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Large numbers of studies of the KdV equation have appeared since the pioneering paper by Gardner, Greene, Kruskal, and Miura in 1967. Most of those works have employed the inverse spectral method for 1D Schrödinger operators or an advanced Fourier analysis. Although algebraic approaches have been discovered by Hirota–Sato and Marchenko independently, those have not been fully investigated and analyzed. The present book offers a new approach to the study of the KdV equation, which treats decaying initial data and oscillating data in a unified manner. The author’s method is to represent the tau functions introduced by Hirota–Sato and developed by Segal–Wilson later in terms of the Weyl–Titchmarsh functions (WT functions, in short) for the underlying Schrödinger operators. The main result is stated by a class of WT functions satisfying some of the asymptotic behavior along a curve approaching the spectrum of the Schrödinger operators at +∞ in an order of -(n-1/2) for the nth KdV equation. This class contains many oscillating potentials (initial data) as well as decaying ones. Especially bounded smooth ergodic potentials are included, and under certain conditions on the potentials, the associated Schrödinger operators have dense point spectrum. This provides a mathematical foundation for the study of the soliton turbulence problem initiated by Zakharov, which was the author’s motivation for extending the class of initial data in this book. A large class of almost periodic potentials is also included in these ergodic potentials. P. Deift has conjectured that any solutions to the KdV equation starting from nearly periodic initial data are almost periodic in time. Therefore, our result yields a foundation for this conjecture. For the reader’s benefit, the author has included here (1) a basic knowledge of direct and inverse spectral problem for 1D Schrödinger operators, including the notion of the WT functions; (2) Sato’s Grassmann manifold method revised by Segal–Wilson; and (3) basic results of ergodic Schrödinger operators.

Functional analysis --- Operational research. Game theory --- Probability theory --- Mathematical physics --- waarschijnlijkheidstheorie --- stochastische analyse --- functies (wiskunde) --- wiskunde --- fysica --- kansrekening --- Mathematical physics. --- Probabilities. --- Functional analysis. --- Mathematical Physics. --- Probability Theory. --- Functional Analysis. --- Equacions diferencials no lineals

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A broad range of phenomena in science and technology can be described by non-linear partial differential equations characterized by systems of conservation laws with source terms. Well known examples are hyperbolic systems with source terms, kinetic equations, and convection-reaction-diffusion equations. This book collects research advances in numerical methods for hyperbolic balance laws and kinetic equations together with related modelling aspects. All the contributions are based on the talks of the speakers of the Young Researchers’ Conference “Numerical Aspects of Hyperbolic Balance Laws and Related Problems”, hosted at the University of Verona, Italy, in December 2021.

Computer science—Mathematics. --- Mathematics—Data processing. --- Neural networks (Computer science). --- Mathematical Applications in Computer Science. --- Computational Mathematics and Numerical Analysis. --- Mathematical Models of Cognitive Processes and Neural Networks. --- Artificial neural networks --- Nets, Neural (Computer science) --- Networks, Neural (Computer science) --- Neural nets (Computer science) --- Artificial intelligence --- Natural computation --- Soft computing --- Equacions diferencials no lineals --- Equacions en derivades parcials --- Solucions numèriques

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Differential equations, Partial. --- Differential equations, Nonlinear. --- Nonlinear differential equations --- Nonlinear theories --- Partial differential equations --- Equacions en derivades parcials --- Equacions diferencials no lineals --- Teories no lineals --- Equacions de Painlevé --- Sistemes no lineals --- EDPs --- Equació diferencial en derivades parcials --- Equacions diferencials en derivades parcials --- Equacions diferencials parcials --- Equacions diferencials --- Dispersió (Matemàtica) --- Equació d'ona --- Equació de Dirac --- Equació de Fokker-Planck --- Equació de Schrödinger --- Equacions de Navier-Stokes --- Equacions de Hamilton-Jacobi --- Equacions de Maxwell --- Equacions de Monge-Ampère --- Equacions de Von Karman --- Equacions diferencials el·líptiques --- Equacions diferencials hiperbòliques --- Equacions diferencials parabòliques --- Equacions diferencials parcials estocàstiques --- Funcions harmòniques --- Laplacià --- Problema de Cauchy --- Problema de Neumann --- Teoria espectral (Matemàtica) --- Equacions de Von Kármán

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Mathematical physics. --- Differential equations, Partial. --- Mathematics --- Data processing. --- Partial differential equations --- Physical mathematics --- Physics --- Equacions diferencials no lineals --- Equacions en derivades parcials --- EDPs --- Equació diferencial en derivades parcials --- Equacions diferencials en derivades parcials --- Equacions diferencials parcials --- Equacions diferencials --- Dispersió (Matemàtica) --- Equació d'ona --- Equació de Dirac --- Equació de Fokker-Planck --- Equació de Schrödinger --- Equacions de Navier-Stokes --- Equacions de Hamilton-Jacobi --- Equacions de Maxwell --- Equacions de Monge-Ampère --- Equacions de Von Kármán --- Equacions diferencials el·líptiques --- Equacions diferencials hiperbòliques --- Equacions diferencials parabòliques --- Equacions diferencials parcials estocàstiques --- Funcions harmòniques --- Laplacià --- Problema de Cauchy --- Problema de Neumann --- Teoria espectral (Matemàtica) --- Teories no lineals --- Equacions de Painlevé --- Sistemes no lineals

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Biology --- Human medicine --- Biomedical engineering --- Biophysics --- Medical physics --- Génie biomédical --- Biophysique --- Physique médicale --- Periodicals --- Périodiques --- Nonlinear theories --- Biophysical Phenomena. --- Biophysics. --- Medical physics. --- Nonlinear theories. --- Niet-lineaire dynamica. --- Niet-lineaire systemen. --- Biologie. --- Geneeskunde. --- Nonlinear problems --- Nonlinearity (Mathematics) --- Health physics --- Health radiation physics --- Medical radiation physics --- Radiotherapy physics --- Radiation therapy physics --- Biological physics --- Biophysical Phenomenon --- Biophysical Process --- Biophysical Concepts --- Biophysical Processes --- Biophysical Concept --- Concept, Biophysical --- Concepts, Biophysical --- Phenomena, Biophysical --- Phenomenon, Biophysical --- Process, Biophysical --- Processes, Biophysical --- Chaos Theory. --- Biology. --- Biomedicine. --- Calculus --- Mathematical analysis --- Mathematical physics --- Physics --- Medical sciences --- Biofísica. --- Física mèdica. --- Teories no lineals. --- No linealitat (Matemàtica) --- Problemes no lineals --- Anàlisi funcional no lineal --- Anàlisi matemàtica --- Càlcul --- Física matemàtica --- Caos (Teoria de sistemes) --- Equacions diferencials no lineals --- Ones no lineals --- Oscil·lacions no lineals --- Sistemes no lineals --- Solitons --- Anàlisi de sistemes --- Biofísica --- Física --- Radiologia mèdica --- Ultrasons en medicina --- Biologia física --- Física biològica --- Biologia --- Absorció (Fisiologia) --- Biologia molecular --- Biònica --- Biomagnetisme --- Biomecànica --- Enginyeria biomèdica --- Física mèdica --- Interfícies biològiques --- Reologia (Biologia)