solutions
:
Burgers'
Shock
Wave
(Purely
dissipative)
vet)
satisfying
1 - + 1 - + - +
•
I(V
... + v...) -
I(v
... - v...)
tanh
[(v
... - v...)1;/4v] •
(25)
and
lim
viti
Itl
.....
±
• v
...
with
v
...
+
>
v~
a
--
------
--
-----
U
oo
\f+
-----
«I:J
o
KdV
Solitary
Wave
(Purely
dispersive)
(26)
92
satisfying
lim
V(I;)
11;1--
and
v..
with
v..
~
0
s z v.. +
a/3.
o
The
Burgers'
shock
wave, as
the
solution
(25)
is
called,
is
seen
to
- +
propagate
with
a
speed
s • (v.. + v..
)/2
that
is
uniquely
determined
by
the
boundary
conditions,
but
is
invariant
with
respect
to an
arbitrary
fixed
spatial
translation.
In
general,
all
solutions
v(x,t)
of
Burgers'
equation
are
invariant
with
respect
to a
Galilean
transformation.
This
solution
links
two
different
constant
states
at
plus
and minus
infinity.
The
KdV
solitary
wave, as
the
solution
(26)
is
called,
is
different
and
is
a
pulse
shaped
wave
that,
relative
to
the
same
constant
value
v..
at
plus
and minus
infinity,
tends
to
zero
together
with
all
its
derivatives
as
11;1--.
Its
speed
of
propagation
relative
to
v..
is
proportional
to
the
amplitude
a,
and
its
width
is
inversely
proportional
to
the
square
root
of
the
amplitude.
In
this
travelling
wave
solution
the
speed
is
not
determined
by
the
boundary
conditions,
but
by
the
amplitude
a > O.
Like
Burgers'
equation,
the
KdV
solitary
wave
is
also
invariant
with
respect
to a
Galilean
transformation.
Z3busky and
Kruskal
found
numerically
that
a
KdV
solitary
wav~
behaves
like
a
particle.
Specifically
they
found
that
when two
different
amplitude
waves
of
this
type
are
such
that
the
one
with
the
greater
amplitude
starts
to
the
left
of
the
one
with
theless
er
amplitude,
then
the
larger
one
overtakes
the
smaller
one and ,
after
interacting
with
it,
the
waves
have
merely
inter-
93
changed
positions.
This
is
a
nonlinear
interaction
yet
the
pulse
shapes
are
preserved
exactly
after
the
interaction
, though
the
ph
ases
of
the
pulses
(the
location
of
their
peaks)
is
affected
by
this
process.
On
account
of
this
Zabusky and Kruska1
invented
the
word
"soliton"
for
a wave
that
preserves
its
identity
exactly
in
this
sense
after
a
nonlinear
interaction.
Thus
KdV
solitary
waves
are
solitons.
The
recent
interest
in
solitons
derives
from
the
fact
that
the
KdV
equation
is
often
found
to
arise
as a
far
field
equation
and,
furthermore,
arbitrary
initial
data
for
the
KdV
equation
evolves
into
a
train
of
solitons
together
with,
possibly
, an
oscillatory
tail.
This
means
that
solitons
are,
in
a
sense,
fundamental
solutions
of
the
KdV
equation.
An
extensive
literature
now
exists
on
this
topic,
and we
refer
to
the
articles
and to
the
references
contained
therein,
in
Jeffrey
and
Kakutani
and
in
the
various
articles
by
Kruskal,
Lax,
Ablowitz,
Newell and Segur
in
the
AMS
publication
Nonlinear
Wave Motion
listed
at
the
end
of
this
lecture.
Many
different
types
of
nonlinear
evolution
equation
have
been
found to
possess
soliton
solutions
and
for
more
information
on
this
topic
we
refer
to
the
review
paper
by
Scott,
Chu_ and McLaughlin
for
both
a good
account
of
some
of
them
and
also
for
the
basic
references,
and
also
to
the
edited
collection
of
papers
by Bul10ugh and Caudrey.
4.
Inverse
Scattering
The
behaviour
of
soliton
solutions
to
the
KdV
equation
is
suggestive
of
linear
behaviour
and
this
motivated
Gardner,
Greene,
Kruskal
and Miura
to
try
to
find
a
linearising
transformation
of
the
type
used
by Hopf and
Cole
to
transform
Burgers'
equation
to
the
heat
equation
(see
Scott,
Chu,
Mclaughlin).
No
such
transformation
was found,
but
during
their
search
they
discovered
an
important
connection
between
the
KdV
equation
and an
eigenvalue
problem
for
the
Schr6dinger
equation
in
terms
of
the
inverse
scattering
method used
in
qu~ntum
mechanics .
It
is
this
result
that
has