Transcript The FPTU Model and Solitary Waves - Mathematics

Mathematics provides physical science with a language
that connects cause and effect. The equations and functions
of mathematics provide a grammar and a vocabulary to
understand and interpret the physical world. Mathematical
language helps uncover hidden connections between seemingly
disparate physical phenomena, and helps us to think about
nature in new ways.
In this lecture, we will illustrate the power of mathematical
thinking by tracing the story of solitary waves, beginning with
their discovery by John Scott Russell in 1834.
Making Waves
with Mathematics
The Solitary Wave
of Translation
(Edinburgh, 1834)
The KdV Equation
(Amsterdam, 1895)
The Birth of
Experimental Mathematics
(Los Alamos, 1955)
Solitary Waves
and Quantum Waves
(Princeton and New York, 1964-68)
New Frontiers
(1972-Present)
Åke Hultkrantz (1920-2006), Swedish Anthropologist
and scholar of the Saami, Shosone, and Arapaho peoples
The Solitary Wave of Translation
John Scott Russell
(1808-1882)
This is a most beautiful and extraordinary phenomenon.
The first day I saw it was the happiest day of my life.
Nobody had ever had the good fortune to see it before or,
at all events, to know what it meant. It is now known as
the solitary wave of translation.
John Scott Russell, 1865
Union Canal, Gyle, Edinburgh, Scotland
“I was observing the motion of a boat which was rapidly drawn
along a narrow channel by a pair of horses, when the boat
suddenly stopped - not so the mass of water in the channel which
it had put in motion; it accumulated round the prow of the vessel
in
a
state of violent agitation, then suddenly leaving it behind, rolled
forward with great velocity, assuming the form of a large
solitary elevation, a rounded, smooth and well-defined heap of
water, which continued its course along the channel apparently
without change of form or diminution of speed.”
Plaque at Hermiston House, Union Canal
“I followed it on horseback, and overtook it still rolling on at
a
rate of some eight or nine miles an hour, preserving its
original
figure some thirty feet long and a foot to a foot and a half in
height. Its height gradually diminished, and after a chase of
one or two miles I lost it in the windings of the channel.
Such,
in the month of August 1834, was my first chance interview
with that singular and beautiful phenomenon which I have
called the Wave of Translation.”
John Scott Russell, Report on Waves, Fourteenth Meeting of
Russell re-created the solitary waves and discovered four facts
about them:
(1) The shape of the wave is described by the hyperbolic secant
function
(2) A larger solitary wave travels faster than a smaller one
(3) Solitary waves can cross each other “without change of
any kind”
(4) A large enough mass of water creates two or more
independent solitary waves
The Hyperbolic Secant Function
The Korteweg-De Vries Equation
Diederik Korteweg
(1848-1941)
Gustav de Vries
(1866-1934)
We are not disposed to recognize this wave (discovered by
Scott
Russell) as deserving the epithets “great” or “primary,” and
utt  g u xx
we
conceive that ever since it was known that the theory of
waves
in shallow water was contained in the equation
the theory of solitary waves has been perfectly well-known.
George Biddle Airy, “Tides and Waves,”
Encyclopedia Metropolitana, 1845
The Search for a Law of Motion
A travelling wave in a narrow channel is described by a
height function u (x , t ) where
x
is the distance along the channel
t
is the elapsed time
u (x , t ) is the height of the wave above the undisplaced
fluid surface
To understand Russell’s wave, one must find a law of motion
that describes how the shape functionu (x , t ) changes with
time, and correctly predicts the four properties of solitary waves
observed by Russell.
Kortweg and de Vries revisited the equations of shallow
water waves in a channel, assuming that the depth of the
channel was small compared to the wave length. They
assumed a wave moving to the right and in place of Airy’s
linear equation utt  g u xx
for the shape function, they
obtained the nonlinear Korteweg-de Vries Equation
u  3u
u
 3  6u
0
t x
x
What is the physical mechanism that generates solitary waves?
u  3u
u
 3  6u
0
t x
x
The rate of change of the shape function is determined by two
different terms:
• A term that tends to make waves disperse
• A term that tends to make waves focus
If one looks for solutions to the KdV equation of the form
u (x , t )  u (x  ct )
one immediately finds a solution that describes a single
solitary wave:
 c
1
2
u (x , t )  c sech 
x  ct
2
 2





explaining Russell’s observations about the wave shape and
the relationship between wave amplitude and speed of
propagation
When both dispersive and nonlinear terms are present,
focussing can competes with dispersion to create stable,
nondispersive waves. In special circumstances, it may be
possible for the competition to “balance” and create a
solitary wave which neither focuses nor disperses.
But under what circumstances, and why?
The Birth of Experimental
Mathematics
Enrico Fermi John Pasta Stanlisaw Ulam Mary Tsingou
(1901-1954) (1918-1984) (1909-1984)
Menzel
(1927- )
It is Fermi who had the genius to propose that
computers could be used to study a problem or
test a physical idea by simulation, instead of
simply performing standard calculus.
Thierry Dauxois
• Fermi, Pasta and Ulam wanted to study how energy is
distributed between normal modes of a one-dimensional
crystal
• The crystal consists of a lattice of collinear atoms each
of which vibrates due to interaction with its two nearest
neighbors
• The system can be modeled as a system of masses
coupled by springs
The spring mass system is linear if the force exerted on
a given mass by its neighbor to the left or right is
proportional to the compression or extension of the
spring.
A linear spring-mass system normal modes of vibration
which, once started, continue repeating the
same motion over and over. FPUT believed that this
would change if the force law for the springs was
changed to a nonlinear law.
The FPTU model consists of:
• N  64 balls on a line, connected by springs
• u n is the displacement of the nth mass from its
rest position
• Newton’s law is applied to each mass to find the
motion
Linear Versus Nonlinear Springs
• In the linear spring model, each mass
moves according to the law

 

m un  k  un  un 1  un 1  un 


• In the nonlinear spring model, each
mass moves according to the law

 


mun  k  un  un 1  un 1  un  1   un 1  un



Fermi, Pasta, Tsingou, and Ulam computed the motion
of the 64 nonlinear oscillators on a computer called the
MANIAC I (Mathematical Analyzer, Numerator,
Integrator, and Computer) based on ideas of John von
Neumann
MANIAC I
• For the linear model, there are normal modes of
oscillation which, once started, persist forever
• Fermi, Pasta, Tsingou, and Ulam expected that in the
nonlinear model, the energy of the system would
be distributed over all accessible modes of vibration
• Instead, they found that, if the system of nonlinear
oscillators was “started” in a low mode, it would
come
back to that mode repeatedly, so that the nonlinear
oscillators were behaving as if they were "really"
linear
oscillators
The FPTU Model and Solitary Waves
Kruskal and Zambusky (1965) sought to understand why
the FPTU nonlinear oscillators exhibited periodic behavior.
They computed a continuum limit of the system studied by
FPTU. The continuum limit means:
• Take the spacing h between oscillators to zero
• Take the number N of oscillators to infinity
• Consider u j to be a discrete "sampling" of a continuous
function,
u ( jh, t )  u j (t )
Approaching The Continuum Limit
N  32
N  64
N  128
• It was long known that the continuum limit for the
linear model gives Airy’s wave equation
 2u
1  2u
 2 2
2
x
c t
• Kruskal and Zambusky found that the continuum limit
of the FPTU nonlinear model gives the Korteweg-de Vries
equation
u  3u
u
 3  6u
0
t x
x
the same equation that describes Russell’s solitary wave!
• They computed numerical solutions to the KdV equation
and reproduced the periodic behavior of FPTU’s
numerical experiment
Kruskal and Zabusky called the new solitary waves
“solitons.” The mechanism produced them remained
mysterious
Solitary Waves
and
Quantum Waves
Clifford Gardner
(1924- )
John Greene
(1928-2007)
Martin Kruskal
(1925-2006)
Robert Miura
(1938- )
Peter Lax
(1926- )
“Theories permit consciousness to `jump over its own
shadow’, to leave behind the given, to represent the
transcendent, yet, as is self-evident, only in symbols.”
Hermann Weyl
• Mathematicians suspected that some as yet
undiscovered conservation laws were responsible
for the existence of solitary waves
• Conservation laws for the total mass and energy
were known, and more conservation laws were
found by hard calculation
Conservation Laws






u dx
(Conservation of mass)
u 2 dx
(Conservation of Momentum)
 3 1 2
 u  2 u x  dx

(Conservation of Energy)
“The next surge of momentum came with the arrival of
Robert Miura who was asked by Kruskal to get his feet
wet by searching for a conservation law at level seven.
He found one and then quickly filled in the missing
sixth. Eight and nine fell quickly…
Miura was challenged to find the tenth. He did it during
a two-week vacation in Canada (There is also a rumor
that he was seen about this time in Mt. Sinai, carrying all
ten).”
Alan C. Newell, Solitons in Mathematics and Physics
The conservation laws pointed to a remarkable connection
between two very different problems:
(1) the initial value problem for the KdV equation
u
 3u
u


6
u
0
3
t
x
x
u (x , 0)  u 0 (x )
(2) Schrödinger’s Equation
 (x )  u 0 (x ) (x )  k 2 (x )
for the “wave function” (x ) of a particle moving along a
Straight line under the influence of a potential u 0 (x )
A quantum mechanical particle in a one-dimensional
potential well u 0 (x ) has two possible states of motion:
(1) “Bound state” motion where the particle stays
localized near the well
(2) “Free motion” where the particle moves away
from the well
The wave function  (x ) is largest in amplitude
where the particle is most likely to be found
The rules of quantum mechanics imply that:
(1) Any potential well of finite depth can have
at most finitely many bound states
(2) The deeper the well, the more bound states
will occur
Using this connection, Gardner, Greene, Kruskal, and
Miura (GGKM) discovered a remarkable method for
solving the KdV equation using ideas from quantum
mechanics
GGKM’s solution method connects two completely
different problems:
(1) The motion of waves in a shallow channel, and
(2) The motion of a quantum-mechanical particle in
one dimension
by a precisely defined mathematical transformation.
Their discovery explained Russell’s third and fourth
observations about solitary waves from the KdV
equation.
Recall those observations:
(3) Solitary waves can cross each other “without change
of any kind”
(4) A sufficiently large mass of water creates two or more
independent solitary waves
The set of all possible quantum states is called the spectrum
of u 0 . There are at most finitely many bound states,
described by the bound state energies. The following
correspondence holds:
Quantum Problem
u 0 is the potential
Water Wave Problem


u 0 (x )dx is the

strength of the potential
Bound states of u 0

u 0 is the initial wave shape

u 0 (x )dx is the mass of the
wave
Solitons for KdV
Moreover, the spectrum ofu (x , t )
uof(x , 0)  u 0 (x )
is the same as the spectrum
We now exploit the following facts:
Fact 1: If u (x , t ) evolves according to the KdV
equation, the number of bound states (and hence, the
number of solitons) stays fixed
Fact 2: The shape of each soliton is determined by the
corresponding bound state energy
Fact 3: So long as



u 0 (x ) dx  0
there is at least one bound state in the quantum
problem.
The larger the integral is, the larger the number of
bound states.
We can now explain Russell’s third and fourth
observations using the connection with quantum
Theory:
(3) Solitary waves can cross each other “without change
of any kind”
The solitary waves correspond to quantum-mechanical
bound states, which are determined by the spectrum and
therefore do not change over time
(4) A sufficiently large mass of water creates two or more
independent solitary waves

The larger the mass of water  u 0 (x ) dx , the larger the
number of quantum-mechanical bound states. Each bound
state will correspond to a soliton.
Quiz Question
Can the waveform shown below generate solitons?
Hint: Remember that the Schrodinger equation has
bound states only when
 u ( x ) dx  0
0
Peter Lax formulated the connection between the two
problems in terms of a Lax Pair of operators, one that
determines the scattering data and the other that
determines how the scattering data evolve in time
Lax showed that the KdV equation can be described
by a Lax Pair of operatorsL (t ) and B (t ) :
2
L (t )  2  u (x , t )
x

3
 
u
B (t )  4 3  6u (x , t )
 C  3 (x , t ) 
x 
x
x

(spectral problem)
(time evolution)
together with a law of motion that is equivalent to the original
KdV equation:
L (t )  B (t ), L (t ) 
Lax’s law of motion is equivalent to the original KdV
Equation, but expresses it in a different form.
Key Fact: If a pair of operators obey Lax’s law of motion,
then the spectrum of
L (t ) is automatically preserved, and
soliton solutions are possible. Systems obeying Lax’s law
of motion are called completely integrable.
Lax’s framework enabled mathematicians to look for
other nonlinear wave equations that were completely
integrable. In 1972, Zakharov and Shabat showed
that the nonlinear Schrödinger equation
2
u
 2u
i
(x , t )  2 (x , t )  2 u (x , t ) u (x , t )  0
t
x
could be solved by the inverse scattering method and
had soliton solutions.
2
u
 2u
i
(x , t )  2 (x , t )  2 u (x , t ) u (x , t )  0
t
x
The nonlinear Schrödinger equation, in a slightly different
guise, governs the conduction of pulses in optical fibers:
2
u
1  2u
i
(z ,  ) 
(z ,  )  u ( z ,  ) u (z ,  )  0
2
z
2 
where τ measures time from the pulse center and z measures
length along the fiber. This equation admits “bright soliton”
solutions.
The one-soliton solution takes the form


u (x , t )   2 sech 2  t  z  


2
The two-soliton solution takes the form
These “bright solitons” can be used to transmit data in
optical fibers at high speed, with one soliton equal to
one bit
New Frontiers
All of the examples discussed so far concern waves
in one dimension. What about two dimensions – e.g.
surface waves in shallow water?
The Kadomtsev-Petviashvili Equation

x
 u
u  3u 
2u
 6u
 33 2 0
 4
t
x x 
y

The solution u (x , y , t ) represents unidirectional
long waves propagating in shallow water. If
u (x , y , t ) does not depend on y then the KP
equation reduces to the KdV equation
u
u  3u
4
 6u
 3 0
t
x x
The KP equation admits line solitons
Line Soliton for the KP Equation
Morning Glory Cloud
Morning Glory clouds occur on a regular basis in the southern
part of North Australia’s Gulf of Carpentaria. They can be up to
1000 km long and up to 1 km high.
KP Line Solitons
Video courtesy of Professor Yuji Kodama,
Ohio State University
KP Line Solitons – Y Type
Movie courtesy of Professor Yuji Kodama,
Ohio State University
KP Line Solitons in the
Gulf of Mexico
Movie courtesy of Mark Ablowitz
Via
Yuji Kodama
The KP equation can be solved, in principal at least,
by the method of inverse scattering. The “potential” for
three line solitons closely resembles the potential for
three quantum-mechanical particles moving in one
dimension.
A fundamental tool in the scattering theory of these
systems is the Weinberg-Van Winter Equation, co-discovered
by Stephen Weinberg and Clasine Van Winter of the
University of Kentucky. It allows one to analyze the 3-particle
system in terms of 2-particle subsystems. This is roughly
equivalent to analyzing multiple line-solitons in terms of
single line solitons.
A thorough analysis of KP line solitons will require
mathematical tools from the following disciplines
within `pure’ mathematics:
• Combinatorics and graph theory
• Harmonic analysis
• Operator theory
• Topology
as well as physical insight from quantum theory
In memory of Clasine Van Winter (1929-2000)
Special Thanks To:
• My parents, Edmund Franklin Perry and Lena Bowers Perry, who
were always both necessary and sufficient
• My family for their love, support, and understanding
• The College of Arts and Sciences for support during sabbatical years
2004-2005 and 2010-2011
• The National Science Foundation for continuing support
• Professor Paul Umbanhower, Northwestern University, for permission
to use images from his oscillon experiments to publicize this lecture
• Amy Hisel, Jennifer Allen and their colleagues for help with
publicity, design, and arrangements
• Professor Yuji Kodama, Ohio State University, for KP line soliton videos
• Professor David Royster for assistance with MathType, Mathematica,
and Power Point