We propose and examine an integrable system of nonlinear equations that generalizes the nonlinear Schrödinger equation to $2+1$ dimensions. This integrable system of equations is a promising starting point to elaborate more accurate models in nonlinear optics and molecular systems within the continuum limit. The Lax pair for the system is derived after applying the singular manifold method. We also present an iterative procedure to construct the solutions from a seed solution. Solutions with one-, two-, and three-lump solitons are thoroughly discussed.

• Received 12 February 2016
• Revised 29 April 2016

DOI:https://doi.org/10.1103/PhysRevE.93.062219