Conserved densities and fluxes for nonlinear Schrödinger equations using scaling invariance approach

Abstract

We have adopted a direct method for the computation of polynomial conservation laws (CLs) of three nonlinear Schrödinger equations (NLSEs). The equations under consideration are firstly converted to evolution forms. Instead of using advanced differential-geometric tools, our method utilizes tools from linear algebra and variational calculus. This method can be implemented on NLSEs which occur in quantum physics, plasma physics, and fluid dynamics. In case of NLSEs with parameters, our method evaluates conditions on the parameters involved in order to find a sequence of conserved densities. The complete integrability of a NLSE can be predicted by the existence of a large number of CLs of the equation. The method utilizes linear combinations of scaling homogeneous terms having undetermined coefficients for the computation of conserved densities. The undetermined coefficients are being evaluated with the help of variational derivative (Euler operator) while Homotopy operator assists in the determination of conserved fluxes.