Attractors of nonlinear Hamiltonian partial differential equations
Abstract
This is a survey of the theory of attractors of nonlinear Hamiltonian partial differential equations since its appearance in 1990. Included are results on global attraction to stationary states, to solitons, and to stationary orbits, together with results on adiabatic effective dynamics of solitons and their asymptotic stability, and also results on numerical simulation. The results obtained are generalized in the formulation of a new general conjecture on attractors of $G$ -invariant nonlinear Hamiltonian partial differential equations. This conjecture suggests a novel dynamical interpretation of basic quantum phenomena: Bohr transitions between quantum stationary states, de Broglie's wave-particle duality, and Born's probabilistic interpretation.
- Publication:
-
Russian Mathematical Surveys
- Pub Date:
- February 2020
- DOI:
- 10.1070/RM9900
- arXiv:
- arXiv:1907.06998
- Bibcode:
- 2020RuMaS..75....1K
- Keywords:
-
- 35B41;
- 35B40;
- 35C08;
- Hamiltonian equations;
- nonlinear partial differential equations;
- wave equation;
- Maxwell equations;
- Klein-Gordon equation;
- limiting amplitude principle;
- limiting absorption principle;
- attractor;
- steady states;
- soliton;
- stationary orbits;
- adiabatic effective dynamics;
- symmetry group;
- Lie group;
- Schrödinger equation;
- quantum transitions;
- wave-particle duality;
- Mathematical Physics;
- Nonlinear Sciences - Pattern Formation and Solitons;
- Quantum Physics
- E-Print:
- 69 pages, 8 figures. arXiv admin note: text overlap with arXiv:1409.2009