The Properties Of Solutions For Several Nonlinear Schr(?)dinger Systems With Different Fractional Laplacian

This thesis mainly deals with the properties of solutions for three different nonlinear Schr(?)dinger equations with the fractional Laplacian under different order by a direct method of moving planes.Using the integral defining the fractional Laplacian,we establish the corresponding key ingredients needed in the method of moving planes for different types,such as narrow region principle and decay at infinity.For the semilinear fractional elliptic systems and the coupled nonlinear fractional Schr(?)dinger system with two equations and Gross-Pitaevskii equations(GPE) we prove the symmetry and monotonicity of solutions.