Spatio-Temporal Chaos, Solitons and NLS

Abstract

Spatio-Temporal Chaos, Solitons and NLS elaborates on the conjecture that computational dynamical systems best perform near the edge of chaos. Following this view, this Chapter presents several fields from nonlinear dynamics that are related to quantum neural computation, including classical and quantum chaos, turbulence and solitons, with the special treatment to nonlinear Schrödinger equation (NLS, the basic model of quantum neural networks). It includes the following sections:

• 4.1 Reaction-Diffusion Processes and Ricci Flow

  This section gives a unique treatment of various reaction-diffusion processes, reactive neurodynamics and dissipative evolution, based on geometrical Ricci flow.

• 4.2 Turbulence and Chaos in PDEs

  This section elaborates on turbulence and spatio-temporal chaos in nonlinear partial differential equations.

• 4.3 Quantum Chaos and Its Control

  This section compares classical and quantum chaos, with emphasize to its optimal control.

• 4.4 Solitons

  This section first gives a brief history of solitons, followed by modern Lie-Poisson bracket methods and their physiological applications.

• 4.5 Dispersive Wave Equations and Stability of Solitons

  This section gives a mathematical treatment of Korteveg-deVries solitons and similar dispersive wave equations, followed by physical inverse scattering methods.

• 4.6 Nonlinear Schrödinger Equation (NLS)

  This section gives mathematical, physical and numerical treatment of this most important equation in the quantum neural computation.