Solitons in coupled nonlinear Schrödinger equations with variable coefficients

• The connection between coupled NLS equation with variable coefficients (VCNLS) and the coupled NLS equation are studied.
• The Bright–Dark soliton and the Bright–Bright soliton solutions of the VCNLS system are obtained.
• The optical super-lattice potentials with parameters is designed for the VCNLS system.
• The hyperbolic cosine potentials with parameters is designed for the VCNLS system.

We study the coupled nonlinear Schrodinger equation with variable coefficients (VCNLS), which can be used to describe the interaction among the modes in nonlinear optics and Bose–Einstein condensation. By constructing an explicit transformation, which maps VCNLS to the classical coupled nonlinear Schrödinger equations (CNLS), we obtain Bright–Dark and Bright–Bright solitons for VCNLS. Furthermore, the optical super-lattice potentials (or periodic potentials) and hyperbolic cosine potentials with parameters are designed, which are two kinds of important potentials in physics. This method can be used to design a large variety of external potentials in VCNLS, which could be meaningful for manipulating solitons experimentally.