A Dressing Method in Mathematical Physics [Paperback]

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.

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 Schr?dinger equation. 4.4 Darboux lƒ¼