Nonlinear Waves, Solitons and Chaos

Transcription

1 Nonlinear Waves, Solitons and Chaos Eryk Infeld Institute for Nuclear Studies, Warsaw George Rowlands Department of Physics, University of Warwick 2nd edition CAMBRIDGE UNIVERSITY PRESS

2 Contents Foreword to thefirst edition Foreword to the second edition page xi xiii 1 Introduction Occurrence of nonlinear waves and instabilities in Nature Nonlinear phenomena in our everyday experience Nonlinear phenomena in the laboratory Universal wave equations The Korteweg-de Vries and Kadomtsev-Petviashvili equations and a first look at solitons The nonlinear Schrödinger equation Nonlinear optics What is a plasma? Wave modes on a water surface Mathematical theory Comments Linear stability analysis and its limitations Nonlinear structures Coherent structures and pattern formation Contents of Chapters Linear waves and instabilities in infinite media Introduction Plasma waves CMA diagrams Instabilities The Vlasov equation Weak instabilities Exercises Convective and non-convective instabilities; group velocity in unstable media Introduction Kinematics of unstable wavepackets 50 V

3 VI Contents 3.3 Moving coordinate Systems Higher dimensional Systems Summary 57 Exercise 57 4 Afirstlook at surface waves and instabilities Introduction Simple surface waves The Rayleigh-Taylor instability The Kelvin-Helmholtz instability Solid-liquid interface instabilities A first look at gravity wave instabilities The small amplitude onset of wave instability Further numerical results Summary 81 Exercises 81 5 Model equations for small amplitude waves and solitons; weakly nonlinear theory Introduction Some physical equations ask for surgery Examples A few model equations as derived by introducing a small Parameter Shallow water, weak amplitude gravity waves Weak amplitude ion acoustic waves in an unmagnetized plasma Weak amplitude ion acoustic waves in a magnetized plasma Weakly nonlinear waves Spreading, Splitting and instabilities The story ofdeep water waves Mystery of the missing term Dynamics of a wavepacket Some generalizations A general look at two families of model equations A natural extension to finite amplitude waves due to Hayes Temporal development of instabilities and wave-wave coupling Concluding remarks 119 Exercises Exact methods for fully nonlinear waves and solitons Introduction Phase plane analysis and other methods 124

4 Contents VII One stationary wave in a dissipationless medium A two-fluid layer soliton pair Weak ion acoustic shock waves in a collisional plasma Solitons generated by laser fields Solitons and domains in dipole chains Discrete equations Bernstein-Greene-Kruskal waves Statistical description of a plasma and BGK waves No trapped particles Various limits Trapped particle equilibria Stability; subsequent developments Lagrangian methods Lagrangian interpolation 159 Exercises Cartesian solitons in one and two space dimensions Introduction The direct method Constants of motion Inverse scattering method Bäcklund transformations Entr'acte Breathers and boundary effects Experimental evidence Plane soliton interaction in two space dimensions Introducing the trace method One and two soliton solutions Some other developments and summary Integrable equations in two space dimensions as treated by the Zakharov-Shabat method Lax pairs and the PDEs they represent Extension to x,y,t How to proceed from the Lax pair to the general Solution An example: the Kadomtsev-Petviashvili equation Summary 199 Exercises Evolution and stability of initially one-dimensional waves and solitons A brief historical survey of large amplitude nonlinear wave studies Solitons Water waves are unstable The geometrical optics limit 207

5 VIII Contents More recent results What the remainder of Chapter 8 is about Four methods as illustrated by the nonlinear Klein-Gordon equation WhithamI Whitham II K expansion Hayes Higher dimensional dynamics Kadomtsev-Petviashvili as analysed by Whitham II Various limits Common features of the weak amplitude and soliton limits for \ji = Group velocity Zakharov-Kuznetsov as analysed by K expansion The variational method A more physical approach leading to an assessment of modeis Form of the waves considered Unmagnetized plasmas, Q 0 = Magnetized plasmas, Q c > Dynamics of nonlinear wave, shock and soliton Solutions to the cubic nonlinear Schrödinger equation Results of a general stability calculation One-dimensional dynamics: \j/ = Oblique and perpendicular propagation of perturbations The direct K method Transverse instability of Zakharov-Kuznetsov solitons The Cahn-Hilliard equation Some general conclusions and possible future lines of investigation 264 Exercises Cylindrical and spherical solitons in plasmas and other media Interest in higher dimensional plasma solitons Unidirectional cylindrical and spherical ion acoustic solitons Model equations in non-cartesian geometry Cylindrical soliton equations CI and CII Spherical solitons Summary Properties of unidirectional soliton equations Integrability by inverse scattering Conservation laws Soliton Solutions as compared with numerics and experiments Exact Solutions to CI Initial value problem and experiments Reflection from the axis (centre) Models 284

6 Contents IX Stability of cylindrical solitons Langmuir solitons Integrability Stability of Langmuir solitons Interacting solitons and some conclusions Epilogue. Some other examples of spherical and cylindrical solitons 292 Exercises Soliton metamorphosis The next step in investigating soliton behaviour Decay oflinekpi solitons in two dimensions Decay of 2D solitons in three dimensions D solitons perturbed perpendicular to the motion D solitons perturbed parallel to the velocity Conclusions 303 Exercises Non-coherent phenomena Introduction Bifurcation sequences and chaos Flows and maps Strange attractors Effect of external noise Experimental evidence for stränge attractors Other theories of turbulence Conclusions 344 Exercises 345 Appendices 346 AI Parameter stretching as suggested by the linear dispersion relations 346 ALI Ion acoustic waves in an unmagnetized plasma, il c = AI.2 Magnetized plasmas, l c > A2 Relation between the trace method and the inverse scattering method 349 A3 Some formulae for perturbed nonlinear ion acoustic waves and solitons 351 A3.1 No magnetic field 351 A3.2 Q c >0 352 A4 Colliding soliton theory 354 A5 A model equation for spherical solitons 356 A6 Stability calculation for 2D KPI soliton in 3D 358 References 360 Author index 379 Subject index 387 Colour plates between pages