In this talk, I will present an approximate first-order hyperbolic model for the hydrodynamic form of the defocusing nonlinear Schrödinger equation (NLS). This Euler-Korteweg type system can be seen as a Euler-Lagrange equation to a Lagrangian submitted to a mass conservation constraint. Due to the presence of dispersive terms, such a Lagrangian depends explicitly on the gradient of density. The idea is to create a new dummy variable that accurately approximates the density via a penalty method. Then, we take its gradient as a new independent variable and apply Hamilton's principle.

I will explain the main ideas behind the method, how the resulting system is hyperbolic, and present some obtained numerical results for gray solitons and dispersive shockwaves.