Self-adjointness and conservation laws of two variable coefficient nonlinear equations of Schrödinger type

In this paper, some recent concepts and results on self-adjointness and conservation laws are applied to two variable coefficient nonlinear equations of Schrödinger type: the generalized variable coefficient nonlinear Schrödinger (GVCNLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients. The two equations are changed to two real systems by a proper transformation. To obtain the formal Lagrangians of the two systems, we discuss their self-adjointness and find that the GVCNLS system is weak self-adjoint and the CQNLS system is quasi self-adjoint. Having performed Lie symmetry analysis for the two systems, we find five nontrivial conservation laws for the GVCNLS system and four nontrivial conservation laws for the CQNLS system by using a general theorem on conservation laws given by Ibragimov.

► Extend definitions of weak self-adjointness and quasi self-adjointness to system.
► The GVCNLS system is weak self-adjoint, the CQNLS system is quasi self-adjoint.
► Formal Lagrangians and adjoint systems for the two equations are obtained.
► Perform Lie group analysis and obtain Lie symmetries for the two systems.
► Some nontrivial conservations laws for the two systems are derived.