Symmetric Ground States Solutions of M-coupled Nonlinear Schrödinger Equations

      i∂tΦ1 +∆Φ1 + g1 (|x|, |Φ1| , . . . , |Φm| ) Φ1 = 0, .. i∂tΦm +∆Φm + gm (|x|, |Φ1| , . . . , |Φm| )Φm = 0, Φi(0, x) = Φ 0 i (x) for 1 ≤ i ≤ m. For 1 ≤ i ≤ m : Φi : R N → C and gi : R ∗ + × R m + → R, Φi : R+ × R N → C, has numerous applications in physical problems. It appears in the study of spatial solitons in nonlinear waveguides [30], the theory of Bose-Einstein condensates [12], interactions of m-wave packets [5], optical pulse propagation in birefringent fibers [25, 26], wavelength division multiplexed optical systems. Physically, the solution Φi is the ith component of the beam in Kerr-like photorefractive media [1]. In the most relevant cases, it is possible to write (1.1) in a vectorial form as follows: (1.2) 