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A Dressing Method in Mathematical Physics Evgeny V. Doktorov

A Dressing Method in Mathematical Physics By Evgeny V. Doktorov

A Dressing Method in Mathematical Physics by Evgeny V. Doktorov

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Summary

This monograph systematically develops and considers the so-called dressing method for solving differential equations (both linear and nonlinear), a means to generate new non-trivial solutions for a given equation from the (perhaps trivial) solution of the same or related equation.

A Dressing Method in Mathematical Physics Summary

A Dressing Method in Mathematical Physics by Evgeny V. Doktorov

This monograph systematically develops and considers the so-called dressing method for solving differential equations (both linear and nonlinear), a means to generate new non-trivial solutions for a given equation from the (perhaps trivial) solution of the same or related equation. Throughout, the text exploits the linear experience of presentation, with special attention given to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions.

A Dressing Method in Mathematical Physics 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 Schroedinger 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)

Table of Contents

INTRODUCTION 1 Mathematical preliminaries 1.1 Intertwine relations. Ladder operators 1.2 Factorization of matrices. 1.3 Factorization of l -matrix. 2 Factorization and dressing 2.1 Left and right division of ordinary differential operators. Bell polynomials. 2.2 Generalized Bell polynomials. 2.3 Division and factorization of differential operators. Generalized Riccati equations. 2.4 Darboux transformation. Generalized Burgers equations. 2.5 Darboux transformations in associative ring with automorphism. Quasideterminants. 2.6 Joint covariance of equations and nonlinear problems. 2.7 Example. Nonabelian Hirota system. 2.8 Second example. Nahm equation. 2.9 On symmetry and supersymmetry. 3 Darboux transformations 3.1 Gauge transformations and general definition of DT. 3.2 Basic notations: algebraic objects. 3.3 Zakharov - Shabat equations for two projectors. Elementary DT. 3.4 Elementary and binary Darboux Transformations for ZS equations with three projectors. 3.5 General case. Elementary and binary Darboux transformations. 3.6 The limit case - analog of Schlesinger transformation. 3.7 N-wave equations. 3.8 Higher combinations. Hirota-Satsuma (integrable CKdV) and KdV-MKdV equations. 3.9 Infinitesimal transforms for iterated DTs. 3.10 Geometric aspect. 4 Applications in linear problems 4.1 Integrable potentials in quantum mechanics. 4.2 Darboux transformations in continuous spectrum. Scattering problem. 4.3 Radial Schroedinger equation. 4.4 Darboux transformations and potentials in multidimensions. 4.5 Zero-range potentials, dressing and electron-molecule scattering. 4.5 Linear Darbouxauto-transformation. 4.6 One-dimensional Dirac equation. 4.7 Non-stationary problems. 4.8 Dressing in classical mechanics. One-dimensional problems. 4.9 Poisson form of dynamics equations. Darboux theorem for classical evolution. 5 Dressing chain equations 5.1 Scalar case. The Boussinesq equation. 5.2 Chain equations for noncommutative field formulation. 5.3 The example of Zakharov-Shabat problem. 5.4 General operator. Stationary equations as eigenvalue problems and chains. 5.5 Finite closures of the chain equations. Finite-gap solutions. 6 The dressing in 2+1 6.1 Laplace transformations. 6.2 Combined Darboux-Laplace transforms. 6.3 Moutard transformation. 6.4 Goursat transformation. 6.5 The addition of the lower level to spectra of matrix and scalar components of d=2 SUSY Hamiltonian. 7 Important links 7.1 Bilinear formalism. The Hirota method. 7.2 Backlund transformations. 7.3 Singular manifold method. 7.4 General ideas for integral equations. 7.5 The Zakharov-Shabat theory. 7.6 Reduction conditions. 8 Dressing via the local Riemann-Hilbert problem 8.1 The RH problem and generation of new solutions. 8.2 The nonlinear Shchroedinger equation. 8.3 The modified NLS equation. 8.4 The Ablowitz-Ladik equation. 8.5 Three wave resonant interaction equations. 8.6 Homoclinic orbits by the dressing method. 9 Dressing via the non-local RH problem 9.1 The Benjamin-Ono equation. 9.2 The Kadomtsev-Petviashvili I equation. 9.3 The Davey-Stewartson I equation. 9.4 The modified Kadomtsev-Petviashvili I equation. 10 Generating new solutions via the d-bar problem 10.1 Elements of the

Additional information

NPB9781402061387
9781402061387
1402061382
A Dressing Method in Mathematical Physics by Evgeny V. Doktorov
Evgeny V. Doktorov
Mathematical Physics Studies
New
Hardback
Springer-Verlag New York Inc.
2007-05-30
383
N/A
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