  We present determinant expressions for vector rogue wave solutions of the
Manakov system, a two-component coupled nonlinear Schr\"odinger equation. As
special case, we generate a family of exact and non-symmetric rogue wave
solutions of the nonlinear Schr\"odinger equation up to third-order, localized
in both space and time. The derived non-symmetric doubly-localized second-order
solution is generated experimentally in a water wave flume for deep-water
conditions. Experimental results, confirming the characteristic non-symmetric
pattern of the solution, are in very good agreement with theory as well as with
numerical simulations, based on the modified nonlinear Schr\"odinger equation,
known to model accurately the dynamics of weakly nonlinear wave packets in
deep-water.
