Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions

The inverse scattering transform for the vector defocusing nonlinear Schrodinger NLS equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros poles of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert RH problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system. © 2006 American Institute of Physics. DOI: 10.1063/1.2209169

[1]  Ronald R. Coifman,et al.  Inverse scattering and evolution equations , 1985 .

[2]  Non‐self‐adjoint Zakharov–Shabat operator with a potential of the finite asymptotic values. I. Direct spectral and scattering problems , 1981 .

[3]  Leon A. Takhtajan,et al.  Hamiltonian methods in the theory of solitons , 1987 .

[4]  M. Lakshmanan,et al.  Bright and dark soliton solutions to coupled nonlinear Schrodinger equations , 1995 .

[5]  Athanassios S. Fokas,et al.  Complex Variables: Contents , 2003 .

[6]  Carlos Tomei,et al.  Direct and inverse scattering on the line , 1988 .

[7]  The Dirac inverse spectral transform: Kinks and boomerons , 1980 .

[8]  Hiroshi Inoue,et al.  Inverse Scattering Method for the Nonlinear Evolution Equations under Nonvanishing Conditions , 1978 .

[9]  Yuri S. Kivshar,et al.  Dark optical solitons: physics and applications , 1998 .

[10]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[11]  Yuri S. Kivshar,et al.  Polarized dark solitons in isotropic Kerr media , 1997 .

[12]  L. Faddeev,et al.  Comparison of the exact quantum and quasiclassical results for a nonlinear Schrödinger equation , 1976 .

[13]  A. Fokas,et al.  Complex Variables: Introduction and Applications , 1997 .

[14]  Y. Kato,et al.  Non‐self‐adjoint Zakharov–Shabat operator with a potential of the finite asymptotic values. II. Inverse problem , 1984 .

[15]  S K Turitsyn,et al.  Vector dark solitons. , 1993, Optics letters.

[16]  M. Boiti,et al.  The spectral transform for the NLS equation with left-right asymmetric boundary conditions , 1982 .

[17]  D. J. Kaup,et al.  The Three-Wave Interaction-A Nondispersive Phenomenon , 1976 .