II Nonlinear wave equations

• Introduction• Solitary waves• Korteweg-deVries (KdV) equation• Nonlinear Schrodinger equation

2.1 Introduction

• Simplest (second order) linear wave equation

utt – c2uxx = 0

• D’Alembert’s solution

u(x,t) = f(x-ct) + g(x+ct)f, g arbitrary functionsDispersionlessDissipationlessDispersion relation = ck

IntroductionLinear wave equations

IntroductionLinear wave equations

• Simplest Linear

ut – cux = 0 or ut + cux = 0

u(x,t) = f(x+ct) or u(x,t) = f(x-ct)• Simplest Dispersive, Dissipationless

ut + cux + auxxx = 0

u(x,t) = exp[i(kx – t)]

= ck - ak3 • Simplest Nondispersive, Dissipative

ut + cux - auxx = 0

u(x,t) = exp[i(kx – t)]

= ck – iak2

IntroductionNonlinear wave equations

• Simplest Nonlinearut + (1+u)ux = 0u(x,t) = f(x-(1+u)t)Sharpens at leading and trailing edges (shock formation)

• Korteweg deVries (KdV) Equation (1895)

ut + (1+u)ux + uxxx = 0

Solitary wave/soliton behaviour

Dispersion and tendency to shock formation in balance

o

2 x-ct -x ( 2

c sech

2

c t)u(x,

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 (1808-1882) made a remarkable scientific discovery.Here is an extract fromJohn Scott Russell’s ‘Report on waves’ 

Solitary wavesRussell’s 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”.

• 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.

0 u 6uu -u xxxxt

2.3 Korteweg deVries (KdV) equation

'wave.dat'

0 20 40 60 80 100 120 14000.511.522.533.54

-0.20

0.20.40.60.81

1.21.41.61.8

Korteweg deVries (KdV) equation Numerical solution (strong dispersive term)

'wave.dat'

0 20 40 60 80 100 120 14000.511.522.533.54

-0.200.20.40.60.811.2

1.41.61.82

Korteweg deVries (KdV) equation Numerical solution (weak dispersive term)

Korteweg deVries (KdV) equation Effect of nonlinear term ut = -(1+u)ux

The sequence of plots at t = 0, t and 2t illustrate how a pulse forms and splits off from theleading edge of a smooth front.

2 4 6 8 10

-1

-0.5

0.5

1

1.5

ux(0)

-(1+u(0))ux(0)u(0)

2 4 6 8 10

-1

-0.5

0.5

1

1.5

ux(t)

-(1+u(t))ux(t)u(t)

2 4 6 8 10

-1

-0.5

0.5

1

1.5

u(2t)

Korteweg deVries (KdV) equation Effect of dispersive term ut = - uxxx

2 4 6 8 10

-1

-0.5

0.5

1

1.5

u(0)-uxxx(0)

2 4 6 8 10

-1

-0.5

0.5

1

1.5

u(0) u(t)

2 4 6 8 10

-1

-0.5

0.5

1

1.5

u(2t)

Combined effects of nonlinear and dispersive terms

Korteweg deVries (KdV) equation Soliton simulations

These simulations come from Klaus Brauer's webpage (Osnabrück)

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).

Korteweg deVries (KdV) equation Analytic solution

•KdV equation

•Let the solution be u = u(x,t) and consider a change of variables = x – ct and = t

•Call the function in new variables f()

•The change in u or f brought about by translations (dx, dt) or () is

dt ddt c -dx d d f d f df

dt tu dx

xu du

0u6uuu xxxxt

• 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

Korteweg deVries (KdV) equation Analytic solution

dt fc- f dx f df

dt f dt c -dx f df

fc- f tu

f xu

Korteweg deVries (KdV) equation Analytic solution

• When transforming the pde from (x, t) to () we must make the replacements

• In the (x, t) variables a soliton moves along the x axis as time advances

•In the () variables a soliton is stationary in time provided we choose c in the transformation to be the soliton velocity

c- t

x

•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     

Korteweg deVries (KdV) equation Analytic solution

0 f 6ff -cf-

0 u 6uu -u xxxxt

ct - x )f( t)u(x,

Korteweg deVries (KdV) equation Analytic solution

B Af f2

c f f

2

1

B Af f2

1 f -f

2

c-

Afd f 3f -cf-fd

A f 3f -cf-

0 f 6ff -cf-d

232

232

2

2

•Integrate twice wrt

Korteweg deVries (KdV) equation Analytic solution

• A and B are constants of integration. In order to have a localised traveling wave packet, we impose boundary conditions: all tend to zero as || goes to infinity.

• To ensure these conditions we set A = B = 0. Solutions also exist at zeros of the polynomial in f.

• The solution with A = B = 0 obeys

• Rearrange to

f,ff,

c 2ff f 22

d c 2ff

df

Korteweg deVries (KdV) equation Analytic solution

dc

2-

tanhc sech 2

c-

tanh sech cd

becomes c 2ff

df

22

2

• Make change of variable

2sech 2

c- f

c

x

c

2 o

• Last term on rhs is constant of integration

Korteweg deVries (KdV) equation Analytic solution

• Rearrange to

• Make back substitution

o

2 x-ct -x ( 2

c sech

2

c f

tanh sech - sech d

d

ee

2 sech

o x- 2

c

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

• The NLS can be derived for wave packets localised in k space for systems where the dispersion relation depends on wave intensity ) (k,

2

0 ψ V xxψ2m

tψi

0 ψψ Q xxψ P tiψ2

2

Nonlinear Schrödinger equationDerivation from dispersion relation

• Consider the superposition of 2 waves of similar wavenumber and frequency

t] -cos[kx t] -k x [ cos 2 ψ ψ

t])( -k)x cos[(k ψ

t])( -k)x cos[(k ψ

21

2

1

• The result is a slow envelope wave with group velocity vg = / k and a rapid carrier wave with velocity /k

• Simulation with /k1and/k20

Nonlinear Schrödinger equationDerivation from dispersion relation

...

...)k(kk

ω

2!

1 )k(k

k

ω ω- ω

2

2

2o

o

2

2

o

o

o

2o

• The NLS is derived from the dispersion relation for the envelope function which has a slow time variation cf the carrier waves

• Suppose that the dispersion relationship isMake a Taylor expansion of this about ko and zero intensity

) (k, 2

Nonlinear Schrödinger equationDerivation from dispersion relation

2

oψ,

o

2ψ2

o,

o

2

2

2

o,

og

oo

ω

ω Q

ωk

ω 2P

ωk

ω

v

k– k K ω - ω Ω

• Let

• Then the Taylor expanded dispersion relation becomes

22g Q PK K v

Nonlinear Schrödinger equationDerivation from dispersion relation

• Consider a wavepacket constructed from a small group of waves in slow variables X = x, T = t <<1

vecarrier wa*ddK -

)]T

- KX

exp[i( )(K, T)(X,

t)]-xexp[i(k* ddK -

)]T

- KX

exp[i( )(K,

ddk -

t)] -exp[i(kx )(k, t)(x,

oo

• The latter is the envelope function in ‘slow’ variables X,T

Nonlinear Schrödinger equationDerivation from dispersion relation

T i

i

T

X iK

iK

X

T)(X,iK

ddK -

)]T

- KX

exp[i( )(K,iK

ddK -

)]T

- KX

exp[i( )(K,X

T)(X,X

Nonlinear Schrödinger equationDerivation from dispersion relation

22g Q PK K v

• The dispersion relation

becomes

22XX

2XgT

222

g

QPvi- i

QX

i-PX

i- vT

i

g

g

v T

X

T Tv-X• Make further change of variables

Nonlinear Schrödinger equationDerivation from dispersion relation

22XX

2XgT QPvi- i

becomes

0qi

0QPi

2

2

• This is the conventional form for the NLS equation. It has an envelope solution with a sech profile. (See handout)

31-nn

3n1n1nn1nn )u(u)u(uAK)uu2K(uum

Nonlinear Schrödinger equationApplication to lattice dynamics

• Hooke’s Law plus additional nonlinear term

3

42

AKr -Krdr

dU(r)- F(r)

r4

AK Kr

2

1 U(r)

• Equation of motion

• Solution and dispersion relation

2

kasinR6A1

2

kasin

m

K2ω

c.c. t)]-exp[i(kna Ru

22

n

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 |R|2. At the website below, use the monatomic chain applet to see some of these localised modes.

•Intrinsic localised modes in lattice dynamics of crystals

0.5 1 1.5 2 2.5 3

k

0.2

0.4

0.6

0.8

1

2

kasin

m

K2ω

2

kasinR6A1

2

kasin

m

K2ω 22

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 = ) (group velocity zero)

• moving ILM composed of groups of waves centred away from the Brillouin zone boundary (group velocity nonzero)

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

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