Nonlinear partial differential equations of mathematical physics are an important subject in physics. One of the well-known such equations is the nonlinear Schrödinger equation, which has applications in such areas as hydrodynamics, nonlinear optics, quantum mechanics, etc. The search for exact solutions to nonlinear partial differential equations plays a significant role in the study of the dynamics of nonlinear phenomena. Currently, there are many efficient and effective methods for finding exact solutions. In this paper, we study the two-dimensional chiral nonlinear Schrödinger equation, which contains the corresponding nonlinear terms. This equation is an extension of the one-dimensional nonlinear Schrödinger equation and is described by the Ablowitz-Kaup-Newell-Segur hierarchy. To obtain exact solutions, the sine-cosine method is applied. It is shown that the sine-cosine method is an effective mathematical tool for finding solutions to nonlinear partial differential equations of mathematical physics. The dynamics of the obtained solutions is shown in the figures.