2015/12/18 by Gino Biondini, Biondini, Gino, Dionyssios Mantzavinos +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1512.06095
The long-time asymptotic behavior of the focusing nonlinear Schr "odinger\n(NLS) equation on the line with symmetric nonzero boundary conditions at\ninfinity is characterized by using the recently developed inverse scattering\ntransform (IST) for such problems and by employing the nonlinear steepest\ndescent method of Deift and Zhou for oscillatory Riemann-Hilbert problems.\nFirst, the IST is formulated over a single sheet of the complex plane without\nintroducing a uniformization variable. The solution of the focusing NLS\nequation with nonzero boundary conditions is thus associated with a suitable\nmatrix Riemann-Hilbert problem whose jumps grow exponentially with time for\ncertain portions of the continuous spectrum. This growth is the signature of\nthe well-known modulational instability within the context of the IST. This\ngrowth is then removed by suitable deformations of the Riemann-Hilbert problem\nin the complex spectral plane. Asymptotically in time, the xt-plane is found\nto decompose into two types of regions: a left far-field region and a right\nfar-field region, where the solution equals the condition at infinity to\nleading order up to a phase shift, and a central region in which the asymptotic\nbehavior is described by slowly modulated periodic oscillations. In the latter\nregion, it is also shown that the modulus of the leading order solution, which\nis initially obtained in the form of a ratio of Jacobi theta functions,\neventually reduces to the well-known elliptic solution of the focusing NLS\nequation. These results provide the first characterization of the long-time\nbehavior of generic perturbations of a constant background in a modulationally\nunstable medium.\n