Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation

A nonlocal nonlinear Schrodinger (NLS) equation was recently introduced and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this paper a detailed study of the inverse scattering transform of this nonlocal NLS equation is carried out. The direct and inverse scattering problems are analyzed. Key symmetries of the eigenfunctions and scattering data and conserved quantities are obtained. The inverse scattering theory is developed by using a novel left–right Riemann–Hilbert problem. The Cauchy problem for the nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads to explicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation is given and brief remarks about nonlocal versions of the modified Korteweg–de Vries and sine-Gordon equations are made.

[1]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[2]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[3]  A. B. Shabat,et al.  Interaction between solitons in a stable medium , 1973 .

[4]  M. Ablowitz,et al.  The Inverse scattering transform fourier analysis for nonlinear problems , 1974 .

[5]  M. Ablowitz,et al.  The inverse scattering transform: Semi‐infinite interval , 1975 .

[6]  Mark J. Ablowitz,et al.  Nonlinear differential−difference equations , 1975 .

[7]  M. Ablowitz,et al.  Nonlinear differential–difference equations and Fourier analysis , 1976 .

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

[9]  H. Inoue,et al.  Eigen Value Problem with Nonvanishing Potentials , 1977 .

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

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

[12]  P. Kulish.  Quantum difference nonlinear Schrödinger equation , 1981 .

[13]  Yuji Ishimori,et al.  An Integrable Classical Spin Chain , 1982 .

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

[15]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

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

[17]  Campbell,et al.  Self-trapping on a dimer: Time-dependent solutions of a discrete nonlinear Schrödinger equation. , 1986, Physical Review B (Condensed Matter).

[18]  N. Papanicolaou.  Complete integrability for a discrete Heisenberg chain , 1987 .

[19]  Govind P. Agrawal,et al.  Nonlinear Fiber Optics , 1989 .

[20]  S. Takeno,et al.  A Propagating Self-Localized Mode in a One-Dimensional Lattice with Quartic Anharmonicity , 1990 .

[21]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

[22]  V. Konotop,et al.  On soliton creation in the nonlinear Schrodinger models: discrete and continuous versions , 1992 .

[23]  C. Bender,et al.  Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , 1997, physics/9712001.

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

[25]  M. Wadati,et al.  Inverse scattering method for square matrix nonlinear Schrödinger equation under nonvanishing boundary conditions , 2006, nlin/0603010.

[26]  Gino Biondini,et al.  Inverse scattering transform for the vector nonlinear Schrödinger equation with nonvanishing boundary conditions , 2006 .

[27]  Yasuhiro Ohta,et al.  Casorati determinant form of dark soliton solutions of the discrete nonlinear Schrödinger equation , 2006 .

[28]  Z. Musslimani,et al.  Theory of coupled optical PT-symmetric structures. , 2007, Optics letters.

[29]  Z. Musslimani,et al.  Optical Solitons in PT Periodic Potentials , 2008 .

[30]  Z. Musslimani,et al.  Beam dynamics in PT symmetric optical lattices. , 2008, Physical review letters.

[31]  R. Morandotti,et al.  Observation of PT-symmetry breaking in complex optical potentials. , 2009, Physical review letters.

[32]  Gino Biondini,et al.  Solitons, boundary value problems and a nonlinear method of images , 2009, 0904.2410.

[33]  M. Segev,et al.  Observation of parity–time symmetry in optics , 2010 .

[34]  M. Ablowitz,et al.  Nonlinear shallow ocean-wave soliton interactions on flat beaches. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  U. Peschel,et al.  Parity–time synthetic photonic lattices , 2012, Nature.

[36]  Francesco Calogero,et al.  Spectral Transform and Solitons , 2012 .

[37]  M. Ablowitz,et al.  Integrable nonlocal nonlinear Schrödinger equation. , 2013, Physical review letters.

[38]  Nick Lazarides,et al.  Gain-driven discrete breathers in PT-symmetric nonlinear metamaterials. , 2012, Physical review letters.

[39]  Mordechai Segev,et al.  Nonlinearly induced PT transition in photonic systems. , 2013, Physical review letters.

[40]  Mohammad-Ali Miri,et al.  Observation of defect states in PT-symmetric optical lattices. , 2013, Physical review letters.

[41]  M. Ablowitz,et al.  Integrable discrete PT symmetric model. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[42]  A. S. Fokas,et al.  Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation , 2016 .

[43]  Sarah Rothstein,et al.  Optical Solitons From Fibers To Photonic Crystals , 2016 .