By Evgeny V. Doktorov, Sergey B. Leble
This monograph systematically develops and considers the so-called "dressing process" for fixing differential equations (both linear and nonlinear), a method to generate new non-trivial options for a given equation from the (perhaps trivial) answer of an identical or comparable equation. the first subject matters of the dressing technique lined listed below are: the Moutard and Darboux changes stumbled on in XIX century as utilized to linear equations; the BÃncklund transformation in differential geometry of surfaces; the factorization strategy; and the Riemann-Hilbert challenge within the shape proposed by means of Shabat and Zakharov for soliton equations, plus its extension by way of the d-bar formalism.Throughout, the textual content exploits the "linear event" of presentation, with certain recognition given to the algebraic elements of the most mathematical structures and to functional ideas of acquiring new recommendations. numerous linear equations of classical and quantum mechanics are solved by way of the Darboux and factorization equipment. An extension of the classical Darboux modifications to nonlinear equations in 1+1 and 2+1 dimensions, in addition to its factorization, also are mentioned intimately. what is extra, the applicability of the neighborhood and non-local Riemann-Hilbert problem-based strategy and its generalization by way of the d-bar technique are illustrated through quite a few nonlinear equations.