If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: g = 0): { ∂ ∂ v ∂ ∂ t + u ∂ ∂ v ∂ ∂ x − − v ∂ ∂ u ∂ ∂ x = 0 ∂ ∂ u ∂ ∂ t + u ∂ ∂ u ∂ ∂ x − − e v v v ∂ ∂ v ∂ ∂ x − − e v s v ∂ ∂ s ∂ ∂ x = 0 ∂ ∂ s ∂ ∂ t + u ∂ ∂ s ∂ ∂ x = 0