Attractors of Dissipative Wave Equations

Our aim in this chapter is to study some nonlinear wave equations and a nonlinear Schrödinger equation, the Ginzburg—Landau equation. The wave equations that we consider are the sine—Gordon equation, a nonlinear wave equation of relativisitic quantum mechanics, and some nonlinear vibration equations in solid mechanics that involve fourth-order differential operators in space variables. Strictly speaking, the nonlinear Schrödinger equation is an evolution equation of the first order in time, and it is studied by the methods of Chapter III; however, this equation is related to wave phenomena and from the physical point of view has some properties in common with wave equations.