Hydrodynamic scales of integrable many-body systems
The words ``hydrodynamic scales'' in the title suggests that this is a study of fluids and their motions. Instead, in this book the author considers the subject in a much broader context, and describes it as based on the ``observation that the motion of a large assembly of strongly interacting particles is constrained by local conservation laws''. The author's intention is to provide an introduction to integrable systems with many degrees of freedom. Indeed, the author states in his overview that all integrable multi-particle systems are structurally alike on a hydrodynamic scale.\N\NGeneralized hydrodynamics is concerned with integrable many particle models where the number of conserved fields is proportional to system size.\N\NThe author begins by establishing the Toda lattice as a guiding backbone, one that is peculiar because it is integrable for every system size, and suggests that the conventional definition of integrability might need to be reconsidered, especially in the sense that the usual definition of integrability for quantum systems fails. He suggests that -- for hydrodynamic considerations -- a system should be called integrable if it has an infinite number of linearly independent local conservation laws.\N\NBesides the Toda lattice, the author examines models for the Dyson Brownian motion, a fluid system with hot rods, Calogero fluids, and the Ablowitz-Ladik discretized nonlinear Schrödinger equation.\N\NThe Toda lattice helps provide a guiding theme, in part because it shares the dynamical features of the Korteweg-deVries equation and the Toda fluid, then later leads to considerations of the quantum mechanical Lieb-Liniger delta-Bose gas and the quantum Toda lattice. The author also deals with random initial data described by generalized Gibbs ensembles with parameters of slow spatial variation. This hydrodynamic scale, as he says, should be understood in the context of the ballistic Euler scale.\N\NIntegrable microscopic particle models are truly very diverse, and a central theme of the book is to identify their structural similarity on hydrodynamic scales.\N\NThis is a book best suited to those with broad expertise in dynamical systems of many varieties. The author pulls together questions from a large collection of systems and approaches, and he assumes considerable familiarity in this area.
Ablowitz-Ladik discretized Schrödinger equation
scientific article; zbMATH DE number 7852571