Uniqueness and structure-preserving numerical simulations for an Ultrahyperbolic Schrödinger equation

In this paper, we investigate a Cauchy problem for an ultrahyperbolic Schrödinger equation posed in an unbounded domain. Ultrahyperbolic Schrödinger models arise in multidimensional wave propagation and have applications in water wave theory, quantum kinetic theory, ferromagnetism, nonlinear optics, and plasma physics. A uniqueness result for the considered Cauchy problem is established using a pointwise Carleman estimate. In addition, a second-order, structure-preserving Crank–Nicolson finite-difference scheme is developed for the numerical solution of the problem. For the homogeneous constant-coefficient problem, its discrete evolution is unitary and conserves the discrete \( ₕ²\) -norm exactly. Numerical experiments illustrate the behaviour of solutions and verify the accuracy and conservation properties of the numerical scheme.