N. Theodorakopoulos- Nonlinear physics (solitons, chaos, discrete breathers)

8/3/2019 N. Theodorakopoulos- Nonlinear physics (solitons, chaos, discrete breathers)

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Nonlinear physics(solitons, chaos, discrete breathers)

N. Theodorakopoulos

Konstanz, June 2006


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Contents

Foreword vi

1 Background: Hamiltonian mechanics 11.1 Lagrangian formulation of dynamics . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Hamiltonian dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Canonical momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.2 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.3 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.4 Canonical transformations . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.5 Point transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Hamilton-Jacobi theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Hamilton-Jacobi equation . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.2 Relationship to action . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.3 Conservative systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.4 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.5 Periodic motion. Action-angle variables . . . . . . . . . . . . . . . . . 5

1.3.6 Complete integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Homogeneity of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.2 Homogeneity of space . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.3 Galilei invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4.4 Isotropy of space (rotational symmetry of Lagrangian) . . . . . . . . . 7

1.5 Continuum field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5.1 Lagrangian field theories in 1+1 dimensions . . . . . . . . . . . . . . . 8

1.5.2 Symmetries and conservation laws . . . . . . . . . . . . . . . . . . . . 8

1.6 Perturbations of integrable systems . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Background: Statistical mechanics 11

2.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Liouvilles theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Averaging over time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.4 Ensemble averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.5 Equivalence of ensembles . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.6 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 The FPU paradox 15

3.1 The harmonic crystal: dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 The harmonic crystal: thermodynamics . . . . . . . . . . . . . . . . . . . . . 16

3.3 The FPU numerical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 17

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4 The Korteweg - de Vries equation 204.1 Shallow water waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1.1 Background: hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 204.1.2 Statement of the problem; boundary conditions . . . . . . . . . . . . . 214.1.3 Satisfying the bottom boundary condition . . . . . . . . . . . . . . . . 214.1.4 Euler equation at top boundary . . . . . . . . . . . . . . . . . . . . . . 224.1.5 A solitary wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1.6 Is the solitary wave a physical solution? . . . . . . . . . . . . . . . . . 24

4.2 KdV as a limiting case of anharmonic lattice dynamics . . . . . . . . . . . . . 244.3 KdV as a field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3.1 KdV Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3.2 Symmetries and conserved quantities . . . . . . . . . . . . . . . . . . . 264.3.3 KdV as a Hamiltonian field theory . . . . . . . . . . . . . . . . . . . . 27

5 Solving KdV by inverse scattering 28

5.1 Isospectral property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Lax pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Inverse scattering transform: the idea . . . . . . . . . . . . . . . . . . . . . . 295.4 The inverse scattering transform . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.4.1 The direct problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4.2 Time evolution of scattering data . . . . . . . . . . . . . . . . . . . . . 315.4.3 Reconstructing the potential from scattering data (inverse scattering

problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4.4 IST summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.5 Application of the IST: reflectionless potentials . . . . . . . . . . . . . . . . . 355.5.1 A single bound state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.5.2 Multiple bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.6 Integrals of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6.1 Lemma: a useful representation of a(k) . . . . . . . . . . . . . . . . . 395.6.2 Asymptotic expansions ofa(k) . . . . . . . . . . . . . . . . . . . . . . 395.6.3 IST as a canonical transformation to action-angle variables . . . . . . 41

6 Solitons in anharmonic lattice dynamics: the Toda lattice 426.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2 The dual lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.2.1 A pulse soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Complete integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7 Chaos in low dimensional systems 48

7.1 Visualization of simple dynamical systems . . . . . . . . . . . . . . . . . . . . 487.1.1 Two dimensional phase space . . . . . . . . . . . . . . . . . . . . . . . 487.1.2 4-dimensional phase space . . . . . . . . . . . . . . . . . . . . . . . . . 507.1.3 3-dimensional phase space; nonautonomous systems with one degree

of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.2 Small denominators revisited: KAM theorem . . . . . . . . . . . . . . . . . . 527.3 Chaos in area preserving maps . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.3.1 Twist maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.3.2 Local stability properties . . . . . . . . . . . . . . . . . . . . . . . . . 547.3.3 Poincare-Birkhoff theorem . . . . . . . . . . . . . . . . . . . . . . . . . 557.3.4 Chaos diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.3.5 The standard map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.3.6 The Arnold cat map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.3.7 The baker map; Bernoulli shifts . . . . . . . . . . . . . . . . . . . . . . 64

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7.3.8 The circle map. Frequency locking . . . . . . . . . . . . . . . . . . . . 66

7.4 Topology of chaos: stable and unstable manifolds, homoclinic points . . . . . 67

8 Solitons in scalar field theories 69

8.1 Definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.1.1 Lagrangian, continuum field equations . . . . . . . . . . . . . . . . . . 69

8.2 Static localized solutions (general KG class) . . . . . . . . . . . . . . . . . . . 71

8.2.1 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8.2.2 Specific potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.2.3 Intrinsic Properties of kinks . . . . . . . . . . . . . . . . . . . . . . . . 73

8.2.4 Linear stability of kinks . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.3 Special properties of the SG field . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.3.1 The Sine-Gordon breather . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.3.2 Complete Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9 Atoms on substrates: the Frenkel-Kontorova model 77

9.1 The Commensurate-Incommensurate transition . . . . . . . . . . . . . . . . . 78

9.1.1 The continuum approximation . . . . . . . . . . . . . . . . . . . . . . 78

9.1.2 The special case = 0: kinks and antikinks . . . . . . . . . . . . . . . 79

9.1.3 The general case > 0: the soliton lattice . . . . . . . . . . . . . . . . 79

9.2 Breaking of analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9.2.1 FK ground state as minimizing periodic orbit of the standard map . . 84

9.2.2 Small amplitude motion . . . . . . . . . . . . . . . . . . . . . . . . . . 85

9.2.3 Free end boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 85

9.3 Metastable states: spatial chaos as a model of glassy structure . . . . . . . . 86

10 Solitons in magnetic chains 8810.1 I ntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10.2 Classical spin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10.2.1 Spin Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

10.2.2 An alternative representation . . . . . . . . . . . . . . . . . . . . . . . 89

10.3 Solitons in ferromagnetic chains . . . . . . . . . . . . . . . . . . . . . . . . . . 90

10.3.1 The continuum approximation . . . . . . . . . . . . . . . . . . . . . . 90

10.3.2 The classical, isotropic, ferromagnetic chain . . . . . . . . . . . . . . . 91

10.3.3 The easy-plane ferromagnetic chain in an external field . . . . . . . . 96

10.4 Solitons in antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10.4.1 Continuum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

10.4.2 The isotropic antiferromagnetic chain . . . . . . . . . . . . . . . . . . 101

10.4.3 Easy axis anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10210.4.4 Easy plane anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

10.4.5 Easy plane anisotropy and symmetry-breaking field . . . . . . . . . . . 106

11 Solitons in conducting polymers 110

11.1 Peierls instabil ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

11.1.1 Electrons decoupled from the lattice . . . . . . . . . . . . . . . . . . . 110

11.1.2 Electron-phonon coupling; dimerization . . . . . . . . . . . . . . . . . 111

11.2 Solitons and polarons in (CH)x . . . . . . . . . . . . . . . . . . . . . . . . . . 114

11.2.1 A continuum approximation . . . . . . . . . . . . . . . . . . . . . . . . 114

11.2.2 Dimerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 16

11.2.3 The soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

11.2.4 The polaron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 19

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12 Solitons in nonlinear optics 122

12.1 Background: Interaction of light with matter, Maxwell-Bloch equations . . . 122

12.1.1 Semiclassical theoretical framework and notation . . . . . . . . . . . . 1221 2 . 1 . 2 D y n a m i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3

12.2 Propagation at resonance. Self-induced transparency . . . . . . . . . . . . . . 123

12.2.1 Slow modulation of the optical wave . . . . . . . . . . . . . . . . . . . 123

12.2.2 Further simplifications: Self-induced transparency . . . . . . . . . . . 125

12.3 Self-focusing off-resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

12.3.1 Off-resonance limit of the MB equations . . . . . . . . . . . . . . . . . 126

12.3.2 Nonlinear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

12.3.3 Space-time dependence of the modulation: the nonlinear Schrodingere q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 8

12.3.4 Soliton solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

13 Solitons in Bose-Einstein Condensates 132

13.1 The Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

13.2 Propagating solutions. Dark solitons . . . . . . . . . . . . . . . . . . . . . . . 132

14 Unbinding the double helix 134

14.1 A nonlinear lattice dynamics approach . . . . . . . . . . . . . . . . . . . . . . 134

14.1.1 Mesoscopic modeling of DNA . . . . . . . . . . . . . . . . . . . . . . . 134

14.1.2 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

14.2 Nonlinear structures (domain walls) and DNA melting . . . . . . . . . . . . . 139

14.2.1 Local equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

14.2.2 Thermodynamics of domain walls . . . . . . . . . . . . . . . . . . . . . 142

15 Pulse propagation in nerve cells: the Hodgkin-Huxley model 144

1 5 . 1 B a c k g r o u n d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 4

15.2 The Hodgkin-Huxley model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

15.2.1 The axon membrane as an array of electrical circuit elements . . . . . 145

15.2.2 Ion transport via distinct ionic channels . . . . . . . . . . . . . . . . . 146

15.2.3 Voltage clamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

15.2.4 Ionic channels controlled by gates . . . . . . . . . . . . . . . . . . . . . 146

15.2.5 Membrane activation is a threshold phenomenon . . . . . . . . . . . . 148

15.2.6 A qualitative picture of ion transport during nerve activation . . . . . 148

15.2.7 Pulse propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

16 Localization and transport of energy in proteins: The Davydov soliton 151

16.1 Background. Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 151

16.1.1 Energy storage in C=O stretching modes. Excitonic Hamiltonian . . . 151

16.1.2 Coupling to lattice vibrations. Analogy to polaron . . . . . . . . . . . 151

16.2 Born-Oppenheimer dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

16.2.1 Quantum (excitonic) dynamics . . . . . . . . . . . . . . . . . . . . . . 152

16.2.2 Lattice motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 53

16.2.3 Coupled exciton-phonon dynamics . . . . . . . . . . . . . . . . . . . . 153

16.3 The Davydov soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

16.3.1 The heavy ion limit. Static Solitons . . . . . . . . . . . . . . . . . . . 153

16.3.2 Moving solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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17 Nonlinear localization in translationally invariant systems: discrete breathers 15717.1 The Sievers-Takeno conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 157

17.2 Numerical evidence of localization . . . . . . . . . . . . . . . . . . . . . . . . 15917.2.1 Diagnostics of energy localization . . . . . . . . . . . . . . . . . . . . . 16017.2.2 Internal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

17.3 Towards exact discrete breathers . . . . . . . . . . . . . . . . . . . . . . . . . 161

A Impurities, disorder and localization 164A.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 64

A.1.1 Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164A.1.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A.2 A single impurity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A.2.1 An exact result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A.2.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

A.3 Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

A.3.1 Electrons in disordered one-dimensional media . . . . . . . . . . . . . 169A.3.2 Vibrational spectra of one-dimensional disordered lattices . . . . . . . 169

Bibliography 173

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Foreword

The fact that most fundamental laws of physics, notably those of electrodynamics and quan-tum mechanics, have been formulated in mathematical language as linear partial differentialequations has resulted historically in a preferred mode of thought within the physics com-munity - a linear theoretical bias. The Fourier decomposition - an admittedly powerfulprocedure of describing an arbitrary function in terms of sines and cosines, but nonethelessa mathematical tool - has been firmly embedded in the conceptual framework of theoreticalphysics. Photons, phonons, magnons are prime examples of how successive generations of

physicists have learned to describe properties of light, lattice vibrations, or the dynamics ofmagnetic crystals, respectively, during the last 100 years.

This conceptual bias notwithstanding, engineers or physicists facing specific problems inclassical mechanics, hydrodynamics or quantum mechanics were never shy of making par-ticular approximations which led to nonlinear ordinary, or partial differential equations.Therefore, by the 1960s, significant expertise had been accumulated in the field of nonlin-ear differential and/or integral equations; in addition, major breakthroughs had occurred onsome fundamental issues related to chaos in classical mechanics (Poincare, Birkhoff, KAMtheorems). Due to the underlying linear bias however, this substantial progress took unusu-ally long to find its way to the core of physical theory. This changed rapidly with the adventof electronic computation and the new possibilities of numerical visualization which accom-panied it. Computer simulations became instrumental in catalyzing the birth of nonlinear

science.This set of lectures does not even attempt to cover all areas where nonlinearity has proved

to be of importance in modern physics. I will however try to describe some of the basicconcepts mainly from the angle of condensed matter / statistical mechanics, an area whichprovided an impressive list of nonlinearly governed phenomena over the last fifty years -starting with the Fermi-Pasta-Ulam numerical experiment and its subsequent interpretationby Zabusky and Kruskal in terms of solitons (paradox turned discovery, in the words ofJ. Ford).

There is widespread agreement that both solitons and chaos have achieved the status oftheoretical paradigm. The third concept introduced here, localization in the absence of dis-order, is a relatively recent breakthrough related to the discovery ofindependent (nonlinear)localized modes (ILMs), a.k.a. discrete breathers.

Since neither the development of the field nor its present state can be described in termsof a unique linear narrative, both the exact choice of topics and the digressions necessary todescribe the wider context are to a large extent arbitrary. The latter are however necessaryin order to provide a self-contained presentation which will be useful for the non-expert, i.e.typically the advanced undergraduate student with an elementary knowledge of quantummechanics and statistical physics.

Konstanz, June 2006

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1 Background: Hamiltonian mechanics

Consider a mechanical system with s degrees of freedom.

The state of the mechanical system at any instant of time is described by the coordinates{Qi(t), i = 1, 2, , s} and the corresponding velocities {Qi(t)}.

In many applications that I will deal with, this may be a set of N point particles whichare free to move in one spatial dimension. In that particular case s = N and the coordinatesare the particle displacements.

The rules for temporal evolution, i.e. for the determination of particle trajectories, aredescribed in terms of Newtons law - or, in the more general Lagrangian and Hamiltonianformulations. The more general formulations are necessary in order to develop and/or makecontact with fundamental notions of statistical and/or quantum mechanics.

1.1 Lagrangian formulation of dynamics

The Lagrangian is given as the difference between kinetic and potential energies. For aparticle system interacting by velocity-independent forces

L({Qi, Qi}) = T V (1.1)

T =

1

2

s

i=1 mi

Q

2

i

V = V({Qi}, t) .where an explicit dependence of the potential energy on time has been allowed. Lagrangian

dynamics derives particle trajectories by determining the conditions for which the actionintegral

S(t, t0) =

tt0

d L({Qi, Qi, }) (1.2)has an extremum. The result is

d

dt

L

Qi=

L

Qi(1.3)

which for Lagrangians of the type (1.2) becomes

miQi = VQi

(1.4)

i.e. Newtons law.

1.2 Hamiltonian dynamics

1.2.1 Canonical momenta

Hamiltonian mechanics, uses instead of velocities, the canonical momenta conjugate to thecoordinates {Qi}, defined as

Pi =L

Qi. (1.5)

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In the case of (1.2) it is straightforward to express the Hamiltonian function (the totalenergy) H = T + V in terms of Ps and Qs. The result is

H({Pi, Qi}) =s

I=1

P2i2mi

+ V({Qi}) . (1.6)

1.2.2 Poisson brackets

Hamiltonian dynamics is described in terms of Poisson brackets

{A, B} =si=1

A

Qi

B

Pi A

Pi

B

Qi

(1.7)

where A, B are any functions of the coordinates and momenta. The momenta are canonicallyconjugate to the coordinates because they satisfy the relationships

1.2.3 Equations of motion

According to Hamiltonian dynamics, the time evolution of any function A({Pi, Qi}, t) isdetermined by the linear differential equations

A dAdt

= {A, H} + At

. (1.8)

where the second term denotes any explicit dependence of A on the time t. Application of

(1.8) to the cases A = Pi and A = Qi respectively leads to

Pi = {Pi, H}Qi = {Qi, H} (1.9)

which can be shown to be equivalent to (1.4). The time evolution of the Hamiltonian itselfis governed by

dH

dt=

H

t

=

V

t

. (1.10)

1.2.4 Canonical transformations

Hamiltonian formalism important because the symplecticstructure of equations of motion(from Greek = crosslink - of momenta & coordinate variables -) remains invari-ant under a class of transformations obtained by a suitable generating function (canoni-caltransformations). Example, transformation from old coordinates & momenta {P, Q} tonew ones {p,q}, via a generating function F1(Q,q,t) which depends on old and new coor-dinates (but not on old and new momenta - NB there are three more forms of generatingfunctions - ):

Pi =F1(q,Q,t)

Qi

pi = F1(q,Q,t)qi

K = H + F1t

(1.11)

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new coordinates are obtained by solving the first of the above eqs., and new momenta byintroducing the solution in the second. It is straightforward to verify that the dynamics

remains form-invariant in the new coordinate system, i.e.

pi = {pi, K}qi = {qi, K} (1.12)

anddK(p,q,t)

dt=

K(p,q,t)

t. (1.13)

Note that if there is no explicit dependence of F1 on time, the new Hamiltonian K is equalto the old H.

1.2.5 Point transformations

A special case of canonical transformations are point transformations, generated by

F2(Q,p,t) =i

fi(Q, t)pi ; (1.14)

New coordinates depend only on old coordinates - not on old momenta; in general new mo-menta depend on both old coordinates and momenta. A special case of point transformationsare orthogonal transformations, generated by

F2(Q, p) =i,k

aikQkpi (1.15)

where a is an orthogonal matrix. It follows that

qi =k

aikQk

pi =k

aikPk . (1.16)

Note that, in the case of orthogonal transformations, coordinates transform among them-selves; so do the momenta. Normal mode expansion is an example of (1.16).

1.3 Hamilton-Jacobi theory

1.3.1 Hamilton-Jacobi equationHamiltonian dynamics consists of a system of 2N coupled first-order linear differential equa-tions. In general, a complete integration would involve 2N constants (e.g. the initial valuesof coordinates and momenta). Canonical transformations enable us to play the followinggame:1 Look for a transformation to a new set of canonical coordinates where the newHamiltonian is zero and hence all new coordinates and momenta are constants of the mo-tion.2 Let (p,q) be the set of original momenta and coordinates in eqs of previous section,

1Hamilton-Jacobi theory is not a recipe for integration of the coupled ODEs; nor does it in general lead toa more tractable mathematical problem. However, it provides fresh insight to the general problem, in-cluding important links to quantum mechanics and practical applications on how to deal with mechanicalperturbations of a known, solved system.

2Does this seem like too many constants? We will later explore what independent constants mean in

mechanics, but at this stage let us just note that the original mathematical problem of integrating the2N Hamiltonian equations does indeed involve 2N constants.

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(, ) the set of new constant momenta and coordinates generated by the generating func-tion F2(q,,t) which depends on the original coordinates and the new momenta. The choice

of K 0 in (1.11) means thatF2t

+ H(q1, qs; F2q1

, , F2qs

; t) = 0 . (1.17)

Suppose now that you can [miraculously] obtain a solution of the first-order -in generalnonlinear-PDE (1.17), F2 = S(q,,t). Note that the solution in general involves s constants{i, i = 1, , s}. The s + 1st constant involved in the problem is a trivial one, because ifS is a solution, so is S+ A, where A is an arbitrary constant.

It is now possible to use the defining equation of the generating function F2

i =S(q,,t)

i

(1.18)

to obtain the new [constant] coordinates {i, i = 1, , s}; finally, turning inside out(1.18)yields the trajectories

qj = qj(,,t) . (1.19)

In other words, a solution of the Hamilton-Jacobi equation (1.17) provides a solution of theoriginal dynamical problem.

1.3.2 Relationship to action

It can be easily shown that the solution of the Hamilton-Jacobi equation satisfiesdS

dt= L , (1.20)

or

S(q,,t) S(q,,t0) =tt0

d L(q, q, ) (1.21)

where the r.h.s involves the actual particle trajectories; this shows that the solution of theHamilton-Jacobi equation is indeed the extremum of the action function used in Lagrangianmechanics.

1.3.3 Conservative systems

If the Hamiltonian does not depend explicitly on time, it is possible to separate out the timevariable, i.e.

S(q,,t) = W(q, ) 0t (1.22)where now the time-independent function W(q) (Hamiltons characteristic function) satisfies

H

q1, qs; W

q1, , W

qs

= 0 , (1.23)

and involves s

1 independent constants, more precisely, the s constants 1,

s dependon 0.

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1.3.4 Separation of variables

The previous example separated out the time coordinate from the rest of the variablesof the HJ function. Suppose q1 andWq1

enter the Hamiltonian only in the combination

1

q1,

Wq1

. The Ansatz

W = W1(q1) + W

(q2, , qs) (1.24)in (1.23) yields

H

q2, qs; W

q2, , W

qs; 1

q1,

W1q1

= 0 ; (1.25)

since (1.25) must hold identically for all q, we have

1

q1, W1

q1

= 1

H

q2, qs; W

q2, , W

qs; 1

= 0 . (1.26)

The process can be applied recursively if the separation condition holds. Note that cycliccoordinates lead to a special case of separability; if q1 is cyclic, then 1 =

Wq1

= W1q1 , and

hence W1(q1) = 1q1. This is exactly how the time coordinate separates off in conservativesystems (1.23).

Complete separability occurs if we can write Hamiltons characteristic function - in someset of canonical variables - in the form

W(q, ) =i

Wi(qi, 1, , s) . (1.27)

1.3.5 Periodic motion. Action-angle variables

Consider a completely separable system in the sense of (1.27). The equation

pi =S

qi=

Wi(qi, 1, , s)qi

(1.28)

provides the phase space orbit in the subspace (qi, pi). Now suppose that the motion in all

subspaces {(qi, pi), i = 1, , s} is periodic - not necessarily with the same period. Note thatthis may mean either a strict periodicity of pi, qi as a function of time (such as occurs inthe bounded motion of a harmonic oscillator), or a motion of the freely rotating pendulumtype, where the angle coordinate is physically significant only mod 2. The action variablesare defined as

Ji =1

2

pidqi =

1

2

dqi

Wi(qi, 1, , s)qi

(1.29)

and therefore depend only on the integration constants, i.e. they are constants of the motion.If we can turn inside out(1.29), we can express W as a function of the Js instead of thes. Then we can use the function W as a generating function of a canonical transformationto a new set of variables with the Js as new momenta, and new anglecoordinates

i = WJi= Wi(qi, J1, , Js)Ji . (1.30)

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In the new set of canonical variables, Hamiltons equations of motion are

Ji = 0

i =H(J)

Ji i(J) . (1.31)

Note that the Hamiltonian cannot depend on the angle coordinates, since the action coordi-nates, the Js, are - by construction - all constants of the motion. In the set of action-anglecoordinates, the motion is as trivial as it can get:

Ji = const

i = i(J) t + const . (1.32)

1.3.6 Complete integrability

A system is called completely integrable in the sense of Liouville if it can be shown to haves independent conserved quantities in involution (this means that their Poisson brackets,taken in pairs, vanish identically). If this is the case, one can always perform a canonicaltransformation to action-angle variables.

1.4 Symmetries and conservation laws

A change of coordinates, if it reflects an underlying symmetry of physical laws, will leave theform of the equations of motion invariant. Because Lagrangian dynamics is derived from anaction principle, any such infinitesimal change which changes the particle coordinates

qi qi = qi + fi(q, t)qi qi = qi + fi(q, t) (1.33)

and adds a total time derivative to the Lagrangian, i.e.

L = L + dF

dt, (1.34)

will leave the equations of motion invariant. On the other hand, the transformed Lagrangianwill generally be equal to

L({qi, qi}) = L({qi, qi})

= L({qi, qi}) +s

i=1

Lqi fi + Lqi fi= L({qi, qi}) +

si=1

d

dt

L

qi

fi +

L

qifi

= L({qi, qi}) +si=1

d

dt

L

qifi

and therefore the quantitysi=1

L

qifi F (1.35)

will be conserved.

Such underlying symmetries of classical mechanics are:

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1.4.1 Homogeneity of time

L = L(t + ) = L(t) + dL/dt, i.e. F = L; furthermore, qi = qi(t + ) = qi + qi, i.e. fi = qi.As a result, the quantity

H =si=1

L

qiqi L (1.36)

(Hamiltonian) is conserved.

1.4.2 Homogeneity of space

The transformation qi qi + (hence fi = 1) leaves the Lagrangian invariant (F = 0). Theconserved quantity is

P =si=1

Lqi

(1.37)

(total momentum).

1.4.3 Galilei invariance

The transformation qi qi t (hence fi = t) does not generally change the potentialenergy (if it depends only on relative particle positions). It adds to the kinetic energy aterm P, i.e. F = miqi. The conserved quantity is

si=1

miqi P t (1.38)

(uniform motion of the center of mass).

1.4.4 Isotropy of space (rotational symmetry of Lagrangian)

Let the position of the ith particle in space be represented by the vector coordinate qi.Rotation around an axis parallel to the unit vector n is represented by the transformationqi qi + fi where fi = n qi. The change in kinetic energy is

i

qi fi = 0 .

If the potential energy is a function of the interparticle distances only, it too remains invariantunder a rotation. Since the Lagrangian is invariant, the conserved quantity (1.35) is

si=1

L

qi fi =

si=1

miqi (n qi) = n I ,

where

I =si=1

mi(qi qi) (1.39)

is the total angular momentum.

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1.5 Continuum field theories

1.5.1 Lagrangian field theories in 1+1 dimensions

Given a Lagrangian in 1+1 dimensions,

L =

dxL(, x, t) (1.40)

where the Lagrangian density L depends only on the field and first space and time deriva-tives, the equations of motion can be derived by minimizing the total action

S =

dtdxL (1.41)

and have the formddt

Lt

+ d

dx

Lx

L

= 0 . (1.42)

1.5.2 Symmetries and conservation laws

The form (1.42) remains invariant under a transformation which adds to the Lagrangiandensity a term of the form

J (1.43)

where the implied summation is over = 0, 1, because this adds only surface boundary termsto the action integral. If the transformation changes the field by , and the derivatives byx, t, the same argument as in discrete systems leads us to conclude that

L

+Lx

x +Lt

t = dJ0

dt+

dJ1dx

(1.44)

which can be transformed, using the equations of motion, to

d

dt

Lt

+

Lt

t +d

dx

Lx

+

Lx

x =

dJ0dt

+dJ1dx

(1.45)

Examples:

1. homogeneity of space (translational invariance)

x

x +

= (x + ) (x) = xt = t(x + ) t(x) = xtx = x(x + ) x(x) = xx

L = dLdx

x =dLdx

J1 = L , J0 = 0 . (1.46)

Eq. (1.45) becomes

d

dt

Lt

x +

Lt

xt +d

dx

L

x

x +

Lx

xx =dLdx

(1.47)

ord

dt Ltx + ddx Lx x L = 0 ; (1.48)

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integrating over all space, this gives

dx Lt x P (1.49)i.e. the total momentum is a constant.

2. homogeneity of time

t t + = (t + ) (t) = t

t = t(t + ) t(t) = ttx = x(t + ) x(t) = xt

L = dLdt

t =dLdt

J0 = L , J1 = 0 . (1.50)

Eq. (1.45) becomes

d

dt

Lt

t +

Lt

tt +d

dx

L

x

t +

Lx

tx =dLdt

(1.51)

ord

dt

Lt

t L

+d

dx

Lx

t

= 0 ; (1.52)

integrating over all space, this givesdx

Lt

t L

H (1.53)

i.e. the total energy is a constant.

3. Lorentz invariance

1.6 Perturbations of integrable systems

Consider a conservative Hamiltonian system H0(J) which is completely integrable, i.e. itpossesses s independent integrals of motion. Note that I use the action-angle coordinates,so that H0 is a function of the (conserved) action coordinates Jj . The angles j are cyclicvariables, so they do not appear in H0.

Suppose now that the system is slightly perturbed, by a time-independent perturbation

Hamiltonian H1( 1) A sensible question to ask is: what exactly happens to the integralsof motion? We know of course that the energy of the perturbed system remains constant -since H1 has been assumed to be time independent. But what exactly happens to the others 1 constants of motion?

The question was first addressed by Poincare in connection with the stability of theplanetary system. He succeeded in showing that there are no analytic invariants of theperturbed system, i.e. that it is not possible, starting from a constant 0 of the unperturbedsystem, to construct quantities

= 0(J) + 1(J, ) + 22(J, ) , (1.54)

where the ns are analytic functions of J, , such that

{, H} = 0 (1.55)

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holds, i.e. is a constant of motion of the perturbed system. The proof of Poincarestheorem is quite general. The only requirement on the unperturbed Hamiltonian is that it

should have functionally independent frequencies j = H0/Jj . Although the proof itselfis lengthy and I will make no attempt to reproduce it, it is fairly straightforward to seewhere the problem with analytic invariants lies.

To second order in , the requirement (1.55) implies

{0 + 1 + 22, H0 + H1} = 0{0, H0} + ({1, H0} + {0, H1}) + 2 ({2, H0} + {1, H1}) = 0 .

The coefficients of all powers must vanish. Note that the zeroth order term vanishes bydefinition. The higher order terms will do so, provided

{1, H0} = {0, H1} (1.56)

{2, H0

}=

{1, H1

}.

The process can be continued iteratively to all orders, by requiring

{n, H0} = {n+1, H1} . (1.57)Consider the lowest-order term generated by (1.57). Writing down the Poisson bracketsgives

sj=1

1i

H0Ji

1Ji

H0i

=

sj=1

0i

H1Ji

0Ji

H1i

. (1.58)

The second term on the left hand side and the first term on the right-hand side vanishbecause the s are cyclic coordinates in the unperturbed system. The rest can be rewrittenas

sj=1

i(J)

1i =

sj=1

0Ji

H1i . (1.59)

For notational simplicity, let me now restrict myself to the case of two degrees of freedom.The perturbed Hamiltonian can be written in a double Fourier series

H1 =n1,n2

An1,n2(J1, J2) cos(n11 + n22) . (1.60)

Similarly, one can make a double Fourier series ansatz for 1,

1 =n1,n2

Bn1,n2(J1, J2) cos(n11 + n22) . (1.61)

Now apply (1.59) to the case 0(J) = J1. Using the double Fourier series I obtain

B(J1)n1,n2 =n1

n11 + n22An1,n2 , (1.62)

which in principle determines the first-order term in the expansion of the constant ofmotion J1 which should replace J1 in the new system. It is straightforward to show, usingthe same process for J2, that the perturbed Hamiltonian can be written in terms of the newconstants J1 as

H = H0(J1, J

2) + O(2) . (1.63)

Unfortunately, what looks like the beginning of a systematic expansion suffers from a fatalflaw. If the frequencies are functionally independent, the denominator in (1.62) will in gen-eral vanish on a denumerably infinite number of surfaces in phase space. This however meansthat 1 cannot be an analytic function of J1, J2. Analytic invariants are not possible. All

integrals of motion - other than the energy - are irrevocably destroyed by the perturbation.

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2 Background: Statistical mechanics

2.1 Scope

Classical statistical mechanics attempts to establish a systematic connection between micro-scopic theory which governs the dynamical motion of individual entities (atoms, molecules,local magnetic moments on a lattice) and the macroscopically observed behavior of matter.

Microscopic motion is described - depending on the particular scale of the problem - eitherby classical or quantum mechanics. The rules of macroscopically observed behavior underconditions of thermal equilibrium have been codified in the study of thermodynamics.

Thermodynamics will tell you which processes are macroscopically allowed, and can es-tablish relationships between material properties. In principle, it can reduce everything -everything which can be observed under varying control parameters ( temperature, pres-sure or other external fields) to the equation of statewhich describes one of the relevantmacroscopic observables as a function of the control parameters.

Deriving the form of the equation of state is beyond thermodynamics. It needs a link tomicroscopic theory - i.e. to the underlying mechanics of the individual particles. This linkis provided by equilibrium statistical mechanics. A more general theory of non-equilibriumstatistical mechanics is necessary to establish a link between non-equilibrium macroscopic

behavior (e.g. a steady state flow) and microscopic dynamics. Here I will only deal withequilibrium statistical mechanics.

2.2 Formulation

A statistical description always involves some kind of averaging. Statistical mechanics isabout systematically averaging over hopefully nonessential details. What are these detailsand how can we show that they are nonessential? In order to decide this you have to lookfirst at a system in full detail and then decide what to throw out - and how to go about itconsistently.

2.2.1 Phase space

An Hamiltonian system with s degrees of freedom is fully described at any given time if weknow all coordinates and momenta, i.e. a total of 2s quantities (=6N if we are dealing withpoint particles moving in three-dimensional space). The microscopic state of the systemcan be viewed as a point, a vector in 2s dimensional space. The dynamical evolution of thesystem in time can be viewed as a motion of this point in the 2s dimensional space (phasespace). I will use the shorthand notation (qi, pi, i = 1, s) to denote a point in phasespace. More precisely, (t) will denote a trajectory in phase space with the initial condition(t0) = 0. 1

1

Note that trajectories in phase space do not cross. A history of a Hamiltonian system is determined bydifferential equations which are first-order in time, and is therefore reversible - and hence unique.

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2.2.2 Liouvilles theorem

Consider an element of volume d0 in phase space; the set of trajectories starting at timet0 at some point 0 d0 lead, at time t to points d. Liouvilles theorem assertsthat d = d0. (invariance of phase space volume). The proof consists of showing that theJacobi determinant

D(t, t0) (q, p)(q0, p0)

(2.1)

corresponding to the coordinate transformation (q0, p0) (q, p), is equal to unity. Usinggeneral properties of Jacobians

(q, p)

(q0, p0)=

(q, p)

(q0, p) (q

0, p)

(q0, p0)=

(q)

(q0)

p=const

(p)(p0)

q=const

(2.2)

and

D(t, t0t

t=t0

=si=1

qiqi

+pipi

t=t0

=si=1

2H

qipi

2H

piqi

= 0 , (2.3)

and noting that D(t0, t0) = 1, it follows that D(t, t0) = 1 at all times.

2.2.3 Averaging over time

Consider a function A() of all coordinates and momenta. If you want to compute its long-time average under conditions of thermal equilibrium, you need to follow the state of thesystem over a long time, record it, evaluate the function A at each instant of time, and takea suitable average. Following the trajectory of the point in phase space allows us to definea long-time average

A = limT

1

T

T0

dtA[(t)] . (2.4)

Since the system is followed over infinite time this can then be regarded as a true equilibriumaverage. More on this later.

2.2.4 Ensemble averaging

On the other hand, we could consider an ensemble of identically prepared systems andattempt a series of observations. One system could be in the state 1, another in the state2. Then perhaps we could determine the distribution of states (), i.e. the probability

(), that the state vector is in the neighborhood ( , + ). The average of A in thiscase would be

< A >=

d()A() (2.5)

Note that since is a probability distribution, its integral over all phase space should benormalized to unity:

d() = 1 (2.6)

A distribution in phase space must obey further restrictions. Liouvilles theorem states thatif we view the dynamics of a Hamiltonian system as a flow in phase space, elements ofvolume are invariant - in other words the fluid is incompressible:

ddt

(, t) = {, H} + t

(, t) = 0 . (2.7)

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For a stationary distribution () - as one expects to obtain for a system at equilibrium -

{, H

}= 0 , (2.8)

i.e. can only depend on the energy2. This is a very severe restriction on the forms ofallowed distribution functions in phase space. Nonetheless it still allows for any functionaldependence on the energy. A possible choice (Boltzmann) is to assume that any point onthe phase space hypersurface defined by H() = E may occur with equal probability. Thiscorresponds to

() =1

(E) {H() E} (2.9)

where

(E) =

d {H() E} (2.10)

is the volume of the hypersurface H() = E. This is the microcanonical ensemble. Other

choices are possible - e.g. the canonical (Gibbs) ensemble defined as

() =1

Z()eH() (2.11)

where the control parameter can be identified with the inverse temperature and

Z() =

deH() (2.12)

is the classical partition function.

2.2.5 Equivalence of ensembles

The choice of ensemble, although it may appear arbitrary, is meant to reflect the actualexperimental situation. For example, the Gibbs ensemble may be derived- in the sensethat it can be shown to correspond to a small (but still macroscopic) system in contactwith a much larger reservoirof energy - which in effect holds the smaller system at a fixedtemperature T = 1/. Ensembles must - and to some extent can - be shown to be equivalent,in the sense that the averages computed using two different ensembles coincide if the controlparameters are appropriately chosen. For example a microcanonical average of a functionA() over the energy surface H() = will be equal with the canonical average at a certaintemperature T if we choose to be equal to the canonical average of the energy at thattemperature, i.e. < A() >micro =< A() >

canonT if =< H() >

canonT .

If ensembles can be shown to be equivalent to each other in this sense, we do not need to

perform the actual experiment of waiting and observing the realization of a large numberof identical systems as postulated in the previous section. We can simply use the mostconvenient ensemble for the problem at hand as a theoretical tool for calculating averages. Ingeneral one uses the canonical ensemble, which is designed for computing average quantitiesas functions of temperature.

2.2.6 Ergodicity

The usage of ensemble averages - and therefore of the whole edifice of classical statisticalmechanics - rests on the implicit assumption that they somehow coincide with the morephysical time averages. Since the various ensembles can be shown to be equivalent (cf.

2

or - in principle - on other conserved quantities; in dealing with large systems it may well be necessary toaccount for other macroscopically conserved quantities in defining a proper distribution function.

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above), it would be sufficient to provide a microscopic foundation for the ensemble mostdirectly accessible to Hamiltonian dynamics, i.e. the microcanonical ensemble. The ergodic

hypothesis states that

limT

1

T

T0

dtA [(t)] =1

(E)

d {H() E} A() (2.13)

i.e. that time averages and microcanonical averages coincide. This requires that as a point moves around phase space, it spends - on the average - equal times on equal areas ofthe energy hypersurface (recall that the phase point must stay on the energy hypersurfacebecause H() is a constant of the motion. This seems like a strong & rather nonobviousassertion; Boltzmann had a rough time when he tried to sell it as a plausible basis for theemerging theory of statistical mechanics.

One of the reasons why (2.13) appears implausible was a theorem proved by Poincare whichstated that if a Hamiltonian system is bounded, its trajectory in phase space - although not

allowed to cross itself - will return arbitrarily close to any point already traveled, providedone waits long enough. Therefore, even statistically improbable microstates may recur. Thecatch is that Poincare recurrence times for rare events in large systems are of order eN andmay easily exceed the age of the universe[1].

In fact, ergodicity was later shown by Birkhoff to hold if the energy surface cannot bedivided in two invariant regions of nonzero measure (i.e. regions such that the trajectoriesin phase space always remain in one of them). The energy surface is then called metricallyindecomposable. One way this decomposition could occur might be if further integrals ofmotion are present.

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3 The FPU paradox

3.1 The harmonic crystal: dynamics

Consider a chain of N point particles, each of unit mass. Each of the particles is coupled toits nearest neighbor via a harmonic spring of unit strength; let Qi be the displacement ofthe ith particle; the Hamiltonian (1.6) is

H(P, Q) = 12

Ni=1

P2i + 12

Ni=0

(Qi+1 Qi)2 , (3.1)

where the canonical momenta are Pi = Qi and the end particles are held fixed, i.e. Q0 =QN+1 = 0 (NB: N degrees of freedom).

The Fourier decomposition

Qi =

2

N + 1

N=1

sin

i

N + 1

A

Pi =

2

N + 1

N

=1sin

i

N + 1

B (3.2)

is a canonical transformation (cf. above) to a new set of coordinate and momenta {A, B}.(NB: exercise, check properties, orthogonality, trigonometric sums, boundary conditionssatisfied). In this new set of coordinates, the Hamiltonian can be written as

H =N=1

H 12

N=1

B2 +

2A

2

(3.3)

where

2 = 4 sin2

2(N + 1)

. (3.4)

This is a case of a separable Hamiltonian, where Hamilton-Jacobi theory can be triviallyapplied, i.e.

1

2

WA

2+

1

22A

2 = = 1, , N. (3.5)

where each is a constant representing the energy stored in the th normal mode. Thesubstitution

A =

2

sin (3.6)

transforms (3.5) toW

=2

cos2 . (3.7)

The corresponding action variable

J = 12BdA = 12 WA dA (3.8)

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3 The FPU paradox

can now be evaluated as

J =1

2

2

20

d cos2 = (3.9)

by integrating over a full cycle of the substitution variable . The Hamiltonian can berewritten in terms of the action variables

H =

=

J (3.10)

The angle variables conjugate to the action variables can be found from (1.30

=W(A, J)

J. (3.11)

It can be shown explicitly that j = j .

The Hamiltonian equations in action-angle variables are

J = 0

=H

J= , (3.12)

i.e. the s are the natural frequencies of the normal modes. Note that we did not needthe explicit form of the solution of the Hamilton-Jacobi equation to derive this.

More explicitly, the time evolution of the normal mode coordinates is

A(t) =2J

1/2 sin t + 0 , (3.13)

with an analogous expression for the momenta B.

In the action-angle representation, the 2N constants of integration are the N actionvariables {J} and the N initial phases {0}.

3.2 The harmonic crystal: thermodynamics

The average energy of the harmonic chain at any given temperature T is given by thecanonical average

< H >=1

Z

deH()/TH() , (3.14)

where Z is the partition function

Z(T) =

deH()/T . (3.15)

It is possible to transform the integrals in both numerator and denominator of (3.14) toaction-angle coordinates (cf. previous section). Because of the separability property of theHamiltonian, the denominator splits into product over all N normal modes

Z =

N=1

Z (3.16)

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3 The FPU paradox

where

Z =

0 dJ 2

0 de

J/T

=2T

(3.17)

whereas the numerator transforms to is a sum of the form

N=1

=

Z

N (3.18)

where

N =

0 dJ 2

0 deJ/T

J

=2T2

. (3.19)

It follows that

< H >=N=1

< >=N=1

N/Z =N=1

T = N T , (3.20)

i.e. each the average energy which corresponds to each degree of freedom is equal to T(equipartition property).

The statistical mechanics of the harmonic chain has a fundamental flaw: althoughcanonical averages are straightforward to obtain, there is obviously no basis for assuming

ergodicity - in the presence of N integrals of motion. Now, this might not be a seriousproblem if one could argue that a tiny generic perturbation, as might arise from e.g. a smallnonlinearity of the interactions, could drive the system away from complete integrability,and into an ergodic regime. If this turned out to be the case, one could still argue that thecomputed canonical averages reflect the intrinsic thermodynamic properties of the harmonicchain, in the programmatic sense of statistical mechanics. Fermi, Pasta and Ulam decidedto put this implicit assumption to a numerical test.

3.3 The FPU numerical experiment

Fermi, Pasta and Ulam (FPU[2]) investigated the Hamiltonian

H(P, Q) =1

2

N1i=1

P2i +1

2

N1i=0

(Qi+1 Qi)2 + 3

N1i=0

(Qi+1 Qi)3 , (3.21)

where the canonical momenta are Pi = Qi and the end particles are held fixed, i.e. Q0 =QN = 0. Their work - undertaken as a suitable test problem for one of the very firstelectronic computers, the Los Alamos MANIAC- is considered as the first numerical ex-periment. In other words, it is the first case where physicists observed and analyzed thenumerical output of Newtons equations, rather than the properties of a mechanical systemdescribed by these same equations.

The dynamics of the Hamiltonian (3.21) was studied as an initial value problem; the initialconfiguration was a half-sine wave Qi = sin(i/N), with N = 32 and all particles at rest;the nonlinearity parameter was chosen as = 1/4. Energy was thus pumped at the lowest

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3 The FPU paradox

Figure 3.1: The quantity plotted is the energy (kinetic plus potential in each of the first fourmodes). The time is given in thousands of computational cycles. Each cycle is 1/2

2

of the natural time unit. The initial form of the string was a single sine wave (mode

1). The energy of the higher modes never exceeded 6% of the total. (from [2]).

Fourier mode, = 1, in the notation of (). The objective of the experiment was to studythe energies stored in the first few Fourier modes, i.e. the quantities

H 12

A2 +

2A

2

(3.22)

where

A =

2

N

Ni=1

sin

i

N

Qi (3.23)

as a function of time, i.e. to test the onset of equipartition. Note that the decomposition ofthe total energy in Fourier modes is not exact - but as long as stays small, H Hwill hold.

Fig. 3.1 shows the time dependence of the energies of the first four modes. After an initialredistribution, all of the energy (within 3%) returns to the lowest mode. The energy residingin higher modes never exceeded 6 % of the total. Longer numerical studies have shown thereturn of the energy to the initial mode to be a periodic phenomenon; the period is about157 times the period of the lowest mode. The phenomenon is known as FPU recurrence.

The results of a more recent numerical study on FPU recurrence[3] are summarized inFig. 3.2.

The Hamiltonian (3.21) is fairly generic. In fact, the original FPU paper describes a fur-ther study with quartic, rather than cubic, anharmonicities which exhibits similar behavior.FPU recurrence has been shown to be a robust phenomenon. The upshot of those exhaustivenumerical observations is that anharmonic corrections to the Hamiltonian, contrary to theoriginal expectation which held them as agents that might help establish ergodicity, actuallyappear to generate new forms of approximately periodic behavior. The process of under-

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3 The FPU paradox

Figure 3.2: FPU recurrence time, divided by N3

vs a scaling variable R = (E/N)

1/2

N

2

whereE/N [B/(2N)]2 is the energy density. Typical values used by FPU correspond toR 1. The asymptotic regime is well described by the relationship Tr/N3 = R1/2(from Ref. [3]).

standing the source of this behavior - also known as the FPU paradox - and relating it toother manifestations of nonlinearity [4] has led to a profound change in theoretical physics.

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4 The Korteweg - de Vries equation

4.1 Shallow water waves

Original context: Wave motion in shallow channels, cf. Scott-Russell1

Mathematical description due to Korteweg and deVries (KdV [6]). The equation arises inwide variety of physical contexts (e.g. plasma physics, anharmonic lattice theory). Hence itcounts as one of the canonical soliton equations.

Long waves (typical length l) in a shallow channel l h.Small amplitude ( h) waves (weak nonlinearity)Two-dimensional fluid flow (motion in lateral dimension of channel neglected)

x: horizontal direction, y: vertical direction

4.1.1 Background: hydrodynamics

Fluid velocityV ux + vy (4.1)

Equations of (Eulerian) incompressible fluid dynamics

continuity equation V = 0 (4.2)

Euler equationV

t+ (V )V = 1

p + g (4.3)

where g = gy plus irrotational flow (no vortices)

V = 0 V = . (4.4)

Using vector identity

(V )V = 12V2 V ( V) (4.5)

in (4.3) (only first term survives due to (4.4) ), and (4.4) in (4.2) transforms hydrodynamicsequations to

1I 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; itaccumulated 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, smoothand well-defined heap of water, which continued its course along the channel apparently without changeof form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate ofsome eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a footand 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 thatsingular and beautiful phenomenon which I have called the Wave of translation.[5]

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1. continuity

= 0 , (4.6)

2. Euler

t+

1

2()2 + p

+ gy = 0 . (4.7)

4.1.2 Statement of the problem; boundary conditions

The above eqs (4.6) and (4.7) must now be solved subject to the boundary conditions

1. bottom: no vertical motion of the fluid

v(x, y = 0) = 0

x (4.8)

2. top: free surface defined as

y = h + (x, t). (4.9)

Velocity of free boundary coincides with fluid velocity,

dy

dt=

t+

x

dx

dthence

v =

t+

xu (4.10)

holds at the free surface.

The solution will involve two steps: first, find a general class of solutions of (4.6) whichsatisfy the bottom BC (4.8), and then use this general class to determine the height profile(4.9) by demanding that the Euler equation (4.7) be satisfied at the free surface, where p = 0holds. The Euler equation can then be used to determine the pressure at any point.

4.1.3 Satisfying the bottom boundary condition

Consider the general form of an expansion (the height O(h) is small in a sense which willbe made precise below) of the type

u = f(x) + f1(x)y + f2(x)y2 + f3(x)y

3 + v = g1(x)y + g2(x)y2 + g3(x)y3 + . (4.11)

The conditions uy

= vx

and ux

= vy

imposed by (4.6) can now be written, respectively,as

f1 + 2f2y + 3f3y2 = g1xy + g2xy

2 (4.12)

and

fx + f1y + f2y2 = g1 2g2y 3g3y2 (4.13)

from which

f1 = 0 (4.14)

2f2 = g1x (4.15)

3f3 = g2x (4.16)

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and

fx

=

g1

(4.17)

f1x = 2g2 (4.18)f2x = 3g3 (4.19)

follow. Using the second set in the first, results in f1 = 0, 2f2 = fxx, 2f3 = 1/2f1xx(= 0);it follows that g2 = 0 and g3 = 1/3f2x = 1/3!fxxx. Collecting terms,

u = f 12

fxxy2 + O(y4) (4.20)

v = fxy + 13!

fxxxy3 . (4.21)

4.1.4 Euler equation at top boundarySet p = 0 in (4.7) and differentiate with respect to x:

u

t+

1

2

x(u2 + v2) + g

t= 0 . (4.22)

The problem is now to solve the system of coupled differential equations (4.22) and (4.10)using the expressions (4.20) and (4.21). Key: follow the scale of variation of the physicalquantities involved. First note that if the water height is not much different from h (smallnonlinearity), it will be useful to set

= h (4.23)

Note is not a parameter of the problem. It simply serves as a tag to let us keep

track of scales. At the end we will have to check the consistency of the assumptions andapproximations made.

According to our assumption, the length scale on which the fluid profile varies along the xdirection is of the order l h. In order to incorporate this assumption in the approximation,I define a rescaled variable via

x = lx . (4.24)

Dimensional consideration determine a natural velocity scale c =

gh. The motion shouldbe slow with respect to that scale - in agreement with small amplitude variations of theprofile. In other words, we expect u c. Note that from the leading orders of (4.20) and(4.21)it follows that v is typically of order h/l smaller than u. It is therefore reasonableto rescale

f = cf (4.25)

u = cu (4.26)

v = cv . (4.27)

Finally I use a rescaled timet = t l/c . (4.28)

With these rescalings, keeping lowest order terms, i.e. of O() and O(2), the rescaledequations (4.20) and (4.21) become - on the surface -

u = f 12

2fxx (4.29)

v = (1 + )fx + 16

2fxxx ; (4.30)

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accordingly, the top boundary condition (4.10) and the Euler equation (4.22) transform to

fx + t + (f)x 1

6 2

fxxx = 0 (4.31)

ft + x +

2(f2)x 1

22fxxt = 0 . (4.32)

First we note that in the absence of nonlinearity ( = 0) and dispersion ( = 0), freewave propagation with unit velocity (in dimensionless units) occurs; in that (zeroth) order,f = . But of course this is hypothetical because and are not parameters of the problem- they just help us keep track of things! However, the zeroth order approximation is usefulin the sense that it suggests a coordinate transformation which absorbs the fastest timedependence; let

= x t (4.33) = t . (4.34)

Keeping terms to first order in and 2, we use the property

x = (4.35)

t = + (4.36)(which holds for f as well) transform the system (4.32) to

f + + (2) 16

2 = 0 (4.37)

f + + + 2

(2) +1

22 = 0 . (4.38)

where we have used the property f = in terms which contain or 2 factors. The sum of(4.38) is

2 +3

2(2) +

1

32 = 0 . (4.39)

The three terms in (4.39) will be of the same order if 2 = O(), i.e. if the nonlinearitybalances the dispersion. We choose = 2/6. Note that the choice must be tested at theend to check whether it satisfies the original requirements (small amplitude, long waves).With this choice and the substitution = 4 I arrive at the canonical KdV form,

+ 6 + = 0 . (4.40)

4.1.5 A solitary wave

At this stage, without recourse to advanced mathematical techniques, it is possible to followthe path of KdV and look for special, exact, propagating solutions of (4.40) of the type (s),where s = . (4.40) becomes

s + 3(2)s + sss = 0 (4.41)which has an obvious first integral

+ 32 + ss = const. (4.42)If we are looking for solutions which vanish at infinity (lims (s) = 0 and lims s(s) =0) the constant will be zero, i.e.

ss = 32 = dd ( 12 2 3) (4.43)

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Multiplying both sides by 2s we can integrate once more, obtaining

2

s

= 2

23 (4.44)

where the integration constant must vanish once again (cf. above). Note that, if a solutionexists, the parameter must be > 0 and < /2. Taking the square root of (4.44) andinverting the fractions I obtain

ds = d

2 (4.45)

which can be integrated directly, resulting in

(s) =

2cosh2

2

(s s0) (4.46)

where s0 is an arbitrary constant. (The plus sign in (4.45) has been chosen for s < s0 and

the minus for s > s0).Note that the properties of the propagating solution (4.46) - except for its initial position,

which is determined by s0 - are all governed by a single parameter. If the velocity isgiven, the amplitude is fixed at /2 and the spatial extent at 21/2. In other words - in thecanonical units of (4.40) - a slow pulse will also have a small amplitude and a large spatialextent.

4.1.6 Is the solitary wave a physical solution?

Eq. (4.46 ) is an exact, propagating, pulse-like solution of (4.40). But is it an acceptablesolution of the original problem? In other words, is the surface profile of low amplitude and isit a long wave? To do this, we have to go back to the original variables, and convince ourselvesthat (4.46) generates (some) acceptable solutions for the original problem (Exercise)

4.2 KdV as a limiting case of anharmonic lattice dynamics

Consider the 1-d anharmonic chain; atomic displacements are denoted by {un}; neighboringatoms of mass m interact via anharmonic potentials of the type

V(r) =1

2kr2 +

1

3kbr3 (4.47)

where r is the distance between nearest neighbors. The equations of motion are

mqn = qn

[V(qn+1 qn) + V(qn qn1]) (4.48)= k(qn+1 + qn1 2qn) kb[(qn+1 qn)2 + (qn qn1)2]= k(qn+1 + qn1 2qn) kb(qn+1 + qn1 2qn)(qn+1 qn1) .

If the displacements do not vary appreciably on the scale of the lattice constant a, we canuse a continuum approximation; keeping terms of fourth order in the lattice constant,

mq qtt = ka2qxx + ka4 24!

qxxxx + kba2qxx 2aqx ,

where x = na is the continuum space variable; defining c2 = ka2/m, this can be written as

1c2

qtt qxx = 112

a2qxxxx + 2qxqxx , (4.49)

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where = ab provides a dimensionless measure of the anharmonicity.

I now look for solutions which vary smoothly in space, i.e. over a typical length of many

lattice spacings, and where the main time dependence is contained in the wave equationpart, i.e. of the form

q(, ) q( x cta

, 0t) , (4.50)

where 0 = c/a =

k/m, 1 and ; the exact dependence of on will be fixedlater.

The relevant derivatives transform according to

qx =

aq

qxx =

a2

q

qxxx =

a3 q

qtt = 20

2q 2q + O(2)

.

Using them in (4.49) gives

2q +1

123q + 2qq = 0 , (4.51)

which, after a rescaling

q(=a

qx) =

4a (4.52)

and setting 2

= 124 3

can be reduced to the canonical KdV form

6 + = 0 . (4.53)

Note that the rescaling of length, i.e. the value of the small parameter is still a matter offree choice, depending on the (initial) conditions of the problem.

The above analysis shows that one may legitimately suspect that nonlinear propagatingsolitary waves will be generic in anharmonic lattices, at least for certain parameter ranges.Again, one has to make sure that the solutions found from solving the KdV equation (4.53)are appropriate for the original problem (4.49) (check consistency of approximations made).

4.3 KdV as a field theory

4.3.1 KdV Lagrangian

The KdV equationut 3(u2x)x + uxxx = 0 (4.54)

can be derived from the Lagrangian

L =

dxL(, t, x, xx) (4.55)

2note that this guarantees as demanded above.

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where

L=

1

2

xt

3x

1

2

2xx . (4.56)

Note that because the Lagrangian density depends on the second derivative of the field,(1.42) contain an extra term

d2

dx2

L

xx

. (4.57)

Minimization of the action leads to the field equations of motion

xt 3(2x)x + xxxx = 0 (4.58)

which reduces to (4.54) upon the substitution

x = u . (4.59)

Continuous symmetries of the Lagrangian will again give rise to an equation like (1.44), withan extra term

Lxx

xx (4.60)

on the left-hand side. The above modifications generate an extra contribution

Lxx

xx d2

dx2

L

xx

(4.61)

to the left-hand side of (1.45).

4.3.2 Symmetries and conserved quantities

For some infinitesimal transformations (cf. section ) one can verify explicitly that xx =d2/dx2. If this is the case, the integral over all space of the extra contribution (4.61) caneasily be seen to vanish (repeated integration by parts of either of the two terms). In thiscase, the standard symmetries are reflected in the same standard conservation (with thesame densities of conserved quantities), as in section .... .

Translational invariance in space

Conservation of the total momentum

P =

dxLt

x = 12

dx 2x = 1

2

dx u2 . (4.62)

Translational invariance in time

Conservation of the total energy

H =

dx

Lt

t L

=

dx

1

22xx +

3x

=

dx 12 u2x + u3 . (4.63)

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Conservation of mass

The symmetry + generates = , and all other variations are zero. From (1.45),conservation ofM =

dxLt

=1

2

dx x =1

2

d x u , (4.64)

the total mass, is deduced.

Galilei invariance

The transformation x x t, (x, t) (x t) x (or in terms of the u-field, u(x, t) u , generates (cf. section ....)

x x t = (x t) (x) x = tx x

t = t(x t) t(x) = txtx = x(x t) x(x) = txx

L = dLdx

x = dLdx

t J1 = tL , J0 = 0 . (4.65)

Owing to xx = ()xx there are no extra terms in the conserved currents. Eq. (1.45)applies. Since x = ()x the two last terms in the left-hand side of (1.45) combine toform a total space derivative; similarly, because of t = ()t, the first two terms combineto form a total time derivative, i.e. the conserved density is

L

t

/ =1

2

x(

tx

x) , (4.66)

or, integrating over all space, and dividing by the total mass M,

X =1

M

dx xu

2=

P

Mt + const. (4.67)

which expresses the fact that the center of mass moves at a constant velocity.

4.3.3 KdV as a Hamiltonian field theory

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5 Solving KdV by inverse scattering

5.1 Isospectral property

Given the KdV equationut 6uux + uxxx = 0 (5.1)

and a well behaved initial condition u(x, 0), which vanishes at infinity, it is possible todetermine the time evolution u(x, t) in terms of a general scheme, which is known as inversescattering theory.

The scheme is based on the following particular property of (5.1):

Given the linear operatorL(t) = 2xx + u(x, t) (5.2)

whose parametric time dependence is governed by (5.1), and the associated eigenvalue equa-tion

L(t)j(x, t) = j(t)j(x, t) , (5.3)

it can be shown thatdjdt

= 0 . (5.4)

5.2 Lax pairs

The isospectral property can be formulated somewhat more generally: Suppose we canconstruct a linear, self-adjoint operator B = B, dependent on u and such that

iLt i dLdt

i lim0

L(t + ) L(t)

= [L, B] (5.5)

holds as an operator identity, i.e.

iLtf = [L, B]f f (5.1) . (5.6)

The operators L and B are then called a Lax pair. The time evolution of L is governed by

L(t) = U(t)L(0)U (5.7)

whereU = eiBt . (5.8)

Consider (5.3) at t = 0, and apply the operator U(t) to both sides from the left, i.e.

U(t)L(0) U(t)U(t)j(0) = j(0)U(t)j(0) (5.9)

where, in addition I have inserted a factor UU = 1. It can be recognized immediately thatthe l.h.s. of (5.9) and (5.3) are identical, provided

j(t) = U(t)j(0) , (5.10)

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and that, in order for the r.h.sides to coincide, I must have

(t) = (0)

t (5.11)

(isospectral property).

The form of the operator B in the KdV case is

B = 4i3xxx 3i (ux + xu) (5.12)(verify explicitly (5.6).

5.3 Inverse scattering transform: the idea

The isospectral property tentatively suggests that it might possible to proceed as follows:

solve the linear problem (5.3) at time t = 0, i.e. determine the eigenvalues

{j}

andthe eigenfunctions {j(x, 0)} from the known u(x, 0).

determine the evolution of the eigenfunctions from (5.10) at a later time t. try to solve the inverse problem of determining the potential u(x, t) from the

known spectra and eigenfunctions at the time t.

In fact, the last step is the well known problem of inverse scattering theory in quantummechanics, where physicists had tried to extract information on the nature of interparticleinteractions from analyzing particle scattering data. The one-dimensional problem (corre-sponding to a spherically symmetric potentials in 3 dimensions) was completely solved inthe 1950s (Gelfand, Levitan & Marchenko). I will present the solution below, but beforedoing that, let me outline some broad features:

Scattering datain the mathematical sense are the asymptotic properties of the solutionof the associated linear problem, i.e. the properties far from the source of scattering, wherethe potential is effectively zero. What GLM have shown is that you can reconstruct thepotential from the scattering data. Furthermore, it turns out that the operator B takesan especially simple form in the asymptotic limit, which allows us to write down an exact,analytic formula for the time evolution of scattering data. Evolution of the scattering datais the easy part of the game. But then if I only need scattering data at time t, and I knowhow these data evolve in time, all the input I need is the scattering data for the potentialu(x, 0). This is exactly the program of the inverse scattering transform (IST). Because it isbased only on the asymptotic part of the solution of the associated linear problem, it canbe written down in closed form. I summarize the IST program schematically:

1. determine the scattering data S of the linear problem (5.3) at time t = 0, from theknown u(x, 0).

2. determine the evolution of the scattering data S(t) at a later time t from the asymptoticfrom of the operator B.

3. do the inverse problem at time t, i.e. determine the potential u(x, t) from the knownscattering data S(t).

5.4 The inverse scattering transform

5.4.1 The direct problem

This is just a summary of properties known from elementary quantum mechanics.

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Jost solutions

The linear eigenvalue problem

x2+ u(x)

(x) = k2(x) (5.13)

has, in general, a discrete and a continuum spectrum, corresponding to imaginary and realvalues of k respectively. For real k there are in general two linearly independent solutions.Such a linearly independent set is provided by the Jost solutions:

f1(x, k) eikx x f2(x, k) eikx x . (5.14)

The Jost solutions of (5.13) satisfy the integral equations

f1(x, k) = eikx

x

dxG(x, x)f1(x, k)

f2(x, k) = eikx +

x

dxG(x, x)f2(x, k) (5.15)

where

G(x, x) =sin k(x x)

ku(x) . (5.16)

Eqs. (5.15) can be analytically continued to the upper half plane of complex k. Someinformation on the analytic properties can be obtained by considering the lowest iteration,where we substitute f1(x

, k) = eikx

in the r.h.s. of the first equation. This gives

f1(x, k) eikx x

dxeik(x

x) eik(xx)2ik

u(x)eikx

eikx eikx 12ik

x

dx {1 e2ik(xx)}u(x) (5.17)

which can be thought of as the beginning of a systematic expansion in inverse powers of k.Note that since x x > 0, the exponential will be convergent in the upper-half plane of k;therefore, if the potential vanishes sufficiently rapidly at infinity, I estimate

g1(x, k) f1(x, k) eikx eikxh(x, k) (5.18)

where h vanishes as 1/k for high values of k.

The propertyf2(x, k) = a(k)f1(k, x) + b(k)f1(k, x) . (5.19)

will be useful.

For bound states, corresponding to k = i, the Jost solutions are degenerate.

Asymptotic scattering data

The asymptotic (scattering) data of (5.3) is defined as follows:

discrete spectrum (bound states)

n = 2n n = 1, , N , (5.20)

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where n > 0;

n(x) = f1(x, k)

enx x = Cnf2(x, k) Cnenx x . (5.21)

I will also need the normalization integral of each bound state

1

n=

dx2n(x) =

dxf21 (x,in) (5.22)

continuous spectrum (scattering states)(k) = k2 < k < . (5.23)

The physical scattering states corresponding to waves incident from the right, are

(x, k) eikx + R(k)eikx x T(k)eikx x . (5.24)

where R(k), T(k) are, respectively, the reflection and transmission coefficients, whichsatisfy

|R(k)|2 + |T(k)|2 = 1 .The Jost solutions are related to the physical solution (5.24) via

(x, k) = T(k)f2(x, k) = f1(x, k) + R(k)f1(x, k) x. (5.25)This identifies a(k) = 1/T(k) and b(k) = R(k)/T(k).

The complete set of scattering data for any one dimensional potential of a Schroedinger-typeequation is

S [{n, Cn, n}, n = 1 , N; T(k), R(k)]. (5.26)In fact, for the purposes of performing the inverse scattering transform I will only need thereduced set 1

S [{n, n}, n = 1 , N; R(k)] (5.27)

5.4.2 Time evolution of scattering data

I promised this will be the easy part. The operator B has the property

lim|x| = B = 4i3xxx . (5.28)

Since

tj(x, t) = iBj(x, t) (5.29)

holds for all eigenfunctions, we can apply in the asymptotic regime, where B B.

In the case of a discrete eigenfunction, this gives

n(x) enx+43nt x Cnenx43nt x , (5.30)

1

Note that if scattering theory is to make sense, the potential must be vanishing at ()infinity. I have notspecified the minimal exact mathematical conditions which satisfy this demand.

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or, in keeping with the agreed normalization of the type (5.21), I multiply with a factor

e43nt, and obtain

n(x) enx x Cn(t)enx x , (5.31)

withCn(t) = Cn(0)e

83nt . (5.32)

In the case of Jost solutions I obtainf1(k, x) eikx+4ik3t x f2(k, x) eikx4ik3t x ; (5.33)

the physical solution therefore evolves according to

(k, x)

eikx4ik

3t + R(k)eikx+4ik3t x

T(k)eikx4ik3t x ,or, multiplying both by a factor e4ik

3t, to keep the standard normalization of ()

(k, x) eikx + R(k, t)eikx x T(k)eikx x ,

N. Theodorakopoulos- Nonlinear physics (solitons, chaos, discrete breathers)

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Transcript of N. Theodorakopoulos- Nonlinear physics (solitons, chaos, discrete breathers)