On the Darboux transformation. II. (English) Zbl 0837.34015

Chrastinová, Veronika

Arch. Math., Brno 31, No. 2, 121-138 (1995).

[For part I see Georgian Math. J. 2, No. 3, 237-240 (1995; Zbl 0834.34019).]
Automorphisms of infinitely prolonged system \(y'' = py\), \(z'' = qz\), \(p' - q' = 0\) regarded for undetermined ordinary differential equations with unknown functions \(y,z,p,q\) of independent variable \(x\) are investigated in order to obtain the most general quadrature-free generalization of the classical Darboux transformation of Sturm-Liouville equations. The result reads \(\widetilde y = (y'/y - z'/z)^{{1 \over 2} (1 + B)} y^{{1 \over 2} ({A \over B} + B)} z^{{1 \over 2} ({A \over B} - B)}\) with \(A = A (\lambda)\), \(B = B (\lambda) \neq 0\) \((\lambda - p - q)\) quite arbitrary. The Darboux transformation appears if \(A = B = 1\). A somewhat unusual reasonings are carried out in the infinite-dimensional space \(\mathbb{R}^\infty\) of all real sequences and the method of proof is of independent interests.

Reviewer: J.Chrastina (Brno)

MSC:

34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
58A10	Differential forms in global analysis

Citations:

Zbl 0834.34019