An analytical solution for a coupled partial differential equation

A variational method given by Ritz has been applied to the Coupled Partial Differential Equation to construct an analytical solution. The solution of Partial Differential Equation gives a good description of both linear and non-linear evolution of instabilities generated in waves due to modulation. The spatially periodic trial function is chosen in the form of combination of Jacobian Elliptic Functions with the dependence of its parameters subject to optimization. This equation reduced to stationary form of Non-linear Schrödinger Equation in the static limit.