|
Showing 1 - 15 of
15 matches in All Departments
This book on integrable systems and symmetries presents new results
on applications of symmetries and integrability techniques to the
case of equations defined on the lattice. This relatively new field
has many applications, for example, in describing the evolution of
crystals and molecular systems defined on lattices, and in finding
numerical approximations for differential equations preserving
their symmetries. The book contains three chapters and five
appendices. The first chapter is an introduction to the general
ideas about symmetries, lattices, differential difference and
partial difference equations and Lie point symmetries defined on
them. Chapter 2 deals with integrable and linearizable systems in
two dimensions. The authors start from the prototype of integrable
and linearizable partial differential equations, the Korteweg de
Vries and the Burgers equations. Then they consider the best known
integrable differential difference and partial difference
equations. Chapter 3 considers generalized symmetries and conserved
densities as integrability criteria. The appendices provide details
which may help the readers' understanding of the subjects presented
in Chapters 2 and 3. This book is written for PhD students and
early researchers, both in theoretical physics and in applied
mathematics, who are interested in the study of symmetries and
integrability of difference equations.
This volume of the CRM Conference Series is based on a carefully
refereed selection of contributions presented at the "11th
International Symposium on Quantum Theory and Symmetries", held in
Montreal, Canada from July 1-5, 2019. The main objective of the
meeting was to share and make accessible new research and recent
results in several branches of Theoretical and Mathematical
Physics, including Algebraic Methods, Condensed Matter Physics,
Cosmology and Gravitation, Integrability, Non-perturbative Quantum
Field Theory, Particle Physics, Quantum Computing and Quantum
Information Theory, and String/ADS-CFT. There was also a special
session in honour of Decio Levi. The volume is divided into
sections corresponding to the sessions held during the symposium,
allowing the reader to appreciate both the homogeneity and the
diversity of mathematical tools that have been applied in these
subject areas. Several of the plenary speakers, who are
internationally recognized experts in their fields, have
contributed reviews of the main topics to complement the original
contributions.
This volume of the CRM Conference Series is based on a carefully
refereed selection of contributions presented at the "11th
International Symposium on Quantum Theory and Symmetries", held in
Montreal, Canada from July 1-5, 2019. The main objective of the
meeting was to share and make accessible new research and recent
results in several branches of Theoretical and Mathematical
Physics, including Algebraic Methods, Condensed Matter Physics,
Cosmology and Gravitation, Integrability, Non-perturbative Quantum
Field Theory, Particle Physics, Quantum Computing and Quantum
Information Theory, and String/ADS-CFT. There was also a special
session in honour of Decio Levi. The volume is divided into
sections corresponding to the sessions held during the symposium,
allowing the reader to appreciate both the homogeneity and the
diversity of mathematical tools that have been applied in these
subject areas. Several of the plenary speakers, who are
internationally recognized experts in their fields, have
contributed reviews of the main topics to complement the original
contributions.
Based on the third International Conference on Symmetries,
Differential Equations and Applications (SDEA-III), this
proceedings volume highlights recent important advances and trends
in the applications of Lie groups, including a broad area of topics
in interdisciplinary studies, ranging from mathematical physics to
financial mathematics. The selected and peer-reviewed contributions
gathered here cover Lie theory and symmetry methods in differential
equations, Lie algebras and Lie pseudogroups, super-symmetry and
super-integrability, representation theory of Lie algebras,
classification problems, conservation laws, and geometrical
methods. The SDEA III, held in honour of the Centenary of Noether's
Theorem, proven by the prominent German mathematician Emmy Noether,
at Istanbul Technical University in August 2017 provided a
productive forum for academic researchers, both junior and senior,
and students to discuss and share the latest developments in the
theory and applications of Lie symmetry groups. This work has an
interdisciplinary appeal and will be a valuable read for
researchers in mathematics, mechanics, physics, engineering,
medicine and finance.
This book shows how Lie group and integrability techniques,
originally developed for differential equations, have been adapted
to the case of difference equations. Difference equations are
playing an increasingly important role in the natural sciences.
Indeed, many phenomena are inherently discrete and thus naturally
described by difference equations. More fundamentally, in subatomic
physics, space-time may actually be discrete. Differential
equations would then just be approximations of more basic discrete
ones. Moreover, when using differential equations to analyze
continuous processes, it is often necessary to resort to numerical
methods. This always involves a discretization of the differential
equations involved, thus replacing them by difference ones. Each of
the nine peer-reviewed chapters in this volume serves as a
self-contained treatment of a topic, containing introductory
material as well as the latest research results and exercises. Each
chapter is presented by one or more early career researchers in the
specific field of their expertise and, in turn, written for early
career researchers. As a survey of the current state of the art,
this book will serve as a valuable reference and is particularly
well suited as an introduction to the field of symmetries and
integrability of difference equations. Therefore, the book will be
welcomed by advanced undergraduate and graduate students as well as
by more advanced researchers.
The NATO Advanced Research Workshop "Painleve Transcendents, their
Asymp totics and Physical Applications," held at the Alpine Inn in
Sainte-Adele, near Montreal, September 2 -7, 1990, brought together
a group of experts to discuss the topic and produce this volume.
There were 41 participants from 14 countries and 27 lectures were
presented, all included in this volume. The speakers presented
reviews of topics to which they themselves have made important
contributions and also re sults of new original research. The
result is a volume which, though multiauthored, has the character
of a monograph on a single topic. This is the theory of nonlinear
ordinary differential equations, the solutions of which have no
movable singularities, other than poles, and the extension of this
theory to partial differential equations. For short we shall call
such systems "equations with the Painleve property." The search for
such equations was a very topical mathematical problem in the 19th
century. Early work concentrated on first order differential
equations. One of Painleve's important contributions in this field
was to develop simple methods applicable to higher order equations.
In particular these methods made possible a complete analysis of
the equation;; = f(y', y, x), where f is a rational function of y'
and y, with coefficients that are analytic in x. The fundamental
result due to Painleve (Acta Math."
Many physical phenomena are described by nonlinear evolution
equation. Those that are integrable provide various mathematical
methods, presented by experts in this tutorial book, to find
special analytic solutions to both integrable and partially
integrable equations. The direct method to build solutions includes
the analysis of singularities a la Painleve, Lie symmetries leaving
the equation invariant, extension of the Hirota method,
construction of the nonlinear superposition formula. The main
inverse method described here relies on the bi-hamiltonian
structure of integrable equations. The book also presents some
extension to equations with discrete independent and dependent
variables.
The different chapters face from different points of view the
theory of exact solutions and of the complete integrability of
nonlinear evolution equations. Several examples and applications to
concrete problems allow the reader to experience directly the power
of the different machineries involved."
Many physical phenomena are described by nonlinear evolution
equation. Those that are integrable provide various mathematical
methods, presented by experts in this tutorial book, to find
special analytic solutions to both integrable and partially
integrable equations. The direct method to build solutions includes
the analysis of singularities a la Painleve, Lie symmetries leaving
the equation invariant, extension of the Hirota method,
construction of the nonlinear superposition formula. The main
inverse method described here relies on the bi-hamiltonian
structure of integrable equations. The book also presents some
extension to equations with discrete independent and dependent
variables.
The different chapters face from different points of view the
theory of exact solutions and of the complete integrability of
nonlinear evolution equations. Several examples and applications to
concrete problems allow the reader to experience directly the power
of the different machineries involved."
The NATO Advanced Research Workshop "Painleve Transcendents, their
Asymp totics and Physical Applications," held at the Alpine Inn in
Sainte-Adele, near Montreal, September 2 -7, 1990, brought together
a group of experts to discuss the topic and produce this volume.
There were 41 participants from 14 countries and 27 lectures were
presented, all included in this volume. The speakers presented
reviews of topics to which they themselves have made important
contributions and also re sults of new original research. The
result is a volume which, though multiauthored, has the character
of a monograph on a single topic. This is the theory of nonlinear
ordinary differential equations, the solutions of which have no
movable singularities, other than poles, and the extension of this
theory to partial differential equations. For short we shall call
such systems "equations with the Painleve property." The search for
such equations was a very topical mathematical problem in the 19th
century. Early work concentrated on first order differential
equations. One of Painleve's important contributions in this field
was to develop simple methods applicable to higher order equations.
In particular these methods made possible a complete analysis of
the equation;; = f(y', y, x), where f is a rational function of y'
and y, with coefficients that are analytic in x. The fundamental
result due to Painleve (Acta Math."
This book shows how Lie group and integrability techniques,
originally developed for differential equations, have been adapted
to the case of difference equations. Difference equations are
playing an increasingly important role in the natural sciences.
Indeed, many phenomena are inherently discrete and thus naturally
described by difference equations. More fundamentally, in subatomic
physics, space-time may actually be discrete. Differential
equations would then just be approximations of more basic discrete
ones. Moreover, when using differential equations to analyze
continuous processes, it is often necessary to resort to numerical
methods. This always involves a discretization of the differential
equations involved, thus replacing them by difference ones. Each of
the nine peer-reviewed chapters in this volume serves as a
self-contained treatment of a topic, containing introductory
material as well as the latest research results and exercises. Each
chapter is presented by one or more early career researchers in the
specific field of their expertise and, in turn, written for early
career researchers. As a survey of the current state of the art,
this book will serve as a valuable reference and is particularly
well suited as an introduction to the field of symmetries and
integrability of difference equations. Therefore, the book will be
welcomed by advanced undergraduate and graduate students as well as
by more advanced researchers.
Based on the third International Conference on Symmetries,
Differential Equations and Applications (SDEA-III), this
proceedings volume highlights recent important advances and trends
in the applications of Lie groups, including a broad area of topics
in interdisciplinary studies, ranging from mathematical physics to
financial mathematics. The selected and peer-reviewed contributions
gathered here cover Lie theory and symmetry methods in differential
equations, Lie algebras and Lie pseudogroups, super-symmetry and
super-integrability, representation theory of Lie algebras,
classification problems, conservation laws, and geometrical
methods. The SDEA III, held in honour of the Centenary of Noether's
Theorem, proven by the prominent German mathematician Emmy Noether,
at Istanbul Technical University in August 2017 provided a
productive forum for academic researchers, both junior and senior,
and students to discuss and share the latest developments in the
theory and applications of Lie symmetry groups. This work has an
interdisciplinary appeal and will be a valuable read for
researchers in mathematics, mechanics, physics, engineering,
medicine and finance.
Starting from Sophus Lie, the invariance of a differential equation
under its continuous group of symmetries has become a major tool
for solving ordinary and partial differential equations, in
particular, nonlinear ones. The proceedings focus on the
application of these techniques to nonlinear partial differential
equations. The state of the art in this field is presented clearly
in a series of comprehensive lectures. Several lectures on
applications point out the physical importance of such methods.
Starting from Sophus Lie, the invariance of a differential equation
under its continuous group of symmetries has become a major tool
for solving ordinary and partial differential equations, in
particular, nonlinear ones. The proceedings focus on the
application of these techniques to nonlinear partial differential
equations. The state of the art in this field is presented clearly
in a series of comprehensive lectures. Several lectures on
applications point out the physical importance of such methods.
A co-publication of the AMS and Centre de Recherches
Mathématiques. The purpose of this book is to serve as a tool for
researchers and practitioners who apply Lie algebras and Lie groups
to solve problems arising in science and engineering. The authors
address the problem of expressing a Lie algebra obtained in some
arbitrary basis in a more suitable basis in which all essential
features of the Lie algebra are directly visible. This includes
algorithms accomplishing decomposition into a direct sum,
identification of the radical and the Levi decomposition, and the
computation of the nilradical and of the Casimir invariants.
Examples are given for each algorithm. For low-dimensional Lie
algebras this makes it possible to identify the given Lie algebra
completely. The authors provide a representative list of all Lie
algebras of dimension less or equal to 6 together with their
important properties, including their Casimir invariants. The list
is ordered in a way to make identification easy, using only basis
independent properties of the Lie algebras. They also describe
certain classes of nilpotent and solvable Lie algebras of arbitrary
finite dimensions for which complete or partial classification
exists and discuss in detail their construction and properties. The
book is based on material that was previously dispersed in journal
articles, many of them written by one or both of the authors
together with their collaborators. The reader of this book should
be familiar with Lie algebra theory at an introductory level.
Difference equations are playing an increasingly important role in
the natural sciences. Indeed many phenomena are inherently discrete
and are naturally described by difference equations. Phenomena
described by differential equations are therefore approximations of
more basic discrete ones. Moreover, in their study it is very often
necessary to resort to numerical methods. This always involves a
discretization of the differential equations involved, thus
replacing them by difference equations. This book shows how Lie
group and integrability techniques, originally developed for
differential equations, have been adapted to the case of difference
ones. Each of the eleven chapters is a self-contained treatment of
a topic, containing introductory material as well as the latest
research results. The book will be welcomed by graduate students
and researchers seeking an introduction to the field. As a survey
of the current state of the art it will also serve as a valuable
reference.
|
|