Gaussian solitary waves for argument-Schrödinger equation

•The logarithmic nonlinear Schrödinger equation has Gaussian shaped solitons.•Solitary solutions under the quadratic potential in one dimension is given.•The dispersion relation is time-dependent due to the imaginary damping potential. We present localized analytical solutions of the logarithmic nonlinear Schrödinger equation, i.e., the so-called the argument-Schrödinger equation. The Gaussian solitary waveform is shown to be the solution, and we obtain the explicit form in a one-dimensional case when the dynamics evolve under a quadratic potential. The dispersion relation becomes time-dependent due to the logarithmic nonlinearity.