In present article the self-contained derivation of eigenvalue inverse problem results is given by using a discrete approximation of the Schrödinger operator on a bounded interval as a finite three-diagonal symmetric Jacobi matrix. This derivation is more correct in comparison with previous works which used only single-diagonal matrix. It is demonstrated that inverse problem procedure is nothin...

The problem of constructing semi-discrete integrable analogues the Liouville type PDE is discussed. We call equation a discretization if these two equations have common integral. For from well-known Goursat list for which integrals minimal order are less than or equal to we presented corresponding versions. contains new examples non-autonomous Darboux chains.

We find the exact analytical solution to a Goursat PDE system governing the kernels of a backsteppingbased boundary control law that stabilizes a constant-coefficient 2 × 2 system of first-order hyperbolic linear PDEs. The solution to the Goursat system is related to the solution of a simpler, explicitly solvable Goursat system through a suitable infinite series of powers of partial derivatives...