Management of the discrete eigenvalues of the nonlinear Schrödinger equation using periodic dispersion perturbation

The effect of the periodic oscillation of the coefficient at the second-order derivative of the nonlinear Schrödinger equation on the soliton content of propagating pulses is studied numerically. Resonant fission of multisoliton pulses and soliton fusion are described in terms of eigenvalues of the Zakharov–Shabat spectral problem. Resonant conditions are connected with internal periodicity of solitons and breathers. The resonance can be changed by variation of initial pulse parameters. When the propagation distance does not exceeds a few oscillation periods of the coefficient, a wide resonances can be obtained. With a large number of the oscillation periods, the output pulse shapes and corresponding set of the complex eigenvalues of the Zakharov–Shabat spectral problem become extremely sensitive to the initial pulse parameters. The change in the pulse propagation process is described as a bifurcation. Two types of bifurcations were are found. The first bifurcation type is connected with the join of two imaginary eigenvalues in the single point and then the split of these eigenvalues in real parts symmetrically, so their imaginary part becomes the same. The second one is described by the joining of two eigenvalues with symmetrical real parts in the single complex point and their subsequent splitting. After splitting, the eigenvalues have zero real parts and different imaginary parts. I have found that transformations of the initial pulses are described by pairwise interaction of the eigenvalues. The eigenvalue bifurcations under effect of the change of the initial pulse amplitude and separation between pulses are studied. A periodically modulated media can be used both to control soliton pulse shapes and eigenvalues of the Zakharov–Shabat spectral problem.