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## II Nonlinear wave equations

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Title: II Nonlinear wave equations

1
II Nonlinear wave equations
2.1 Introduction
• Introduction
• Solitary waves
• Korteweg-deVries (KdV) equation
• Nonlinear Schrodinger equation

2
IntroductionLinear wave equations
• Simplest (second order) linear wave equation
• utt c2uxx 0
• DAlemberts solution
• u(x,t) f(x-ct) g(xct)
• f, g arbitrary functions
• Dispersionless
• Dissipationless
• Dispersion relation w ck

3
IntroductionLinear wave equations
• Simplest Linear
• ut cux 0 or ut cux 0
• u(x,t) f(xct) or u(x,t) f(x-ct)
• Simplest Dispersive, Dissipationless
• ut cux auxxx 0
• u(x,t) expi(kx wt)
• w ck - ak3
• Simplest Nondispersive, Dissipative
• ut cux - auxx 0
• u(x,t) expi(kx wt)
• w ck iak2

4
IntroductionNonlinear wave equations
• Simplest Nonlinear
• ut (1u)ux 0
• u(x,t) f(x-(1u)t)
• Sharpens at leading and trailing edges (shock
formation)
• Korteweg deVries (KdV) Equation (1895)
• ut (1u)ux uxxx 0
• Solitary wave/soliton behaviour
• Dispersion and tendency to shock formation in
balance

5
2.2 Solitary waves
Over one hundred and fifty years ago, while
conducting experiments to determine the most
efficient design for canal boats, a young
Scottish engineer named John Scott Russell
discovery. Here is an extract from John Scott
Russells Report on waves
6
Solitary wavesRussells report on waves
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. 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.
7
2.3 Korteweg deVries (KdV) equation
• The wave of translation (or solitary wave)
observed by John Scott Russell is described by a
nonlinear wave equation known as the
Korteweg-deVries (KdV) equation.
• We review various possible types of nonlinearity
in wave equations before studying two specific
equations the KdV and the nonlinear Schrodinger
(NLS) equations.

8
Korteweg deVries (KdV) equation Numerical
solution (strong dispersive term)
9
Korteweg deVries (KdV) equation Numerical
solution (weak dispersive term)
10
Korteweg deVries (KdV) equation Effect of
nonlinear term ut -(1u)ux
The sequence of plots at t 0, Dt and 2Dt
illustrate how a pulse forms and splits off from
the leading edge of a smooth front.
11
Korteweg deVries (KdV) equation Effect of
dispersive term ut - uxxx
Combined effects of nonlinear and dispersive terms
12
Korteweg deVries (KdV) equation Soliton
simulations
These simulations come from Klaus Brauer's
webpage (Osnabrück)
13
Korteweg deVries (KdV) equationSolution for PBC
and sinusoidal initial conditions
This animation by K. Takasaki shows the
sinusoidal initial state breaking up into a
soliton train. Zabusky and Kruskal (1966).
14
Korteweg deVries (KdV) equation Analytic solution
• KdV equation
• Let the solution be u u(x,t) and consider a
change of variables x x ct and t t
• Call the function in new variables f(x,t)
• The change in u or f brought about by
translations (dx, dt) or (x, t) is

15
Korteweg deVries (KdV) equation Analytic solution
• If we convert the change in f brought about by
translations through (dx, dt) into changes in f
brought about by translations through (dx, dt)
• Since u and f represent the same function the
same translation (dx, dt) must produce the same
change in either. Hence

16
Korteweg deVries (KdV) equation Analytic solution
• When transforming the pde from (x, t) to (x, t)
we must make the replacements
• In the (x, t) variables a soliton moves along
the x axis as time advances
• In the (x, t) variables a soliton is stationary
in time provided we choose c in the
transformation to be the soliton velocity

17
Korteweg deVries (KdV) equation Analytic solution
• The conventional form for the KdV equation is
•
•
• Travelling wave solutions have the form
•
• c is the wave velocity
•
• Substituting for u in the KdV equation and
setting the time derivative to zero we obtain
•
•

18
Korteweg deVries (KdV) equation Analytic solution
• Integrate twice wrt x

19
Korteweg deVries (KdV) equation Analytic solution
20
Korteweg deVries (KdV) equation Analytic solution
• Make change of variable
• Last term on rhs is constant of integration

21
Korteweg deVries (KdV) equation Analytic solution
• Rearrange to
• Make back substitution

22
2.4 Nonlinear Schrödinger equation
• The naming of the nonlinear Schrödinger (NLS)
equation becomes obvious when it is compared to
the time-dependent Schrödinger equation from
quantum mechanics

23
Nonlinear Schrödinger equationDerivation from
dispersion relation
• Consider the superposition of 2 waves of similar
wavenumber and frequency
• The result is a slow envelope wave with group
velocity
• vg ?w/ ? k and a rapid carrier wave with
velocity w/k
• Simulation with Dw/Dk 1 and w/k 20

24
Nonlinear Schrödinger equationDerivation from
dispersion relation
25
Nonlinear Schrödinger equationDerivation from
dispersion relation
• Let
• Then the Taylor expanded dispersion relation
becomes

26
Nonlinear Schrödinger equationDerivation from
dispersion relation
• Consider a wavepacket constructed from a small
group of waves in slow variables X ex, T et
e ltlt1
• The latter is the envelope function in slow
variables X,T

27
Nonlinear Schrödinger equationDerivation from
dispersion relation
28
Nonlinear Schrödinger equationDerivation from
dispersion relation
• The dispersion relation

becomes
• Make further change of variables

29
Nonlinear Schrödinger equationDerivation from
dispersion relation
becomes
• This is the conventional form for the NLS
equation. It has an envelope solution with a
sech profile. (See handout)

30
Nonlinear Schrödinger equationApplication to
lattice dynamics
• Hookes Law plus additional nonlinear term
• Equation of motion
• Solution and dispersion relation

31
Nonlinear Schrödinger equationApplication to
lattice dynamics
• We have just seen that introduction of a
nonlinear term in the force law for a 1-D chain
of atoms leads to a dispersion relation which
depends on R2. At the website below, use the
monatomic chain applet to see some of these
localised modes.
• Intrinsic localised modes in lattice dynamics of
crystals

32
Nonlinear Schrödinger equationApplication to
lattice dynamics
• Click on monatomic 1-D chains and then on the
link in the title to the page (works best with
Internet Explorer)
• You will find stationary ILM with
• envelope function (c.f. solutions of NLS
equation) is composed of groups of waves centred
on the Brillouin zone boundary (k p) (group
velocity zero)
• moving ILM composed of groups of waves centred
away from the Brillouin zone boundary (group
velocity nonzero)

33
Nonlinear Schrödinger equationApplication to
lattice dynamics
• You will also find
• molecular dynamics simulations showing ILM in
3-D crystals (click on 3-D Ionic crystals)
• Simulations showing ILM in 1-D chains of
interacting spins

34
Nonlinear Schrödinger equationApplication to
optical communications
• Read the introductory articles on
• Solitons in optical communications by Ablowitz
et al.
• Historical aspects of optical solitons by
Hasegawa
• Soliton propagation in optical fibres