Series and Rational Solutions of the Second Kind Painlevé Equations by Using the Quantum Pseudospectral Method

The Painlevé equations and their series and rational solutions are essential in applied, pure mathematics and theoretical physics. Recently, quantum algorithms have helped to implement numerical algorithms more easily by performing linear algebra in our working. This article uses a hybrid of quantum computing schemes and spectral methods for the second Painlevé equation. Two approaches are investigated: first, a series solution is obtained, and then the rational solutions. The successive linearization method is used for the linearization of the Painlevé-II equation. In the computer implementation, the solution value of the Painlevé-II equation is considered as a final quantum state. In each iterative scheme, by adding the current quantum state, we can compute the final quantum state. We need different quantum models for each approach, series, and rational solutions. Numerical examples illustrate the efficiency of this method, and reasonable solutions are obtained for a wide range of parameter values.