Linear stability analysis of walking vector solitons.
The linear stability analysis of two-parameter families of walking vector solitons of coupled nonlinear Schrödinger equations is performed. It is shown that the eigenvalues of the corresponding linearized problem can be complex valued in certain regions of the parameter space. The complex nature of the associated Lyapunov eigenvalues leads to a quite complicated pattern of instability regions of lowest-order soliton types. This pattern includes two typical situations: (i) the relevant eigenmode has a purely imaginary eigenvalue that passes through zero at the critical point and then becomes purely real, and (ii) the interplay between two discrete eigenmodes having purely imaginary eigenvalues leads to a bifurcation scenario where two imaginary eigenvalues merge together and become complex at the bifurcation point. It is shown that all known, lowest-order soliton types, namely slow, fast, in-phase vector, and out-of-phase vector, are dynamically stable in certain regions of the parameter space.