By Evgeny V. Doktorov, Sergey B. Leble

This monograph systematically develops and considers the so-called "dressing technique" for fixing differential equations (both linear and nonlinear), a way to generate new non-trivial strategies for a given equation from the (perhaps trivial) resolution of an identical or similar equation. all through, the textual content exploits the "linear adventure" of presentation, with distinct awareness given to the algebraic features of the most mathematical buildings and to useful principles of acquiring new solutions.

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42) and the formula for the DT D(φN −1 , . . ” Recall that Crum [94] used the Jacobi identity to prove the determinant formulas for the iterated DT for Abelian entries. 22. The non-Abelian algorithm to exclude sm s hints at the definition of quasideterminants as a function of submatrices apm (compare with the (m) results in Sect. 1). Namely, it is enough to change ψp → apm . 42) (with inserted sm ) and, next, linked to the DT for the potentials ak [N ]. 19): n N −1 n a[N ]k ψ (N +k) + ψ[N ]t = k=0 (sm ψ (m) )(k) .

An1 . . anj . . a1n ... . ain . . . 76) For a 2×2 block matrix A = (aij ), i, j = 1, 2, there are four quasideterminants: |A|11 = a11 − a12 · a−1 22 · a21 , |A|12 = a12 − a11 · a−1 21 · a22 , |A|21 = a21 − a22 · a−1 12 · a11 , |A|22 = a22 − a21 · a−1 11 · a12 . We see that each of the quasideterminants |A|11 , |A|12 , |A|21 , and |A|22 is defined whenever the corresponding elements a22 , a21 , a12 , and a11 are invertible. For a generic n × n matrix (in the sense that all square submatrices of A are invertible) there exist n2 quasideterminants of A.

2) for left Bell polynomials and + Bn+ (s) = −DBn−1 (s) + s, n = 1, 2, . . 3) for right Bell polynomials with the “initial condition” B0 (s) = e. 2. If an element ϕ ∈ G satisfies the equation Dϕ = sϕ, then Dn ϕ = Bn (s)ϕ, n = 0, 1, 2, . . 3. 1), then Dn φ = (−1)n φBn+ (s), n = 0, 1, 2, . . 4. The left and right Bell polynomials are connected by the following relations: Bn (s)∗ = Bn+ (s∗ ), Bn+ (s)∗ = Bn (s∗ ). If the ring is Abelian, left and right polynomials coincide. 5. 4 means that a duality takes place for the Bell polynomials: any relation for right polynomials can be transformed to the corresponding relation for left ones, and vice versa.

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