Shadow Lagrangian dynamics for superfluidity

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended shadow Lagrangian density quantum states of superfluids. The contains additional field variable that is forced to follow the wave function state through rapidly oscillating harmonic oscillator. By considering adiabatic limit large frequencies oscillator, we can derive two equations motions, Schrodinger-type equation and variable. are coupled in nonlinear way, but each individually linear with respect it defines. computational advantage new system be easily discretized using time stepping methods, where propose use Crank-Nicolson-type Schrodinger leapfrog scheme equation. Furthermore, difference between defines consistency error should go zero if frequency tends infinity. coupling time-step size our discretization oscillator extract computable indicator used estimate numerical without costs. findings illustrated experiments.