Alan C. Newell

Solitons in mathematics and physics

The History of the Soliton Derivation of the Korteweg-de Vries, Nonlinear Schrodinger and Other Important and Canonical Equations of Mathematical Physics Soliton Equation Families and Solution Methods The -Function, the Hirota Method, the Painleve Property and Backlund Transformations for the Korteweg-de Vries Family of Soliton Equations Connecting Links among the Miracles of Soliton Mathematics.

Finite bandwidth, finite amplitude convection

The main purpose of this work is to show how a continuous finite bandwidth of modes can be readily incorporated into the description of post-critical Rayleigh-Benard convection by the use of slowly varying (in space and time) amplitudes. Previous attempts have used a multimodal discrete analysis. We show that in addition to obtaining results consistent with the discrete mode approach, there is a larger class of stable and realizable solutions. The main feature of these solutions is that the amplitude and wave-number of the motion is that of the most unstable mode almost everywhere, but, depending on external and initial conditions, the roll couplets in different parts of space may be 180° out of phase. The resulting discontinuities are smoothed by hyperbolic tangent functions. In addition, it is clear that the mechanism for propagating spatial nonuniformities is diffusive in character.

An exact solution for a derivative nonlinear Schrödinger equation

A method of solution for the ’’derivative nonlinear Schrodinger equation’’ iqt=−qxx±i (q*q2)x is presented. The appropriate inverse scattering problem is solved, and the one‐soliton solution is obtained, as well as the infinity of conservation laws. Also, we note that this equation can also possess ’’algebraic solitons.’’

Monodromy- and spectrum-preserving deformations. I

A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformations and with classical and recent work on monodromy preserving deformations are discussed. Specific new results include the reduction of the general initial value problem for the Painlevé equations of the second type and a special case of the third type to a system of linear singular integral equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painlevé equation family is shown to be −d/dx ln(Δ+/Δ−), where Δ+ and Δ− are determinants. We also demonstrate that each of these equations is an exactly integrable Hamiltonian system. The basic ideas presented here are applicable to a broad class of ordinary and partial differential equations; additional results will be presented in a sequence of future papers.

A weak turbulence theory for incompressible mhd

We derive a weak turbulence formalism for incompressible MHD. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. We find energy spectra solution of the kinetic equations. The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schro¨dinger equation

The nonlinear Schrodinger (NLS) equation iΨt + ∇2Ψ + α⋎Ψ⋎sΨ = 0 is a canonical and universal equation which is of major importance in continuum mechanics, plasma physics and optics. This paper argues that much of the observed solution behavior in the critical case sd = 4, where d is dimension and s is the order of nonlinearity, can be understood in terms of a combination of weak turbulence theory and condensate and collapse formation. The results are derived in the broad context of a class of Hamiltonian systems of which NLS is a member, so that the reader can gain a perspective on the ingredients important for the realization of the various equilibrium spectra, thermodynamic, pure Kolmogorov and combinations thereof. We also present time-dependent, self-similar solutions which describe the relaxation of the system towards these equilibrium states. We show that the number of particles lost in an individual collapse event is virtually independent of damping. Our numerical simulation of the full governing equations is the first to show the validity of the weak turbulence approximation. We also present a mechanism for intermittency which should have widespread application. It is caused by strongly nonlinear collapse events which are nucleated by a flow of particles towards the origin in wavenumber space. These highly organized events result in a cascade of particle number towards high wavenumbers and give rise to an intermittency and a behavior which violates many of the usual Kolmogorov assumptions about the loss of statistical information and the statistical independence of large and small scales. We discuss the relevance of these ideas to hydrodynamic turbulence in the conclusion.

A weak turbulence theory for incompressible magnetohydrodynamics

We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0 ê‖. Numerically and analytically, we find energy spectra E± ∼ k± ⊥ , such that n+ +n− = −4, where E± are the spectra of the Elsässer variables z± = v ± b in the two-dimensional case (k‖ = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

Convection patterns in large aspect ratio systems

Abstract A theory, which should have widespread application, is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio Rayleigh-Benard boxes. For case of presentation the theory is developed using model equations. Wavenumber selection, the shape of patterns, stability and the time dependence resulting from long wavelength instabilities are discussed. The effects of adding the local mean drift important in low Prandtl number situations are investigated. Our theory includes the notion of the Busse stability balloon, reduces near critical values of the stress parameter to the Newell-Whitehead-Segel equations, and contains the Pomeau-Manneville phase equation. It also gives a detailed description of the way in which the field amplitude, which is slaved to the phase gradient away from threshold, becomes an independent order parameter near the critical point.

Wave turbulence and intermittency

In the early 1960s, it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive properties of the waves and the large separation of linear and nonlinear time scales. One is thereby led to kinetic equations for the redistribution of spectral densities via three- and four-wave resonances together with a nonlinear renormalization of the frequency. The kinetic equations have equilibrium solutions which are much richer than the familiar thermodynamic, Fermi–Dirac or Bose–Einstein spectra and admit in addition finite flux (Kolmogorov–Zakharov) solutions which describe the transfer of conserved densities (e.g. energy) between sources and sinks. There is much one can learn from the kinetic equations about the behavior of particular systems of interest including insights in connection with the phenomenon of intermittency. What we would like to convince you is that what we call weak or wave turbulence is every bit as rich as the macho turbulence of 3D hydrodynamics at high Reynolds numbers and, moreover, is analytically more tractable. It is an excellent paradigm for the study of many-body Hamiltonian systems which are driven far from equilibrium by the presence of external forcing and damping. In almost all cases, it contains within its solutions behavior which invalidates the premises on which the theory is based in some spectral range. We give some new results concerning the dynamic breakdown of the weak turbulence description and discuss the fully nonlinear and intermittent behavior which follows. These results may also be important for proving or disproving the global existence of solutions for the underlying partial differential equations. Wave turbulence is a subject to which many have made important contributions. But no contributions have been more fundamental than those of Volodja Zakharov whose 60th birthday we celebrate at this meeting. He was the first to appreciate that the kinetic equations admit a far richer class of solutions than the fluxless thermodynamic solutions of equilibrium systems and to realize the central roles that finite flux solutions play in non-equilibrium systems. It is appropriate, therefore, that we call these Kolmogorov–Zakharov (KZ) spectra.

Theory and simulation on the threshold of water breakdown induced by focused ultrashort laser pulses

A comprehensive model is developed for focused pulse propagation in water. The model incorporates self-focusing, group velocity dispersion, and laser-induced breakdown in which an electron plasma is generated via cascade and multiphoton ionization processes. The laser-induced breakdown is studied first without considering self-focusing to give a breakdown threshold of the light intensity, which compares favorably with existing experimental results. The simple study also yields the threshold dependence on pulse duration and input spot size, thus providing a framework to view the results of numerical simulations of the full model. The simulations establish the breakdown threshold in input power and reveal qualitatively different behavior for picoand femto-second pulses. For longer pulses, the cascade process provides the breakdown mechanism, while for shorter pulses the cooperation between the self-focusing and the multiphoton plasma generation dominates the breakdown threshold.