Nonlinear Dispersive Equations

Nonlinear Dispersive Equations are partial differential equations that naturally arise in physical settings where dispersion dominates dissipation, notably hydrodynamics, nonlinear optics, plasma physics and BoseEinstein condensates. The topic has traditionally been approached in different ways, from the perspective of modeling of physical phenomena, to that of the theory of partial differential equations, or as part of the theory of integrable systems.
This monograph offers a thorough introduction to the topic, uniting the modeling, PDE and integrable systems approaches for the first time in book form. The presentation focuses on three "universal" families of physically relevant equations endowed with a completely integrable member: the BenjaminOno, DaveyStewartson, and KadomtsevPetviashvili equations. These asymptotic models are rigorously derived and qualitative properties such as soliton resolution are studied in detail in both integrable andnon-integrable models. Numerical simulations are presented throughout to illustrate interesting phenomena. By presenting and comparing results from different fields, the book aims to stimulate scientific interactions and attract new students and researchers to the topic. To facilitate this, the chapters can be read largely independently of each other and the prerequisites have been limited to introductory courses in PDE theory.

First book uniting the modeling, PDE, and integrable systems points of view Presents recent results on Kortewegde Vries, DaveyStewartson and BenjaminOno equations Includes many numerical simulations to illustrate key features

Autorentext

Christian Klein is Professor of mathematical physics at the Université de Bourgogne in Dijon, France, and a senior member of the Institut Universitaire de France. He works on nonlinear dispersive PDEs, numerical approaches, integrable systems, applied algebraic geometry and general relativity. His main interest is the numerical study of zones of rapid oscillations in the solutions to nonlinear dispersive equations, so-called dispersive shock waves, and a loss of regularity, a so-called blow-up of the solutions.Jean-Claude Saut is Emeritus Professor in the Laboratoire de Mathématiques of the Université Paris-Saclay. He works on the analysis of nonlinear dispersive equations and on their rigorous derivation as asymptotic models of general systems. His recent works concern a general class of Boussinesq systems, the analysis of weakly dispersive perturbations of the Burgers equation, and higher order models in the modulation regime of water waves.

Klappentext

Nonlinear Dispersive Equations are partial differential equations that naturally arise in physical settings where dispersion dominates dissipation, notably hydrodynamics, nonlinear optics, plasma physics and Bose Einstein condensates. The topic has traditionally been approached in different ways, from the perspective of modeling of physical phenomena, to that of the theory of partial differential equations, or as part of the theory of integrable systems. This monograph offers a thorough introduction to the topic, uniting the modeling, PDE and integrable systems approaches for the first time in book form. The presentation focuses on three "universal" families of physically relevant equations endowed with a completely integrable member: the Benjamin Ono, Davey Stewartson, and Kadomtsev Petviashvili equations. These asymptotic models are rigorously derived and qualitative properties such as soliton resolution are studied in detail in both integrable andnon-integrable models. Numerical simulations are presented throughout to illustrate interesting phenomena. By presenting and comparing results from different fields, the book aims to stimulate scientific interactions and attract new students and researchers to the topic. To facilitate this, the chapters can be read largely independently of each other and the prerequisites have been limited to introductory courses in PDE theory.

Inhalt
Acronyms.- Glossary.- 1 General Introduction.- 2 Generalities and Basic Facts.- 3 BenjaminOno and Intermediate Long Wave Equations: Modeling, IST and PDE.- 4 DaveyStewartson and Related Systems.- 5 KadomtsevPetviashvili and Related Equations.- 6 NovikovVeselov and Derivative Nonlinear Schrödinger Equations.- Index.