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  • © 2007

A Dressing Method in Mathematical Physics

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  • Unifying concept for solving linear and nonlinear equations of mathematical physics

  • Current status os research in the field

Part of the book series: Mathematical Physics Studies (MPST, volume 28)

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  • ISBN: 978-1-4020-6140-0
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Table of contents (10 chapters)

  1. Front Matter

    Pages I-XXIV
  2. Mathematical preliminaries

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 1-30
  3. Factorization and classical Darboux transformations

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 31-65
  4. From elementary to twofold elementary Darboux transformation

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 67-108
  5. Dressing chain equations

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 109-140
  6. Dressing in 2+1 dimensions

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 141-160
  7. Applications of dressing to linear problems

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 161-198
  8. Important links

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 199-223
  9. Dressing via local Riemann–Hilbert problem

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 225-275
  10. Dressing via nonlocal Riemann–Hilbert problem

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 277-317
  11. Generating solutions via ∂ problem

    • Evgeny V. Doktorov, Sergey B. Leble
    Pages 319-353
  12. Back Matter

    Pages 355-378

About this book

The monograph is devoted to the systematic presentation of the so called "dressing method" for solving differential equations (both linear and nonlinear) of mathematical physics. The essence of the dressing method consists in a generation of new non-trivial solutions of a given equation from (maybe trivial) solution of the same or related equation. The Moutard and Darboux transformations discovered in XIX century as applied to linear equations, the Bäcklund transformation in differential geometry of surfaces, the factorization method, the Riemann-Hilbert problem in the form proposed by Shabat and Zakharov for soliton equations and its extension in terms of the d-bar formalism comprise the main objects of the book. Throughout the text, a generally sufficient "linear experience" of readers is exploited, with a special attention to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions. Various linear equations of classical and quantum mechanics are solved by the Darboux and factorization methods. An extension of the classical Darboux transformations to nonlinear equations in 1+1 and 2+1 dimensions, as well as its factorization are discussed in detail. The applicability of the local and non-local Riemann-Hilbert problem-based approach and its generalization in terms of the d-bar method are illustrated on various nonlinear equations.

Keywords

  • Potential
  • Schrödinger equation
  • Soliton
  • Transformation
  • algebra
  • mathematical physics
  • operator
  • wave equation

Reviews

From the reviews:

"This book is a collection of the work of the two authors over many years. … The authors connect nicely the newest contributions and the 150 years of history of Darboux transformations and solvable potentials of the linear Schrödinger equation. … The book can be recommended to graduate students … ." (Dmitry E. Pelinovsky, Mathematical Reviews, Issue 2008 k)

"The term ‘dressing’ generally implies a construction that contains a transformation from a simpler state to a more advanced state of a system. … It provides new insight into this significant research area of great importance. … The book can be of very great use to graduate students and professional mathematicians and physicists. … It provides a good introduction to the dressing method, and it is definitely a good job." (Ma Wen-Xiu, Zentralblatt MATH, Vol. 1142, 2008)

Authors and Affiliations

  • Institute of Physics, Minsk, Belarus

    Evgeny V. Doktorov

  • University of Technology, Gdansk, Poland

    Sergey B. Leble

Bibliographic Information

Buying options

eBook
USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-1-4020-6140-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 159.99
Price excludes VAT (USA)
Hardcover Book
USD 169.99
Price excludes VAT (USA)