Exact Results in Quantum Theory

I present the derivation of the hydrodynamics for a cold Bose gas at T = 0 from the microscopic platform based on the many-body Schrödinger equation. The energy is obtained as a functional of both fast inner quantum mode and slow macroscopic mode. The equations governing the fast and slow modes are obtained from this functional by their independent variations. The fast mode is the many-body wave function in the stationary state at local density, which can be ground, excited with nonzero atom momenta, or a superposition of more than one state. The energy eigenvalue of this local wave function determines the nonlinearity of the hydrodynamic equation. For zero inner momenta and particular choices of this eigenvalue as a function of gas density, this equation reduces to the known equations. If, however, the inner momenta are nonzero, the equation includes the interaction between these momenta and the slow mode. It is shown that the type-2 homogeneous excitations of the Lieb-Liniger gas can be unstable and emit supersonic solitons. References:V. M. Pergamenshchik, Phys. Rev. A, 110, 033308 (2024)