Nonlinear physical systems : spectral analysis, stability and bifurcations / edited by Oleg N. Kirillov, Dmitry E. Pelinovsky.

Format:

Book

Contributor:

Kirillov, Oleg N.

Pelinovsky, Dmitry.

Language:

English

Subjects (All):

Nonlinear systems.

Differential equations, Nonlinear.

Differential equations, Partial.

Dynamics.

Physical Description:

1 online resource (449 pages) : illustrations, photographs

Edition:

1st ed.

Place of Publication:

London, England ; Hoboken, New Jersey : ISTE Ltd : John Wiley & Sons, 2014.

Language Note:

English

Summary:

Bringing together 18 chapters written by leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics, this book presents state-of-the-art approaches to a wide spectrum of new and challenging stability problems. Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations focuses on problems of spectral analysis, stability and bifurcations arising in the nonlinear partial differential equations of modern physics. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics, and dissipation-induced instabilities are treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. Each chapter contains mechanical and physical examples, and the combination of advanced material and more tutorial elements makes this book attractive for both experts and non-specialists keen to expand their knowledge on modern methods and trends in stability theory. Contents 1. Surprising Instabilities of Simple Elastic Structures, Davide Bigoni, Diego Misseroni, Giovanni Noselli and Daniele Zaccaria. 2. WKB Solutions Near an Unstable Equilibrium and Applications, Jean-François Bony, Setsuro Fujiié, Thierry Ramond and Maher Zerzeri, partially supported by French ANR project NOSEVOL. 3. The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems, Richard Cushman, Johnathan Robbins and Dimitrii Sadovskii. 4. Dissipation Effect on Local and Global Fluid-Elastic Instabilities, Olivier Doaré. 5. Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field, Sergey Yu. Dobrokhotov and Anatoly Yu. Anikin. 6. Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials, Nir Dror and Boris A. Malomed. 7. Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation, Yasuhide Fukumoto, Makoto Hirota and Youichi Mie. 8. Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1: 1 Resonance, Igor Hoveijn and Oleg N. Kirillov. 9. Index Theorems for Polynomial Pencils, Richard Kollár and Radomír Bosák. 10. Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches, Paolo Luzzatto-Fegiz and Charles H.K. Williamson. 11. Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows, Sherwin A. Maslowe. 12. Continuum Hamiltonian Hopf Bifurcation I, Philip J. Morrison and George I. Hagstrom. 13. Continuum Hamiltonian Hopf Bifurcation II, George I. Hagstrom and Philip J. Morrison. 14. Energy Stability Analysis for a Hybrid Fluid-Kinetic Plasma Model, Philip J. Morrison, Emanuele Tassi and Cesare Tronci. 15. Accurate Estimates for the Exponential Decay of Semigroups with Non-Self-Adjoint Generators, Francis Nier. 16. Stability Optimization for Polynomials and Matrices, Michael L. Overton. 17. Spectral Stability of Nonlinear Waves in KdV-Type Evolution Equations, Dmitry E. Pelinovsky. 18. Unfreezing Casimir Invariants: Singular Perturbations Giving Rise to Forbidden Instabilities, Zensho Yoshida and Philip J. Morrison. About the Authors Oleg N. Kirillov has been a Research Fellow at the Magneto-Hydrodynamics Division of the Helmholtz-Zentrum Dresden-Rossendorf in Germany since 2011. His research interests include non-conservative stability problems of structural mechanics and physics, perturbation theory of non-self-adjoint boundary eigenvalue problems, magnetohydrodynamics, friction-induced oscillations, dissipation-induced instabilities and non-Hermitian problems of optics and microwave physics. Since 2013 he has served as an Associate Editor for the journal Frontiers in Mathematical Physics. Dmitry E. Pelinovsky has been Professor at McMaster University in Canada since 2000. His research profile includes work with nonlinear partial differential equations, discrete dynamical systems, spectral theory, integrable systems, and numerical analysis. He served as the guest editor of the special issue of the journals Chaos in 2005 and Applicable Analysis in 2010. He is an Associate Editor of the journal Communications in Nonlinear Science and Numerical Simulations. This book is devoted to the problems of spectral analysis, stability and bifurcations arising from the nonlinear partial differential equations of modern physics. Leading experts in dynamical systems, operator theory, partial differential equations, and solid and fluid mechanics present state-of-the-art approaches to a wide spectrum of new challenging stability problems. Bifurcations and stability of solitary waves, geometrical optics stability analysis in hydro- and magnetohydrodynamics and dissipation-induced instabilities will be treated with the use of the theory of Krein and Pontryagin space, index theory, the theory of multi-parameter eigenvalue problems and modern asymptotic and perturbative approaches. All chapters contain mechanical and physical examples and combine both tutorial and advanced sections, making them attractive both to experts in the field and non-specialists interested in knowing more about modern methods and trends in stability theory.

Contents:

Cover

Title Page

Table of Contents

Preface

Chapter 1. Surprising Instabilities of Simple Elastic Structures

1.1. Introduction

1.2. Buckling in tension

1.3. The effect of constraint's curvature

1.4. The Ziegler pendulum made unstable by Coulomb friction

1.5. Conclusions

1.6. Acknowledgments

1.7. Bibliography

Chapter 2. WKB Solutions Near an Unstable Equilibrium and Applications

2.1. Introduction

2.2. Connection of microlocal solutions near a hyperbolic fixed point

2.2.1. A model in one dimension

2.2.2. Classical mechanics

2.2.3. Review of semi-classical microlocal analysis

2.2.4. The microlocal Cauchy problem - uniqueness

2.2.5. The microlocal Cauchy problem - transition operator

2.3. Applications to semi-classical resonances

2.3.1. Spectral projection and Schrödinger group

2.3.2. Resonance-free zone for homoclinic trajectories

2.4. Acknowledgment

2.5. Bibliography

Chapter 3. The Sign Exchange Bifurcation in a Family of Linear Hamiltonian Systems

3.1. Statement of problem

3.2. Bifurcation values of γ

3.3. Versal normal forms near the bifurcation values

3.3.1. Normal forms

3.3.2. Linear Hamiltonian Hopf bifurcation γ±

3.3.3. The Switch twist bifurcation at γ+

3.3.4. Sign exchange bifurcation

3.4. Infinitesimally symplectic normal form

3.4.1. Normal form of Xγ at γ±

3.4.2. Normal form of Xγ at γ±

3.5. Global issues

3.5.1. Invariant Lagrange planes

3.5.2. Symplectic signs

3.6. Bibliography

Chapter 4. Dissipation Effect on Local and Global Fluid-Elastic Instabilities

4.1. Introduction

4.2. Local and global stability analyses

4.2.1. Local analysis

4.2.2. Global analysis

4.3. The fluid-conveying pipe: a model problem

4.4. Effect of damping on the local and global stability of the fluid-conveying pipe.

4.4.1. Local stability

4.4.2. Global stability

4.5. Application to energy harvesting

4.6. Conclusion

4.7. Bibliography

Chapter 5. Tunneling, Librations and Normal Forms in a Quantum Double Well with a Magnetic Field

5.1. Introduction

5.2. 1D Landau-Lifshitz splitting formula and its analog for the ground states

5.3. The splitting formula in multi-dimensional case

5.4. Normal forms and complex Lagrangian manifolds

5.4.1. Normal form in the classically allowed and forbidden regions

5.4.2. Complex continuation of integrals

5.4.3. Almost invariant complex Lagrangian manifolds

5.5. Constructing the asymptotics for the eigenfunctions in tunnel problems

5.5.1. Complex WKB-method

5.5.2. WKB-methods with real and pure imaginary phases

5.5.3. Variational methods

5.6. Splitting of the eigenvalues in the presence of magnetic field

5.7. Proof of main theorem (a sketch)

5.7.1. Lifshitz-Herring formula

5.7.2. Instanton splitting formula

5.7.3. Asymptotic behavior of the libration action

5.7.4. Reduction to the 1D splitting problem

5.7.5. Asymptotic behavior of the Floquet exponents

5.7.6. Finishing the proof

5.8. Conclusion

5.9. Acknowledgments

5.10. Bibliography

Chapter 6. Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials

6.1. Introduction

6.2. The model

6.3. Solitons in the first bandgap: the SF nonlinearity

6.3.1. Solution families

6.3.2. Stability of solitons in the first finite bandgap

6.3.3. Bound states of solitons in the first bandgap

6.4. Stability GSs in the second bandgap

6.5. Conclusions

6.6. Bibliography

Chapter 7. Representation of Wave Energy of a Rotating Flow in Terms of the Dispersion Relation

7.1. Introduction

7.2. Lagrangian approach to wave energy

7.3. Kelvin waves.

7.4. Wave energy in terms of the dispersion relation

7.5. Conclusion

7.6. Bibliography

Chapter 8. Determining the Stability Domain of Perturbed Four-Dimensional Systems in 1:1 Resonance

8.1. Introduction

8.1.1. Physical motivation

8.1.2. Setting

8.1.3. Main question and examples

8.2. Methods

8.2.1. Centralizer unfolding

8.2.2. Stability domain

8.2.3. Mapping into the centralizer unfolding

8.3. Examples

8.3.1. Modulation instability

8.3.2. Non-conservative gyroscopic system

8.4. Conclusions

8.5. Bibliography

Chapter 9. Index Theorems for Polynomial Pencils

9.1. Introduction

9.2. Krein signature

9.3. Index theorems for linear pencils and linearized Hamiltonians

9.4. Graphical interpretation of index theorems

9.4.1. Algebraic calculation of Z and Z

9.5. Conclusions

9.6. Acknowledgments

9.7. Bibliography

Chapter 10. Investigating Stability and Finding New Solutions in Conservative Fluid Flows Through Bifurcation Approaches

10.1. Introduction

10.2. Counting positive-energy modes from IVI diagrams

10.3. An approximate prediction for the onset of resonance in 2D vortices

10.4. An example: three corotating vortices

10.4.1. Building a family of solutions from vorticity-preserving rearrangements

10.4.2. Computing signatures for one member of the family

10.4.3. The velocity-impulse diagram

10.4.4. Uncovering bifurcations by introducing imperfections

10.4.5. Counting positive-energy modes from turning points in impulse

10.4.6. Recovering the underlying bifurcation structure

10.4.7. An approximate prediction for resonance

10.5. Comparison with exact eigenvalues and discussion

10.6. Conclusions

10.7. Bibliography

Chapter 11. Evolution Equations for Finite Amplitude Waves in Parallel Shear Flows

11.1. Introduction

11.2. Wave packets.

11.2.1. Conservative systems

11.2.2. Applications to hydrodynamic stability

11.2.3. The Ginzburg-Landau equation

11.3. Critical layer theory

11.3.1. Asymptotic theory of the Orr-Sommerfeld equation

11.3.2. Nonlinear critical layers

11.3.3. The wave packet critical layer

11.4. Nonlinear instabilities governed by integro-differential equations

11.4.1. The zonal wave packet critical layer

11.5. Concluding remarks

11.6. Bibliography

Chapter 12. Continuum Hamiltonian Hopf Bifurcation I

12.1. Introduction

12.2. Discrete Hamiltonian bifurcations

12.2.1. A class of 1 + 1 Hamiltonian multifluid theories

12.2.2. Examples

12.2.3. Comparison and commentary

12.3. Continuum Hamiltonian bifurcations

12.3.1. A class of 2 + 1 Hamiltonian mean field theories

12.3.2. Example of the CHH bifurcation

12.4. Summary and conclusions

12.5. Acknowledgments

12.6. Bibliography

Chapter 13. Continuum Hamiltonian Hopf Bifurcation II

13.1. Introduction

13.2. Mathematical aspects of the continuum Hamiltonian Hopf bifurcation

13.2.1. Structural stability

13.2.2. Normal forms and signature

13.3. Application to Vlasov-Poisson

13.3.1. Structural stability in the space Cn(R) L1(R)

13.3.2. Structural stability in W 1,1

13.3.3. Dynamical accessibility and structural stability

13.4. Canonical infinite-dimensional case

13.4.1. Negative energy oscillator coupled to a heat bath

13.5. Commentary: degeneracy and nonlinearity

13.6. Summary and conclusions

13.7. Acknowledgments

13.8. Bibliography

Chapter 14. Energy Stability Analysis for a Hybrid Fluid-Kinetic Plasma Model

14.1. Introduction

14.2. Stability and the energy-Casimir method

14.3. Planar Hamiltonian hybrid model

14.3.1. Planar hybrid model equations of motion

14.3.2. Hamiltonian structure.

14.3.3. Casimir invariants

14.4. Energy-Casimir stability analysis

14.4.1. Equilibrium variational principle

14.4.2. Stability conditions

14.5. Conclusions

14.6. Acknowledgments

14.7. Appendix A: derivation of hybrid Hamiltonian structure

14.8. Appendix B: Casimir verification

14.9. Bibliography

Chapter 15. Accurate Estimates for the Exponential Decay of Semigroups with Non-Self-Adjoint Generators

15.1. Introduction

15.2. Relevant quantities for sectorial operators

15.3. Natural examples

15.3.1. An example related to linearized equations of fluid mechanics

15.3.2. Kramers-Fokker-Planck operators

15.4. Artificial examples

15.4.1. Adiabatic evolution of quantum resonances in the one-dimensional case

15.4.2. Optimizing the sampling of equilibrium distributions

15.5. Conclusion

15.6. Bibliography

Chapter 16. Stability Optimization for Polynomials and Matrices

16.1. Optimization of roots of polynomials

16.1.1. Root optimization over a polynomial family with a single affine constraint

16.1.2. The root radius

16.1.3. The root abscissa

16.1.4. Examples

16.1.5. Polynomial root optimization with several affine constraints

16.1.6. Variational analysis of the root radius and abscissa

16.1.7. Computing the root radius and abscissa

16.2. Optimization of eigenvalues of matrices

16.2.1. Static output feedback

16.2.2. Numerical methods for non-smooth optimization

16.2.3. Numerical results for some SOF problems

16.2.4. The Diaconis-Holmes-Neal Markov chain

16.2.5. Active derogatory eigenvalues

16.3. Concluding remarks

16.4. Acknowledgments

16.5. Bibliography

Chapter 17. Spectral Stability of Nonlinear Waves in KdV-Type Evolution Equations

17.1. Introduction

17.2. Historical remarks and examples

17.3. Proof of theorem 17.1.

17.4. Generalization of theorem 17.1 for a periodic nonlinear wave.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references at the end of each chapters and index.

Description based on print version record.

ISBN:

9781118577547

111857754X

9781118577608

1118577604

9781118577578

1118577574

OCLC:

865652835