Darboux transformations on a space scale

- We extend the Darboux transformations on a space scale that contains Darboux transformations for discrete and continuous differential equations.
- We extend Darboux's and Crum's theorems on a space scale.
- We deduce the Korteweg-de Vries-like system of equations on a space scale.
- We construct the solution of the system mentioned above by using the Darboux transformations.

Using the time/space-scale calculus introduced by Hilger, we describe a Darboux transformation for partial differential equations on a space scale, and extend the classical theorems of Darboux and Crum to this setting. As an application, we use the Lax equation to deduce a system of Korteweg-de Vries equations on a space scale, and obtain a solution to this system by using the Darboux transformation.