ABSTRACT
This chapter summarizes the work on the nonlinear dynamics of pattern formation in a broad range of dissipative and Hamiltonian systems. It focuses on four central issues: the laws of motion of fields and surfaces; the consequences of geometrical or topological constraints; the existence of variational principles; and the mechanisms of pattern selection. These are illustrated in the context of three classes of problems: the motion of shapes governed by bending elasticity, interface dynamics in two-dimensional systems with nonlocal interactions, and integrable curve dynamics. The chapter discusses the mathematical structure of curve motion in the plane, and a dynamical formalism for shape evolution with global geometric constraints. It considers some two-dimensional systems with emphasis on the complex configuration space which exists when short-range and long-range interactions compete in the presence of constraints. The chapter shows the mathematics of some integrable soliton system which shed light on the geometry of Euler's equation.
