Superposition of quasi coherent Bohm Madelung waves

We examine the problem of superposition of stationary quantum states in the Bohm Madelung formulation, where amplitude and phase obey coupled nonlinear equations and linear superposition is not generally valid. In the quasi-coherent regime of near degenerate stationary branches, the dynamics separates into a hierarchical structure: the mean amplitude evolves according to an Ermakov-Pinney equation governed by a Wronskian invariant, while the difference amplitude obeys a parametrically driven Hill Mathieu equation determined by the energy splitting. Despite this intrinsic nonlinearity, a linear spectral structure re-emerges through a Jacobi Anger expansion, yielding a Fourier Bessel representation with square-summable coefficients and translation covariant weights. Applications to aperture geometries and spatially separated sources demonstrate how amplitude modulation and phase induced sidebands organise interference patterns within a nonlinear amplitude phase framework. Keywords Bohm Madelung, Ermakov Pinney, nonlinear superposition, quasi coherent states, Mathieu Hill, Fourier Bessel, quantum interference.