Softcover ISBN:  9781470472627 
Product Code:  STML/100 
List Price:  $59.00 
Individual Price:  $47.20 
eBook ISBN:  9781470473105 
Product Code:  STML/100.E 
List Price:  $59.00 
Individual Price:  $47.20 
Softcover ISBN:  9781470472627 
eBook: ISBN:  9781470473105 
Product Code:  STML/100.B 
List Price:  $118.00 $88.50 
Softcover ISBN:  9781470472627 
Product Code:  STML/100 
List Price:  $59.00 
Individual Price:  $47.20 
eBook ISBN:  9781470473105 
Product Code:  STML/100.E 
List Price:  $59.00 
Individual Price:  $47.20 
Softcover ISBN:  9781470472627 
eBook ISBN:  9781470473105 
Product Code:  STML/100.B 
List Price:  $118.00 $88.50 

Book DetailsStudent Mathematical LibraryVolume: 100; 2023; 347 ppMSC: Primary 35; 14; 37;
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 halfcentury 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 algebrogeometric 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 \(\wp\)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 thoughtprovoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of Mathematica® 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 noncommutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition.
ReadershipUndergraduate and graduate students and researchers interested in theory of solitons.

Table of Contents

Chapters

Differential equations

Developing PDE intuition

The story of solitons

Elliptic curves and KdV traveling waves

KdV $n$solitons and $\tau $functions

Multiplying and factoring differential operators

Eigenfunctions and isospectrality

Lax form for KdV and other soliton equations

The KP equation and bilinear KP equation

$\Gamma _{2,4}$ and the bilinear KP equation

Pseudodifferential operators and the KP hierarchy

$\Gamma {k,n}$ and the bilinear KP hierarchy

Concluding remarks

Mathematica guide

Complex numbers

Ideas for independent projects


Additional Material

Reviews

This is the second edition of a book first reviewed here. It continues to be a very attractive text. Its subject is at the crossroads of an amazing collection of mathematical areas including partial differential equations, elliptic curves, and the algebra of differential operators, as well as striking applications in science and engineering. All of this is made accessible to undergraduates with multivariable calculus and linear algebra. The author sets out the history of the area clearly and motivates his approach very well.
Bill Satzer, MAA Reviews


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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 halfcentury 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 algebrogeometric 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 \(\wp\)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 thoughtprovoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of Mathematica® 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 noncommutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition.
Undergraduate and graduate students and researchers interested in theory of solitons.

Chapters

Differential equations

Developing PDE intuition

The story of solitons

Elliptic curves and KdV traveling waves

KdV $n$solitons and $\tau $functions

Multiplying and factoring differential operators

Eigenfunctions and isospectrality

Lax form for KdV and other soliton equations

The KP equation and bilinear KP equation

$\Gamma _{2,4}$ and the bilinear KP equation

Pseudodifferential operators and the KP hierarchy

$\Gamma {k,n}$ and the bilinear KP hierarchy

Concluding remarks

Mathematica guide

Complex numbers

Ideas for independent projects

This is the second edition of a book first reviewed here. It continues to be a very attractive text. Its subject is at the crossroads of an amazing collection of mathematical areas including partial differential equations, elliptic curves, and the algebra of differential operators, as well as striking applications in science and engineering. All of this is made accessible to undergraduates with multivariable calculus and linear algebra. The author sets out the history of the area clearly and motivates his approach very well.
Bill Satzer, MAA Reviews