An algebraic geometric classification of the solutions of the 1D Gross-Pitaevskii equation

The stationary solutions of the Schrödinger equation with box or periodic boundaries show a clear correspondence to solutions found for the non-linear Gross-Pitaevskii equation commonly used to model Bose-Einstein condensates. However, in the non-linear case there exists an additional class of solutions for periodic boundaries first identified by L.D. Carr et al. [1]. These nodeless solutions have no corresponding counterpart in the linear case. To fully classify these solutions and to understand their origin, we study the underlying algebraic geometry. Therefore, we treat both equations in the hydrodynamic framework, resulting in a first-order differential equation for the density determined by a quadratic polynomial in the linear case and by a cubic polynomial in the non-linear case, respectively. Our approach allows for a clear geometric interpretation and complete classification of the solution space in terms of the nature and location of the roots of these polynomials. Furthermore, we consider possible generalizations of our method towards higher dimensional systems and beyond-mean-field corrections.

[1] L.D. Carr, C.W. Clark, W.P. Reinhardt, Phys. Rev. A 62, 063610 & 063611 (2000)