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A Dressing Method in Mathematical Physics

  • Evgeny V. Doktorov
  • Sergey B. Leble

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

Table of contents

  1. Front Matter
    Pages I-XXIV
  2. Evgeny V. Doktorov, Sergey B. Leble
    Pages 1-30
  3. Evgeny V. Doktorov, Sergey B. Leble
    Pages 31-65
  4. Evgeny V. Doktorov, Sergey B. Leble
    Pages 67-108
  5. Evgeny V. Doktorov, Sergey B. Leble
    Pages 109-140
  6. Evgeny V. Doktorov, Sergey B. Leble
    Pages 141-160
  7. Evgeny V. Doktorov, Sergey B. Leble
    Pages 161-198
  8. Evgeny V. Doktorov, Sergey B. Leble
    Pages 199-223
  9. Evgeny V. Doktorov, Sergey B. Leble
    Pages 225-275
  10. Evgeny V. Doktorov, Sergey B. Leble
    Pages 277-317
  11. Evgeny V. Doktorov, Sergey B. Leble
    Pages 319-353
  12. Back Matter
    Pages 355-378

About this book

Introduction

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

Authors and affiliations

  • Evgeny V. Doktorov
    • 1
  • Sergey B. Leble
    • 2
  1. 1.Institute of PhysicsMinskBelarus
  2. 2.University of TechnologyGdanskPoland

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