Chapter 12 The nonlinear Schrödinger equation as both a PDE and a dynamical system

Abstract Nonlinear dispersive wave equations provide excellent examples of infinite dimensional dynamical systems which possess diverse and fascinating phenomena including solitary waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclinic orbits, the existence of temporally chaotic waves in deterministic systems, dispersive turbulence and the propagation of spatiotemporal chaos. Nonlinear dispersive waves occur throughout physical and natural systems wherever dissipation is weak. Important applications include nonlinear optics and long distance communication devices such as transoceanic optical fibers, waves in the atmosphere and the ocean, and turbulence in plasmas. Examples of nonlinear dispersive partial differential equations include the Korteweg-de Vries equation, nonlinear Klein-Gordon equations, nonlinear Schroumldinger equations, and many others. In this survey article, we choose a class of nonlinear Schroumldinger equations (NLS) as prototypal examples, and we use members of this class to illustrate the qualitative phenomena described above. Our viewpoint is one of partial differential equations on the one hand, and infinite dimensional dynamical systems on the other. In particular, we will emphasize global qualitative information about the solutions of these nonlinear partial differential equations which can be obtained with the methods and geometric perspectives of dynamical systems theory. The article begins with a brief description of a spectacular success in pde of this dynamical systems viewpoint - the complete understanding of the remarkable properties of the soliton through the realization that certain nonlinear wave equations are completely integrable Hamiltonian systems. This complete integrability follows from a deep connection between certain special nonlinear wave equations (such as the NLS equation with cubic nonlinearity in one spatial dimension) and the linear spectral theory of certain differential operators (the “Zakharov-Shabat” or “Dirac” operator in the NLS case). From this connection the “inverse spectral transform” has been developed and used to represent integrable nonlinear waves. These representations have provided a full solution of the Cauchy initial value problem for several types of boundary conditions, a thorough understanding of the remarkable properties of the soliton, descriptions of quasi-periodic wave trains, and descriptions of the formation and propagation of oscillations as slowly varying nonlinear wavetrains. In addition, more recent developments are described, including: (i) the formation of singularities and their relationship to dispersive turbulence; (ii) weak turbulence theory; (iii) the persistence of periodic, quasi-periodic, and homoclinic solutions, by methods including normal forms for pde's, Melnikov measurements, and geometric singular perturbation theory; (iv) temporal and spatiotemporal chaos; (v) long-time and small dispersion behavior of integrable waves through Riemann-Hilbert spectral methods. For each topic, the description is necessarily brief; however, references will be selected which should enable the interested reader to obtain more mathematical detail.