LAUSR

where i is the imaginary unit. It has applications in many physics and nonlinear optics, see Ref. 1. Physically, the solution Φ j denotes the jth component of the beam in Kerr-like photorefractive media, see Ref. 2. μ j is for self-focusing in the jth component, and the coupling constant β is the interaction between the two components of the beam. (2) is also called the Bose-Einstein condensates system since it arises in the Hartree-Fock theory for a double condensate, see Ref. 3 and references therein. To obtain solitary wave solutions of system (2), we set Φ j(x , t) = eiλ j tu j(x) for j = 1, 2, then it will be reduced to system (1). The existence of the least energy and other finite energy solutions was studied in Refs. 4–6 and references therein. The existence and the multiplicity of positive and sign-changing solutions were studied in Refs. 7–9 and references therein. Later, the general k-coupled case attracts more and more interest because of its many more possible properties of solutions: −∆u j +λ ju j = μ juj + ∑