On the mapping connecting the cylindrical nonlinear von Neumann equation with the standard von Neumann equation

The Wigner transformation is used to define the quasidistribution (Wigner function) associated with the wave function of the cylindrical nonlinear Schrödinger equation (CNLSE) in a way similar to that of the standard nonlinear Schrödinger equation (NLSE). The phase-space equation, governing the evolution of such quasidistribution, is a sort of nonlinear von Neumann equation (NLvNE), called here the ‘cylindrical nonlinear von Neumann equation’ (CNLvNE). Furthermore, the phase-space transformations, connecting the Wigner function and the NLvNE with the ‘cylindrical Wigner function’ and the CNLvNE, are found by extending the configuration space transformations that connect the NLSE and the CNLSE. Some examples of phase-space soliton solutions are given analytically and evaluated numerically.