Symmetry, Conservation Law, Exact Solutions for Nonlinear Schrödinger Equation With Linear Waditi Potential

ABSTRACT In this paper, we study a nonlinear Schrödinger equation with linear Wadati potential, which deeply exhibits the interplay between the nonlinearity and non‐Hermiticity. A complete classification of Lie symmetry groups, including point symmetry and contact symmetry, is presented. Thereby the transformation groups acting on solution are presented. Through Noether's theorem, we derive the conservation laws that arise from the Lagrangian structure of the original equation. Moreover the method of multiplier is employed to find all local conservation laws of lower order, as well as first order. From the admitted point symmetry groups, the optimal systems of one‐dimensional subalgebras are acquired. Afterwards, the reductions to ordinary differential equation are accomplished, which give rise to all explicit group‐invariant solutions for the nonlinear Schrödinger equation with linear Wadati potential, including the ones obtained under the actions of transformation groups. The analytic properties of these solutions are investigated, some of which exhibit periodic, dispersive, blow‐up, and singular.