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Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs, Second Edition
 
Alex Kasman College of Charleston, Charleston, SC
Softcover ISBN:  978-1-4704-7262-7
Product Code:  STML/100
List Price: $59.00
Individual Price: $47.20
Sale Price: $38.35
eBook ISBN:  978-1-4704-7310-5
Product Code:  STML/100.E
List Price: $59.00
Individual Price: $47.20
Sale Price: $38.35
Softcover ISBN:  978-1-4704-7262-7
eBook: ISBN:  978-1-4704-7310-5
Product Code:  STML/100.B
List Price: $118.00 $88.50
Sale Price: $76.70 $57.53
Click above image for expanded view
Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs, Second Edition
Alex Kasman College of Charleston, Charleston, SC
Softcover ISBN:  978-1-4704-7262-7
Product Code:  STML/100
List Price: $59.00
Individual Price: $47.20
Sale Price: $38.35
eBook ISBN:  978-1-4704-7310-5
Product Code:  STML/100.E
List Price: $59.00
Individual Price: $47.20
Sale Price: $38.35
Softcover ISBN:  978-1-4704-7262-7
eBook ISBN:  978-1-4704-7310-5
Product Code:  STML/100.B
List Price: $118.00 $88.50
Sale Price: $76.70 $57.53
  • Book Details
     
     
    Student Mathematical Library
    Volume: 1002023; 347 pp
    MSC: 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 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 \(\wp\)-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 thought-provoking 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 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.

    Readership

    Undergraduate 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
    • Pseudo-differential operators and the KP hierarchy
    • $\Gamma {k,n}$ and the bilinear KP hierarchy
    • Concluding remarks
    • Mathematica guide
    • Complex numbers
    • Ideas for independent projects
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 1002023; 347 pp
MSC: 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 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 \(\wp\)-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 thought-provoking 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 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.

Readership

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
  • Pseudo-differential 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
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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