"The Nonlinear Schrödinger Equation as Both a PDE and a Dynamical System", (with David Cai and Kenneth T. R. McLaughlin), to appear Handbook of Dynamical Systems (2001).

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 whenever 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 Schrödinger equations, and many others.

In this survey article, we choose a class of nonlinear Schrödinger 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 the most 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.

"Chaotic and Turbulent Behavior of Unstable One-Dimensional Nonlinear Dispersive Waves", (with David Cai), Journal of Mathematical Physics, 41 nb. 6, June 2000

Abstract:
In this article we use one-dimensional nonlinear Schrödinger equations (NLS) to illustrate chaotic and turbulent behavior of nonlinear dispersive waves. It begins with a brief summary of properties of NLS with focusing and defocusing nonlinearities. In this summary we stress the role of the modulational instability in the formation of solitary waves and homoclinic orbits, and in the generation of temporal chaos and of spatiotemporal chaos for the nonlinear waves. Dispersive wave turbulence for a class of one-dimensional NLS equations is then described in detail-emphasizing distinctions between focusing and defocusing cases, the role of spatially localized, coherent structures, and their interaction with resonant waves in setting up the cycles of energy transfer in dispersive wave turbulence through direct and inverse cascades. In the article we underline that these simple NLS models provide precise and demanding tests for the closure theories of dispersive wave turbulence. In the conclusion we emphasize the importance of effective stochastic representations for the prediction of transport and other macroscopic behavior in such deterministic chaotic nonlinear wave systems.

"Spectral Bifurcations in Dispersive Wave Turbulence", (with David Cai, Andrew J. Majda, and Esteban G. Tabak) PNAS, 96, No. 25, 14216-14221 (Dec 1999).

Abstract:
Dispersive wave turbulence is studied numerically for a class of one-dimensional nonlinear wave equations. Both deterministic and random (white noise in time) forcings are studied. Four distinct stable spectra are observed - the direct and inverse cascades of weak turbulence (WT) theory, thermal equilibrium, and a fourth spectrum (MMT; Majda, McLaughlin, Tabak). Each spectrum can describe long-time behavior, and each can be only metastable (with quite diverse lifetimes) - depending on details of nonlinearity, forcing, and dissipation. Cases of a long-lived MMT transient state decaying to a state with WT spectra, and vice-versa, are displayed. In the case of freely decaying turbulence, without forcing, both cascades of weak turbulence are observed. These WT states constitute the clearest and most striking numerical observations of WT spectra to date - over four decades of energy, and three decades of spatial, scales. Numerical experiments that study details of the composition, coexistence, and transition between spectra are then discussed, including: (i) for deterministic forcing, sharp distinctions between focusing and defocusing nonlinearities, including the role of long wavelength instabilities, localized coherent structures, and chaotic behavior; (ii) the role of energy growth in time to monitor the selection of MMT or WT spectra; (iii) a second manifestation of the MMT spectrum as it describes a self-similar evolution of the wave, without temporal averaging; (iv) coherent structures and the evolution of the direct and inverse cascades; and (v) nonlocality (in k-space) in the transferral process.

"The Semiclassical Limit of the Defocusing NLS Hierarchy" (with D. Levermore and S. Jin), Comm. Pure Appl. Math. LII, 613-654 (1999).

Abstract:
We establish the semiclassical limit of the one-dimensional defocusing cubic nonlinear Schrödinger (NLS) equation. Complete integrability is exploited to obtain a global characterization of the weak limits of the entire NLS hierarchy of conserved densities as the field evolves from the reflectionless initial data under all the associated commuting flows. Consequently, this also establishes the zero-dispersion limit of the modified Korteweg-de Vries equation that resides in that hierarchy. We have adapted and clarified the strategy introduced by Lax and Levermore to study the zero-dispersion limit of the Korteweg-de Vries equation, expanding it to treat entire integrable hierarchies and strengthening the limits obtained. A crucial role is played by the convexity of the underlying log-determinant with respect to the times associated with the commuting flows.

"A One-Dimensional Model for Dispersive Wave Turbulence", (with A. Majda and E. Tabak), J. Nonlinear Science 7, 9-44 (1997)

Abstract:
A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number.

It is established that the quasi-Gaussian random phase hypothesis of weak turbulence theory is an excellent approximation in the numerical statistical steady state. Nevertheless, the predictions of weak turbulence theory fail and yield a much flatter ( ) spectrum compared with the steeper ( ) spectrum observed in the numerical statistical steady state. The reasons for the failure of weak turbulence theory in this context are elucidated here. Finally, an inertial range closure and scaling theory is developed which successfully predicts the inertial range exponents observed in the numerical statistical steady states.

"Homoclinic Orbits for Pde's" (with J. Shatah), eds. R. Spigler, S. Venakides (Proceedings of conference held in Venice in honor of P. D. Lax and L. Nirenberg 70^th birthdays) AMS 281-299 (1996).

Abstract:
In this lecture, the construction of whiskered tori and homoclinic orbits is summarized for the sine-Gordon equation as an example of such constructions for completely integrable nonlinear wave equations. The construction uses Backlund transform ations which are realized through Lax Pairs. The persistence of these homoclinic orbits under small perturbations of the wave equations, of both dissipative and conservative type, is then established. An analytic perturbation method based on time dependent scattering theory, and Fredholm theory, is used to establish persistence. The estimates are given in space-time function spaces, with a certain time decay, which is required for the existence of a homoclinic orbit.

"Persistent Homoclinic Orbits for a Perturbed Nonlinear Schrödinger Equation" (with Y. Li, J. Shatah and S. Wiggins), Comm. Pure Appl. Math. XLIX, 1175-1255 (1996).

Abstract:
The persistence of homoclinic orbits for certain perturbations of the integrable nonlinear Schrödinger equation under even periodic boundary conditions is established. More specifically, the existence of a symmetric pair of homoclinic orbits is established for the perturbed NLS equation through an argument that combines Melnikov analysis with a geometric singular perturbation theory for the PDE.

"Whiskered Tori and Chaotic Behavior in Nonlinear Waves", Proceedings of International Congress of Mathematicians, Zürich, 1484-1493 (1995)

"Dispersive Wave Turbulence in One Dimension", (with David Cai, Andrew J. Majda, and Esteban G. Tabak ), Physica D 26, 55-81 (1995).

Abstract:
In this article, we study numerically a one-dimensional model of dispersive wave turbulence. The article begins with a description of the model which we introduced earlier, followed by a concise summary of our p revious results about it. In those previous studies, in addition to the spectra of weak turbulence (WT) theory, we also observed another distinct spectrum (the "MMT spectrum"). Our new results, presented here, include: (i) A detailed description of coexistence of spectra at distinct spatial scales, and the transitions between them at different temporal scales; (ii) The existence of a stable MMT front in k-space which separates the WT cascades from the dissipation range, for various forms of strong damping including "selective dissipation"; (iii) The existence of turbulent cycles in the one-dimensional model with focusing nonlinearity, induced by the interaction of spatially localized coherent structures with the resonant quartets of dispersive wave radiation; (iv) The detailed composition of these turbulent cycles including the self-similar formation of focusing events (distinct in the forced and freely decaying cases), and the transport by the WT direct and inverse cascades of excitations between spatial scales. This one-dimensional model admits a very precise and detailed realization of these turbulent cycles and their components. Our numerical experiments demonstrate that a complete theory of dispersive wave turbulence will require a full description of the turbulent field over all spatial scales (including those of the forcing and dissipation), and over extremely long times (as the nonlinear turnover time becomes very long in the weakly nonlinear limit). And, in the focusing case, a complete theory must also incorporate the interaction of localized coherent structures with resonant radiation.