Variational model for liquid-like solitons

We present a variational framework for computing stationary profiles of liquid-like solitons governed by two-dimensional Nonlinear Schrödinger Equations with competing nonlinearities, focusing on the cubic-quintic optical system and the quantum droplet model with logarithmic nonlinearity. By introducing a hypergaussian ansatz with a tunable shape index, we derive closed-form expressions for the norm, amplitude, radius, and surface tension as functions of the nonlinear eigenvalue, covering the full range from smooth sech-like profiles to sharp flat-top distributions. Analytical constraints from the Maxwell coexistence condition and Pohozaev’s identity ensure physical consistency across both regimes. The variational predictions show a good agreement with numerical solutions obtained by standard relaxation methods, demonstrating that the proposed framework accurately captures the scaling laws and steep interfaces characteristic of the liquid phase without requiring a computational grid.