Conservation laws in higher-order nonlinear Schrödinger equations

Conservation laws of the nonlinear Schrödinger equation are studied in the presence of higher-order optical effects including the third-order dispersion and the self-steepening. In a context of group theory, we derive general expressions for infinitely many conserved currents and charges of a coupled higher-order nonlinear Schrödinger equation. The first few currents and associated charges are also presented explicitly. Due to the higher-order effects, the conservation laws of the nonlinear Schrödinger equation are violated in general. The differences between the types of the conserved currents for the Hirota and the Sasa-Satsuma equations imply that the higher-order terms determine the inherent types of conserved quantities for each integrable case of the higher-order nonlinear Schrödinger equation.