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.

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The following theorem is a noncommutative analog of the famous Hirota identities. 13. For n ≥ 2 Tn+1 (φ) = Tn (D2 φ) − Tn (Dφ) · [(Tn−1 (D2 φ)−1 − Tn (φ)−1 ]−1 · Tn (Dφ). The proof follows from the noncommutative Sylvester identity [174]. 3 Noncommutative orthogonal polynomials The results described in this subsection were obtained in [175]. Let S0 , S1 , S2 , . . be elements of a skew field R and x be a commutative variable. Define a sequence of elements Pi (x) ∈ R[x], i = 0, 1, . . by setting P0 = S0 and Sn S Pn (x) = n−1 ...

84) Therefore, the Cauchy-type integral defines a sectionally continuous function which is regular oﬀ the contour and continuous when tending to the contour both from above and from below. 10 The Riemann–Hilbert problem 25 k C+ λ Re k Cλ-iε Fig. 2. Contour of integration (bold line) when the point k crosses the real axis axis to the point λ − iǫ, ǫ > 0. To evaluate φ+ (λ − iǫ), we deform the contour γ as in Fig. 2. z ℓ − λ + iǫ Note that the last term does not contain the factor 1/2 because the contour encloses the point λ − iǫ almost entirely.

36) as a generalized Burgers equation. 18. Suppose an invertible function ϕ is a solution to the linear diﬀerential equation D0 ϕ = Lϕ. 36). 19. 37) for an invertible function ϕ. 38) is a solution of the equation ˜ ˜ ψ. 39) The last statement accomplishes the proof of the Matveev theorem for diﬀerential polynomials [314] in its non-Abelian version. 35) gives a representation of the transformed operator in terms of the generalized Bell polynomials. 41) k = 0, . . , N − 1. 5 Iterations and quasideterminants via Darboux transformation Here we would like to revisit the non-Abelian iterated DT formulas following the ideas of the pioneering paper of Matveev [313], where the basic formulas were derived.