A Mathematical Frameworks for Singular, Nonlinear Phenomena: Applications to Nematocyst Firing and Inhomogeneous NLS with Coulomb Potential

Nematocysts are specialized cellular organelles found in all cnidarians, including corals and jellyfish, as well as in some single-celled protists such as dinoflagellates. These organelles display remarkable diversity in morphology and function, playing roles in prey capture and defense. The firing of a nematocyst is one of the fastest accelerations in nature, yet the underlying physical mechanisms remain not fully understood. In this work, we address key questions: how sufficient force is generated to overcome the fluid boundary layer, whether fluid–structure interaction models can reproduce observed dynamics, and what mechanisms trigger discharge. Our research investigates models based on osmotic pressure forces and their transfer to nematocyst structures. Earlier implementations used finite-difference methods in MATLAB. Here, we adopt the Immersed Boundary Method (IBM), which provides a flexible framework for simulating fluid–structure interactions. This allows us to explore alternative firing mechanisms, including coil-based release and pressurized discharge. Since nematocyst firing occurs on very small spatial and temporal scales, direct experimental observation is limited, making computational modeling essential for understanding the process. In addition, we study an inhomogeneous nonlinear Schrödinger equation (INLS) with a Coulomb-type potential. We establish a local well-posedness theory in the energy space, particularly in the repulsive regime, and extend results to Sobolev spaces without restrictions on the sign of the potential, allowing both attractive and repulsive cases. This work extends results on nonlinear Schrödinger equations with singular potentials by addressing the inhomogeneous setting and higher spatial dimensions. The analysis is challenging due to spatial inhomogeneity and the singular Coulomb potential.