Vladimir E. Zakharov

STABILITY OF PERIODIC WAVES OF FINITE AMPLITUDE ON THE SURFACE OF A DEEP FLUID

We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables: the shape of the surface η(r, t) and the hydrodynamic potential ψ(r, t) at the surface are expressed in terms of these variables. By introducing canonical variables, we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion [3,4]. The resuits of the rest of the paper are also easily applicable to the general case.In section 2, using a method similar to van der Pohls method, we obtain simplified equations describing nonlinear waves in the small amplitude approximation. These equations are particularly simple if we assume that the wave packet is narrow. The equations have an exact solution which approximates a periodic wave of finite amplitude.In section 3 we investigate the instability of periodic waves of finite amplitude. Instabilities of two types are found. The first type of instability is destructive instability, similar to the destructive instability of waves in a plasma [5, 6], In this type of instability, a pair of waves is simultaneously excited, the sum of the frequencies of which is a multiple of the frequency of the original wave. The most rapid destructive instability occurs for capillary waves and the slowest for gravitational waves. The second type of instability is the negative-pressure type, which arises because of the dependence of the nonlinear wave velocity on the amplitude; this results in an unbounded increase in the percentage modulation of the wave. This type of instability occurs for nonlinear waves through any media in which the sign of the second derivative in the dispersion law with respect to the wave number (d2ω/dk2) is different from the sign of the frequency shift due to the nonlinearity.As announced by A. N. Litvak and V. I. Talanov [7], this type of instability was independently observed for nonlinear electromagnetic waves.

What is integrability

Why Are Certain Nonlinear PDEs Both Widely Applicable and Integrable?.- Summary.- 1. The Main Ideas in an Illustrative Context.- 2. Survey of Model Equations.- 3. C-Integrable Equations.- 4. Envoi.- Addendum.- References.- Painleve Property and Integrability.- 1. Background.- 1.1 Motivation.- 1.2 History.- 2. Integrability.- 3. Riccati Example.- 4. Balances.- 5. Elliptic Example.- 6. Augmented Manifold.- 7. Argument for Integrability.- 8. Separability.- References.- Integrability.- 1. Integrability.- 2. Introduction to the Method.- 2.1 The WTC Method for Partial Differential Equations.- 2.2 The WTC Method for Ordinary Differential Equations.- 2.3 The Nature of ?.- 2.4 Truncated Versus Non-truncated Expansions.- 3. The Integrable Henon-Heiles System: A New Result.- 3.1 The Lax Pair.- 3.2 The Algebraic Curve and Integration of the Equations of Motion.- 3.3 The Role of the Rational Solutions in the Painleve Expansions.- 4. A Mikhailov and Shabat Example.- 5. Some Comments on the KdV Hierarchy.- 6. Connection with Symmetries and Algebraic Structure.- 7. Integrating the Nonintegrable.- References.- The Symmetry Approach to Classification of Integrable Equations.- 1. Basic Definitions and Notations.- 1.1 Classical and Higher Symmetries.- 1.2 Local Conservation Laws.- 1.3 PDEs and Infinite-Dimensional Dynamical Systems.- 1.4 Transformations.- 2. The Burgers Type Equations.- 2.1 Classification in the Scalar Case.- 2.2 Systems of Burgers Type Equations.- 2.3 Lie Symmetries and Differential Substitutions.- 3. Canonical Conservation Laws.- 3.1 Formal Symmetries.- 3.2 The Case of a Vector Equation.- 3.3 Integrability Conditions.- 4. Integrable Equations.- 4.1 Scalar Third Order Equations.- 4.2 Scalar Fifth Order Equations.- 4.3 Schrodinger Type Equations.- Historical Remarks.- References.- Integrability of Nonlinear Systems and Perturbation Theory.- 1. Introduction.- 2. General Theory.- 2.1 The Formal Classical Scattering Matrix in the Solitonless Sector of Rapidly Decreasing Initial Conditions.- 2.2 Infinite-Dimensional Generalization of Poincares Theorem. Definition of Degenerative Dispersion Laws.- 2.3 Properties of Degenerative Dispersion Laws.- 2.4 Properties of Singular Elements of a Classical Scattering Matrix. Properties of Asymptotic States.- 2.5 The Integrals of Motion.- 2.6 The Integrability Problem in the Periodic Case. Action-Angle Variables.- 3. Applications to Particular Systems.- 3.1 The Derivation of Universal Models.- 3.2 Kadomtsev-Petviashvili and Veselov-Novikov Equations.- 3.3 Davey-Stewartson-Type Equations. The Universality of the Davey-Stewartson Equation in the Scope of Solvable Models.- 3.4 Applications to One-Dimensional Equations.- Appendix I.- Proofs of the Local Theorems (of Uniqueness and Others from Sect.2.3).- Appendix II.- Proof of the Global Theorem for Degenerative Dispersion Laws.- Conclusion.- References.- What Is an Integrable Mapping?.- 1. Integrable Polynomial and Rational Mappings.- 1.1 Polynomial Mapping of C: What Is Its Integrability?.- 1.2 Commuting Polynomial Mappings of ?N and Simple Lie Algebras.- 1.3 Commuting Rational Mappings of ?Pn.- 1.4 Commuting Cremona Mappings of ?2.- 1.5 Euler-Chasles Correspondences and the Yang-Baxter Equation.- 2. Integrable Lagrangean Mappings with Discrete Time.- 2.1 Hamiltonian Theory.- 2.2 Heisenberg Chain with Classical Spins and the Discrete Analog of the C. Neumann System.- 2.3 The Billiard in Quadrics.- 2.4 The Discrete Analog of the Dynamics of the Top.- 2.5 Connection with the Spectral Theory of the Difference Operators: A Discrete Analogue of the Moser-Trubowitz Isomorphism.- Appendix A.- Appendix B.- References.- The Cauchy Problem for the KdV Equation with Non-Decreasing Initial Data.- 1. Reflectionless Potentials.- 2. Closure of the Sets B(??2).- 3. The Inverse Problem.- References.

Soliton stability in plasmas and hydrodynamics

Abstract The stability of solitons is reviewed for nonlinear conservative media. The main attention is paid to the description of the methods: perturbation theory, inverse scattering transform, Lyapunov method. Its applications are demonstrated in detail for the nonlinear Schrodinger equation, the KdV equation, and their generalizations. Applications to problems in plasma physics and hydrodynamics are considered.

Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction

Abstract Explicit analytic formulae for two-dimensional solitons are given. It is proved that, unlike one-dimensional solitons, two-dimensional ones do not interact at all.

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.

Localized sources of water vapour on the dwarf planet (1) Ceres

The ‘snowline’ conventionally divides Solar System objects into dry bodies, ranging out to the main asteroid belt, and icy bodies beyond the belt. Models suggest that some of the icy bodies may have migrated into the asteroid belt. Recent observations indicate the presence of water ice on the surface of some asteroids, with sublimation a potential reason for the dust activity observed on others. Hydrated minerals have been found on the surface of the largest object in the asteroid belt, the dwarf planet (1) Ceres, which is thought to be differentiated into a silicate core with an icy mantle. The presence of water vapour around Ceres was suggested by a marginal detection of the photodissociation product of water, hydroxyl (ref. 12), but could not be confirmed by later, more sensitive observations. Here we report the detection of water vapour around Ceres, with at least 1026 molecules being produced per second, originating from localized sources that seem to be linked to mid-latitude regions on the surface. The water evaporation could be due to comet-like sublimation or to cryo-volcanism, in which volcanoes erupt volatiles such as water instead of molten rocks.

On the integrability of classical spinor models in two-dimensional space-time

Well known classical spinor relativistic-invariant two-dimensional field theory models, including the Gross-Neveu, Vaks-Larkin-Nambu-Jona-Lasinio and some other models, are shown to be integrable by means of the inverse scattering problem method. These models are shown to be naturally connected with the principal chiral fields on the symplectic, unitary and orthogonal Lie groups. The respective technique for construction of the soliton-like solutions is developed.

Hamiltonian approach to the description of non-linear plasma phenomena

Abstract This review discusses the methods of description of non-linear processes. We demonstrate the usefulness of the Hamiltonian approach to these problems. We show the existence of Hamiltonian structures for a number of plasma situations. The choice of normal variables results in a standard form of equations for all kinds of problems. The actual physics involved changes only dispersion laws and the structure of the matrix elements. This approach makes it possible to consider a number of problems in a unique way. We discuss the stability of monochromatic waves and the statistical description of a plasma. The connection between decay and modulational instability growth rates and matrix elements is demonstrated. The standard form of the equations enables us to introduce a statistical description in a very simple way. We discuss the usual kinetic wave equations and their generalization for inhomogeneous turbulence and turbulence excited by a coherent pump. We pay special attention to the problem of Langmuir turbulence. The average dynamical equations are deduced in a consistent way and we present a detailed discussion of the limits of this description.

Yang-Mills equations as inverse scattering problem

Non-linear self-duality equations F v = rpm are shown to be conditions of compatibility of two linear equations.rAll the N-instanton fields are constructed explicity.

The Inverse Scattering Method

The present article is devoted to a systematic exposition of different methods of obtaining equations which are integrable by the inverse scattering method. The exposition begins with elementary methods and concludes with the method of dressing operator families. Many results (this refers both to the elementary part and in particular to the method of dressing) are original and published for the first time.