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II Nonlinear wave equations

2.1 Introduction

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

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

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

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

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 from John Scott

Russells Report on waves

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.

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.

Korteweg deVries (KdV) equation Numerical

solution (strong dispersive term)

Korteweg deVries (KdV) equation Numerical

solution (weak dispersive term)

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.

Korteweg deVries (KdV) equation Effect of

dispersive term ut - uxxx

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 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

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

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

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

Korteweg deVries (KdV) equation Analytic solution

- Integrate twice wrt x

Korteweg deVries (KdV) equation Analytic solution

Korteweg deVries (KdV) equation Analytic solution

- Make change of variable

- Last term on rhs is constant of integration

Korteweg deVries (KdV) equation Analytic solution

- Rearrange to
- Make back substitution

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

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

Nonlinear Schrödinger equationDerivation from

dispersion relation

Nonlinear Schrödinger equationDerivation from

dispersion relation

- Let

- Then the Taylor expanded dispersion relation

becomes

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

Nonlinear Schrödinger equationDerivation from

dispersion relation

Nonlinear Schrödinger equationDerivation from

dispersion relation

- The dispersion relation

becomes

- Make further change of variables

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)

Nonlinear Schrödinger equationApplication to

lattice dynamics

- Hookes Law plus additional nonlinear term

- Equation of motion

- Solution and dispersion relation

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

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)

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