MATHEMATICAL STRUCTURES IN CONTINUOUS DYNAMICAL SYSTEMS

Transcription

1 MATHEMATICAL STRUCTURES IN CONTINUOUS DYNAMICAL SYSTEMS Poisson Systems and complete integrability with applications from Fluid Dynamics E. van Groesen Dept. of Applied Mathematics University oftwente Enschede, The Netherlands E.M. de Jager, Emeritus Dept. of Mathematics and Computer Science University of Amsterdam Amsterdam, The Netherlands NORTH-HOLLAND

2 Contents Contents Preface v vii Part I - Poisson structures in Fluid Dynamics Introduction to Part I 3 1. Poisson Structures 9 Introduction General definitions and outline Finite dimensional Systems 18 Tangent and cotangent space, vector fields 18 Hamiltonian vector fields and structure map 20 Poisson dynamics 21 Symplectic structure and Hamiltonian Systems Examples 22 Canonical Hamiltonian Systems 22 Lagrangian Systems 24 Particle motion in plane fluid flows 26 Kirchhofes equations for vortex points 26 Complex canonical structure 28 Degenerate Hamiltonian Systems 28 Darboux' Theorem 29 Rigid body rotations 30 Magnetic spin Systems Summary of variational calculus 35 First Variation and variational derivative 35 Stationarity condition 38 vii

3 Euler-Lagrange equation for unconstrained problems. 39 Variational principles in mathematical physics 40 Second Variation for unconstrained problems 42 Constrained problems and Lagrange's multiplier rule. 44 Constrained second Variation Infinite dimensional Poisson Systems 47 Poisson bracket and state equation 47 Canonical continuum description 48 Lagrangian Systems 50 Structure maps 50 Symplectic structures Examples from mathematical physics 53 Maxwell's equations 54 Sine-Gordon equation 55 Schrödinger's equation 56 Nonlinear Schrödinger equation Wave equations 57 Uni-directional wave equations 59 Bi-directional wave equations 63 References Surface Waves 69 Introduction Hamiltonian structure of the basic equations 70 Basic equations 70 Hamiltonian formulation 72 Transformation of basic variables Approximations for the kinetic energy 75 Linear theory: infinitesimal surface waves 76 Tidal wave approximation 78 Boussinesq approximation Uni-directionalization to KdV-equation 80 Transformation to uni-directional variables 81 Uni-directionalization Modifications for varying bottom 86 Derivation of bi-directional equation 87 Transformation to uni-directional variables 89 Uni-directional infinitesimal waves without dispersion. 90 Generalized KdV-equation 97 References Eulehan Fluid Dynamics 103 Introduction Transformation to Eulerian description 107 Compressible isentropic flows 107 Non-isentropic flows 112 Incompressible flows 114 Vorticity formulation 117 viii

4 3.2. Incompressible plane flows 118 Velocity formulation 119 Vorticity formulation 119 Enstrophy Casimir functionals 120 Consistent derivation of Kirchhoff's equations Incompressible axially Symmetrie flows 122 Casimir functionals 124 References Consistent Modelling 127 Introduction Transformations 132 Transformation of the bracket 133 Transformation of the State equation 135 Poisson maps 139 Hamiltonian flows are Poisson maps 140 Canonical transformations Reduction 142 Decomposition 142 Poisson submanifold 144 Reduced manifolds Restricted dynamics on a given manifold 147 Manifolds of parameterized funetions 147 Restricted dynamics 148 Projected dynamics 149 Truncated dynamics 149 Reparametrization of the manifold The restricted dynamics coupled with the remainder 153 Decomposition based on a submanifold 153 Decomposition of the evolution equation 155 Decomposition of the Poisson strueture 156 Restricting the dynamics to the manifold 157 Dirac dynamics 158 Restriction of first integrals Unbalanced diagonal strueture maps 161 Diagonal strueture maps 161 Regulär perturbation problems 162 Singular perturbation problems Large and small scale interactions 166 Direct Fourier Separation 166 Truncations for 1D wave equations 168 Unbalanced contribution from the small and large scales 169 Regulär perturbation of the truncated BBM-equation. 171 Singular perturbation of the truncated KdV-equation. 171 References Poisson Dynamics 175 Introduction 175 ix

5 5.1. Hamiltonian Systems with a cyclic variable 183 Equilibrium Solutions 184 Reduced dynamics 184 Equilibria of the reduced dynamics 185 Relatieve equilibria 185 Relative equilibrium Solutions 187 Stability Relative equilibria 190 First integrals and Casimir functionals 191 Constrained Hamiltonian extremizers 192 Branches of constrained extremizers and value function 195 Manifold of relative equilibria 197 Relative equilibrium Solutions 198 Relative equilibria with more integrals 200 Dynamical invariance of the critical point set of integrals Symmetries and commuting flows 204 Symmetries of evolution equations 205 Commuting flows 205 The Lie-bracket of Hamiltonian vector fields Reduction procedure 210 Reduction with one integral 211 Reduction with more integrals in involution 214 Complete integrability Stability 217 Lyapunov stability of equilibria and invariant sets Lyapunov functionals 219 Geometrie squeezing property of C 220 Stability of a non-degenerate minimizer 222 Stability of invariant sets 223 Energetics for Poisson Systems 225 Constrained stability 227 Unconstrained stability 229 References Coherent Structures as Relative Equilibria 233 Introduction Vortex points 236 Momentum integrals and their flows 237 Relative equilibria 238 Two point vortices 239 Three point vortices Travelling waves 244 KdV-type of equations 245 KdV-solitons 248 KdV-cnoidal waves 248 KdV - Solitary wave interaction 249 BBM-equation 253 Optical solitons 254 x

6 6.3. Vortices in plane bounded domains 255 Casimir functionals 257 Basic variational formulations 257 Intermezzo: Convexity methods 258 Dual formulation 260 Confinement and non-differentiability 260 Patches 261 Confined monopole vortices unfolded from Taylor vortices 263 Dipolar vortices Confined vortices in the plane 267 Renormalization of functionals 269 Integrals and their flows 271 Basic variational formulation 272 Circular patches 274 Leith' vortices 278 Clockwise rotating, maximum energy monopoles Counter-clockwise rotating, minimum energy monopoles Swirling pipe flows 283 Cross-sectional energy and helicity integrals 285 The basic variational principle 286 Swirling flows as relative equilibria 287 Bragg-Hawthorne equation 289 References Poisson Perturbation Methods 293 Introduction Dissipative Poisson Systems 300 Dissipative structures 301 Dissipative Poisson structures 304 Rayleigh dissipation functional Thermodynamic Systems 307 Constrained dissipative Systems 307 Thermodynamic Systems Approximations with relative equilibria 311 Quality of approximations measured by functionals Approximations using relative equilibria 313 Equation for the error Decaying and self-exciting surface waves 319 Dissipative linear wave equations 319 Dissipating KdV-cnoidal waves, numerics 323 Dissipating KdV-cnoidal waves, theory 326 Self-excited KdV-cnoidal waves Self-organization in plane viscous fluids 332 Formulation of the plane flow problem 333 Planar Taylor vortices 333 Self-organization process 335 Flow of averaged spectral energy and enstrophy 338 xi

7 7.6. Viscous decay of monopoles on the plane 340 Viscous decay of integrals 341 Decay of circular patches 342 Inconsistent evolution along Leith' vortices 343 Viscous instability of maximum energy monopoles Viscous stability of minimum energy monopoles 347 Summary 350 References 350 Part II - Mathematical Introduction to the Theory of Solitons Introduction to Part II Solitons in Physics and Mathematics History The Fermi, Pasta, Ulam experiments Periodic Solutions of the periodic Toda chain Solitary wave Solutions of the infinite Toda chain Finite Toda chain as integrable Hamilton-Poisson System Introduction A Poisson-Hamilton structure on a matrix Lie algebra Decomposition of matrix Lie algebras and restricted dynamical Systems Application to the finite Toda chain The finite Toda chain in the Lax formalism The infinite Toda chain in the Lax formalism The KdV-equation for long waves in a canal The derivation by Korteweg and de Vries The derivation of the Korteweg-de Vries equation using multiple scales KdV as a continuum approximation of a discrete System The theory of P. Lax An algebraic approach by Chern and Peng Wave Solutions of the KdV-equation A.K.N.S. Systems and Soliton Equations An eigenvalue problem The family of the Korteweg-de Vries equations Other families of soliton equations The Sine-Gordon equation A.K.N.S. Systems and Lax pairs NLS, a physical model and a soliton Solution Wave Solutions of the Sine-Gordon equation Scattering, Inverse Scattering and Solitons Introduction Direct scattering 452 xii

8 3.3. Inverse scattering The time evolution of the scattering data Initial value problems and solitons The method of bilinearization The bilinearization The iv-soliton Solution for the KdV-equation and a generalization A further generalization The T-function as universal Solution of the KdV-hierarchy Bäcklund-Transformations Introduction Bäcklund transformation for Sine-Gordon equation Nonlinear superposition principle for Sine-Gordon The stationary "kink" Solution The nonlinear superposition principle and the "two-kink Solution" The Bäcklund transformation for KdV Nonlinear superposition principle for KdV Bäcklund transformations and inverse scattering The addition theorem of Deift and Trubowitz Proof of the addition theorem Summary and consequences Bäcklund transformations by inverse scattering The T-function and its Bäcklund transformation Introduction Application The KdV-Hierarchy as a Hierarchy of Hamiltonian Systems Hamiltonian Systems of ordinary differential equations Hamiltonian Systems of evolution equations Hamiltonian representation of the KdV-equations Conserved functionals The Hamitonian structure of the KdV-family Bi-Hamiltonian structures and Integrability of KdV-equations Prolongation Structures Cartan prolongations A Prolongation of the KdV-equation The general Prolongation of the KdV-equation A Prolongation of the Sine-Gordon-equation A Prolongation of the nonlinear Schrödinger-equation The two-dimensional linear Prolongation The isospectral parameter A 593 xiii

9 Subject index References ßg? ßQ3