arXiv:2511.07469v1 Announce Type: new
Abstract: We present a quaternion wavefunction formulation that reduces the incompressible Euler equations to a single nonlinear Schr”odinger-type equation. The velocity field emerges from a complex quaternion wavefunction $Psi in mathbb{C} otimes mathbb{H}$ satisfying a constrained Gross-Pitaevskii equation, with incompressibility enforced through a holomorphic constraint on quaternion space. This formulation preserves all conservation laws through a natural Lagrangian structure and reduces the system from four coupled nonlinear equations (three velocity components plus pressure) to one quaternion field equation with an algebraic constraint. We demonstrate the utility of this approach by deriving analytical solutions for three-dimensional flow past a sphere, obtaining the drag coefficient $C_D approx 0.47$ at Reynolds number Re $= 1000$ and predicting the onset of vortex shedding at $text{Re}_c approx 270$, in excellent agreement with experiments. The formulation provides a new mathematical framework for inviscid fluid dynamics and suggests efficient numerical algorithms exploiting quaternion structure.