Nonlocal Nonlinear Schr\"odinger Equations and Their Soliton Solutions

We study standard and nonlocal nonlinear Schr"{o}dinger (NLS) equations obtained from the coupled NLS system of equations (Ablowitz-Kaup-Newell-Segur (AKNS) equations) by using standard and nonlocal reductions respectively. By using the Hirota bilinear method we first find soliton solutions of the coupled NLS system of equations then using the reduction formulas we find the soliton solutions of the standard and nonlocal NLS equations. We give examples for particular values of the parameters and plot the function $\mathrm{\mid }q(t,x){\mathrm{\mid }}^2$ for the standard and nonlocal NLS equations.