Dieckmann, Simon: Dynamics of patterns in equivariant Hamiltonian partial differential equations. 2017
Content
- Introduction
- 1 Equivariant Hamiltonian Systems
- 1.1 Hamiltonian Ordinary Differential Equations
- 1.2 Abstract Hamiltonian Systems
- 1.3 Partial Differential Equations as Hamiltonian Systems
- 2 Analysis of the Freezing Method
- 2.1 Derivation of the PDAE Formulation
- 2.2 Preliminaries and Spectral Hypotheses
- 2.3 Stability of the PDAE Formulation
- 2.4 Application to the NLS
- 2.5 Application to the NLKG
- 3 Preservation of Solitary Waves and Their Stability
- 3.1 Motivating Examples
- 3.2 Abstract Setting
- 3.3 Positivity Estimates
- 3.4 Existence of Discrete Steady States
- 3.5 Stability of Discrete Steady States
- 3.6 Verification of the Hypotheses
- 4 Truncation and Discretization for the NLS
- 5 Numerical Computations
- 5.1 Nonlinear Schrödinger Equation
- 5.2 Nonlinear Klein-Gordon Equation
- 5.3 Korteweg-de Vries Equation
- Conclusions and Perspectives
- A Auxiliaries