A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed The relationship of the scattering theory and Backlund transformations is brought out In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems

The Inverse scattering transform fourier analysis for nonlinear problems

In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation.

In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.

Integrals of Nonlinear Equations of Evolution and Solitary Waves

Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation u$\_{t}$ + u$\_{x}$ + uu$\_{x}$ - u$\_{xxt}$ = 0, (a) whose solution u(x,t) is considered in a class of real nonperiodic functions defined for -$\infty $ < x < $\infty $, t $\geq $ 0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation u$\_{t}$ + u$\_{x}$ + uu$\_{x}$ + u$\_{xxx}$ = 0, (b) with which (a) is to be compared in various ways. It is contended that (a) is in important respects the preferable model, obviating certain problematical aspects of (b) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics. In section 2 the origins and immediate properties of equations (a) and (b) are discussed in general terms, and the comparative shortcomings of (b) are reviewed. In the remainder of the paper (section section 3, 4) - which can be read independently of the preceding discussion - an exact theory of (a) is developed. In section 3 3 the existence of classical solutions is proved; and following our main result, theorem 1, several extensions and sidelights are presented. In section 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of (a). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of (a) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of section 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in section 3 is established.

Model Equations for Long Waves in Nonlinear Dispersive Systems

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the i...

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle

(1895). XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 39, No. 240, pp. 422-443.

/pdf/xli-on-the-change-of-form-of-long-waves-advancing-in-a-1mqrzu8aza.pdf

D. J. de Korteweg

Papers

XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves