By Francesco Calogero, A. Degasperis
Read Online or Download Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations PDF
Best functional analysis books
A panorama of harmonic analysis
Tracing a direction from the earliest beginnings of Fourier sequence via to the newest learn A landscape of Harmonic research discusses Fourier sequence of 1 and a number of other variables, the Fourier remodel, round harmonics, fractional integrals, and singular integrals on Euclidean area. The climax is a attention of principles from the viewpoint of areas of homogeneous variety, which culminates in a dialogue of wavelets.
This e-book introduces most crucial facets of recent research: the idea of degree and integration and the speculation of Banach and Hilbert areas. it's designed to function a textual content for first-year graduate scholars who're already accustomed to a few research as given in a e-book just like Apostol's Mathematical research.
Lineare Funktionalanalysis: Eine anwendungsorientierte Einführung
Die lineare Funktionalanalysis ist ein Teilgebiet der Mathematik, das Algebra mit Topologie und research verbindet. Das Buch führt in das Fachgebiet ein, dabei bezieht es sich auf Anwendungen in Mathematik und Physik. Neben den vollständigen Beweisen aller mathematischen Sätze enthält der Band zahlreiche Aufgaben, meist mit Lösungen.
- Handbook of Mathematical Functions
- Fourier analysis : an introduction
- Complex variables
- Discovering Evolution Equations with Applications, Volume 1-Deterministic Equations (Chapman & Hall CRC Applied Mathematics & Nonlinear Science)
Extra resources for Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations
Example text
1 17 Direct spectral problem solution of (1) characterized by the asymptotic boundary conditions (3a) (3b) +( x , k ) T( k)exp( -i kx), x+ - 00 , +( x , k ) exp( -i kx ) +R ( k ) exp(i kx ) , + -+ x+ + co This asymptotic behaviour is clearly consistent with (2), and it identifies uniquely the eigenfunction +(x, k), as well as the transmission coefficient T( k ) and the reflection coefficient R( k). , N To each of these eigenvalues there corresponds a solution f , ( x ) of (l), uniquely identified by the asymptotic boundary condition (5) lim [exp( p,x)f,(x)] = 1, n= 1 , 2 , .
Then associate with u ( ' ) ( x ) a new function, say u(12)(x),obtained applying the Backlund transformation BT2; and with d 2 ) ( x ) a function d2')( x) obtained applying the Backlund transformation BT 1. 1 the spectral space, (21) R ( ' ~ k) () = R ( ~ ' )k() , which is instead trivially valid, since (10) implies that with Thus the commutativity of Backlund transformations, in the spectral space, is merely the commutativity of ordinary multiplication. Here we see once more the advantage, in terms of simplicity, to work in the spectral space rather than in configuration space.
1. Clearly this behaviour constitutes a distinctive and remarkable feature of this class of nonlinear evolution equations (and of other classes solvable by analogous technique, based on different spectral problems; see below). The physical, or more generally natural, interpretation of these results-for instance the emergence of the solitons as localized individual entities-does of course depend on the particular physical, or more generally natural, phenomenon that the nonlinear evolution equation under consideration is supposed, perhaps approximately, to represent; be it in fluid dynamics or in demography, in solid state physics or in epidemiology, in the investigation of signal transmission through nervous fibres or of models of elementary particles or of plasma disturbances.
