psipy

psipy bridges symbolic manipulation via SymPy with numerical computation using NumPy and SciPy to analyze partial differential equations (PDEs) and pseudo-differential operators (ΨDOs). It is designed for researchers in mathematical physics to explore phase-space geometry, caustics, and spectral properties in 1D and 2D. Key capabilities include: - Pseudo-Differential Operators: Symbolic calculus for composition, commutators, and formal adjoints, alongside microlocal analysis tools. - PDE Solving: Spectral (FFT) discretization for 1D and 2D linear/nonlinear PDEs using ETD-RK4 time stepping. - Asymptotic Evaluation: Automatic detection and evaluation of oscillatory integrals (stationary phase, Laplace, saddle point) with support for Morse, Airy, and Pearcey singularities. - Semiclassical Analysis: Multidimensional WKB approximations, Van Vleck–Pauli–Morette wavefunction assembly, and ray-based caustic detection with Arnold classification and Maslov index tracking. - Hamiltonian & Symplectic Mechanics: Tools for Legendre transforms, symplectic integrators (Euler, Verlet), Poisson brackets, and stability analysis. It includes a catalog of over 500 pre-defined symbolic Hamiltonians. - Riemannian Geometry: Computation of metrics, Christoffel symbols, curvature, and geodesics in 1D/2D. - Visualization: Specialized tools for phase-space portraits, caustic overlays, and wavefunction density/phase plots.