The nonlinear Schrodinger equation (NLSE) is a fundamental equation in quantum mechanics with applications in optical fibers, plasma physics, and biomolecule dynamics. The focus of this paper is on four types of nonlinear Schrodinger equations, including the cubic nonlinear Schrodinger equation (CNLSE) and the Chen-Lee-Liu equation (CLLE). We present the existence of a Lagrangian and the invariant variational principle for two coupled equations of these types. Our investigation involves the approximation solutions of these equations, achieved by selecting a trial function with single or two nontrivial variational parameters in the rectangular box region in three cases. By using these trial functions, we found the functional integral and the Lagrangian of the system without loss. The general case for the two-box potential can be obtained based on a different ansatz. We approximate the Jost function by quadratic polynomials instead of a piece-wise linear function and then approximate it by the tanh function. Using the amplitude ansatz method, we derive the exact bright soliton, dark soliton, bright-dark solitary wave solutions, rational dark-bright solutions, and the periodic solitary wave solutions for the proposed equations. The solutions obtained from these techniques have a wide range of applications in various areas of physics and other applied sciences. The results will be presented in graphical representations including 2D, 3D, and contour plots, which highlight their effectiveness. These techniques can be utilized to solve numerous other nonlinear models that arise in mathematical physics and different other applied sciences fields.
