Exact solutions of the nonlinear Schr�dinger equation in time-dependent parabolic density profiles

Conformal symmetry of the coupled Chern-Simons and gauged nonlinear Schrödinger equations

“…We conclude that ξ-preserving conformal transformations of M are indeed symmetries of the reduced system. This generalizes the result known for the ungauged non-linear Schrödinger equation [8]. Note that the value p = 1 in the last term |Ψ| 2p Ψ of (1.1) corresponds to the one dictated by relativistic conformal invariance of the above mentioned non-linear Klein-Gordon equation in n = 4 dimensions.…”

Edge Solitons in the Quantum Hall Effect

“…is integrable, and its solutions are obtained from those of the "free" NLS by the transformation (20). Let us mention that the covariance w. r. t. chronoprojective transformations was used before [16] for solving the NLS in oscillator and uniform-field backgrounds. Now the constant-coefficient, damped, driven NLS,…”