Softcover ISBN: 978-1-4704-7262-7
Student Mathematical Library Volume: 100
2023; 347 pp
Solitons are nonlinear waves which behave like interacting particles. When first proposed in the 19th century, leading mathematical physicists denied that such a thing could exist. Now they are regularly observed in nature, shedding light on phenomena like rogue waves and DNA transcription. Solitons of light are even used by engineers for data transmission and optical switches. Furthermore, unlike most nonlinear partial differential equations, soliton equations have the remarkable property of being exactly solvable. Explicit solutions to those equations provide a rare window into what is possible in the realm of nonlinearity.
Glimpses of Soliton Theory reveals the hidden connections discovered over the last half-century that explain the existence of these mysterious mathematical objects. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant explanation of something seemingly miraculous.
Assuming only multivariable calculus and linear algebra, the book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass ?
-functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Hierarchy, and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians.
Notable features of the book include: careful selection of topics and detailed explanations to make the subject accessible to undergraduates, numerous worked examples and thought-provoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of MathematicaR to facilitate computation and animate solutions.
The second edition refines the exposition in every chapter, adds more homework exercises and projects, updates references, and includes new examples involving non-commutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition.
Differential equations
Developing PDE intuition
The story of solitons
Elliptic curves and KdV traveling waves
KdV n
-solitons and Ñ
-functions
Multiplying and factoring differential operators
Eigenfunctions and isospectrality
Lax form for KdV and other soliton equations
The KP equation and bilinear KP equation
¡2,4
and the bilinear KP equation
Pseudo-differential operators and the KP hierarchy
¡k,n
and the bilinear KP hierarchy
Concluding remarks
Mathematica guide
Complex numbers
Ideas for independent projects
Undergraduate and graduate students and researchers interested in theory of solitons.
Hardcover ISBN: 978-1-4704-7245-0
Product Code: GSM/230
Expected availability date: July 09, 2023
Graduate Studies in Mathematics, Volume: 230
As the title of the book indicates, this is primarily a book on partial differential equations (PDEs) with two definite slants: toward inverse problems and to the inclusion of fractional derivatives. The standard paradigm, or direct problem, is to take a PDE, including all coefficients and initial/boundary conditions, and to determine the solution. The inverse problem reverses this approach asking what information about coefficients of the model can be obtained from partial information on the solution. Answering this question requires knowledge of the underlying physical model, including the exact dependence on material parameters.
The last feature of the approach taken by the authors is the inclusion of fractional derivatives. This is driven by direct physical applications: a fractional derivative model often allows greater adherence to physical observations than the traditional integer order case.
The book also has an extensive historical section and the material that can be called "fractional calculus" and ordinary differential equations with fractional derivatives. This part is accessible to advanced undergraduates with basic knowledge on real and complex analysis. At the other end of the spectrum, lie nonlinear fractional PDEs that require a standard graduate level course on PDEs.
Graduate students and researchers interested in Partial Differential Equations.
Softcover ISBN: 978-1-4704-6804-0
Product Code: CONM/783
Contemporary Mathematics Volume: 783
2023; 153 pp
MSC: Primary 20; 22; 53; 68;
This volume contains the proceedings of the virtual workshop on Computational Aspects of Discrete Subgroups of Lie Groups, held from June 14 to June 18, 2021, and hosted by the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island.
The major theme deals with a novel domain of computational algebra: the design, implementation, and application of algorithms based on matrix representation of groups and their geometric properties. It is centered on computing with discrete subgroups of Lie groups, which impacts many different areas of mathematics such as algebra, geometry, topology, and number theory. The workshop aimed to synergize independent strands in the area of computing with discrete subgroups of Lie groups, to facilitate solution of theoretical problems by means of recent advances in computational algebra.
David Gabai, Robert Meyerhoff, Nathaniel Thurston and Andrew Yarmola - Enumerating Kleinian Groups
Willem A. de Graaf - Exploring Lie theory with GAP
A. S. Detinko, D. L. Flannery and A. Hulpke - Freeness and S
-arithmeticity of rational Mobius groups
Jane Gilman - Computability Models: Algebraic, Topological and Geometric Algorithms
William M. Goldman - Compact components of planar surface group representations
Alexander Hulpke - Proving infinite index for a subgroup of matrices
Michael Kapovich - Geometric algorithms for discreteness and faithfulness
Michael Kapovich, Alla Detinko and Alex Kontorovich - List of problems on discrete subgroups of Lie groups and their computational aspects
Alice Mark, Julien Paupert and David Polletta - Picard modular groups generated by complex reflections
J. Maxwell Riestenberg - Verifying the Straight-and-spaced Condition
T. N. Venkataramana - Unipotent Generators for Arithmetic Groups
1st Edition - July 12, 2023
Paperback ISBN: 9780443187056
Partial Differential Equations and Applications: A Bridge for Students and Researchers in Applied Sciences offers a unique approach to this key subject by connecting mathematical principles to the latest research advances in select topics. Beginning with very elementary PDEs, such as classical heat equations, wave equations and Laplace equations, the book focuses on concrete examples. It gives students basic skills and techniques to find explicit solutions for partial differential equations. As it progresses, the book covers more advanced topics such as the maximum principle and applications, Greenfs representation, Schauderfs theory, finite-time blowup, and shock waves. By exploring these topics, students gain the necessary tools to deal with research topics in their own fields, whether proceeding in math or engineering areas.
Graduate students and research mathematicians interested in computational algebra, geometry, topology, and their applications.