Deterministic Vacuum Instanton Prediction via q = 0.5 Modified Gross-Pitaevskii Condensate Dynamics

We formulate a non-perturbative, deterministic framework predicting vacuum instanton transitions byintroducing a fractional power-law non-linearity (q = 0.5) into the Gross-Pitaevskii equation (GPE). StandardGross-Pitaevskii dynamics (q = 1.0) describe cubic self-interacting Bose-Einstein condensates but suffer fromthermal stochasticity near non-linear tunneling thresholds. By scaling the order-parameter non-linearity to|ψ|2q = |ψ|, we prove that the effective action functional exhibits non-analytic, sub-linear potential gradientsthat induce singular turning points in imaginary time. Solving the Euclidean action yields an exact expressionfor deterministic bounce configurations, removing stochastic nucleation dependencies in false vacuum decaymodels. We demonstrate numerical stability using a split-operator pseudo-spectral scheme and provideexplicit critical amplitudes for experimental verification in ultra-cold atomic condensates