Integrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter model

Abstract We introduce a new unified two-parameter { ( ϵ x , ϵ t ) | ϵ x , t = ± 1 } wave model (simply called Q ϵ x , ϵ t ( n ) model), connecting integrable local and nonlocal vector nonlinear Schrodinger equations. The two-parameter ( ϵ x , ϵ t ) family also brings insight into a one-to-one connection between four points ( ϵ x , ϵ t ) (or complex numbers ϵ x + i ϵ t ) with { I , P , T , PT } symmetries for the first time. The Q ϵ x , ϵ t ( n ) model is shown to possess a Lax pair and infinite number of conservation laws, and to be PT symmetric. Moreover, the Hamiltonians with self-induced potentials are shown to be PT symmetric only for Q − 1 , − 1 ( n ) model and to be T symmetric only for Q + 1 , − 1 ( n ) model. The multi-linear form and some self-similar solutions are also given for the Q ϵ x , ϵ t ( n ) model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions.

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

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

[3]  Shachar Klaiman,et al.  Visualization of branch points in PT-symmetric waveguides. , 2008, Physical review letters.

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

[5]  D. Mihalache,et al.  Defect solitons in two-dimensional photonic lattices with parity-time symmetry , 2014 .

[6]  Hui Cao,et al.  Coherent perfect absorbers: Time-reversed lasers , 2010, CLEO/QELS: 2010 Laser Science to Photonic Applications.

[7]  P. Würtz,et al.  High-resolution scanning electron microscopy of an ultracold quantum gas , 2008 .

[8]  Zhenya Yan,et al.  Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and nonlinearities. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Hong Chen,et al.  Experimental demonstration of a coherent perfect absorber with PT phase transition. , 2014, Physical review letters.

[10]  J. Main,et al.  Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential , 2014, 1403.6742.

[11]  Zhenya Yan.  Financial Rogue Waves , 2009, 0911.4259.

[12]  Peter A. Clarkson,et al.  THE DIRECT METHOD IN SOLITON THEORY (Cambridge Tracts in Mathematics 155) , 2006 .

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

[14]  B. Malomed,et al.  Spatiotemporal optical solitons , 2005 .

[15]  Vladimir E. Zakharov,et al.  A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I , 1974 .

[16]  E. K. Sklyanin,et al.  O(N)-invariant nonlinear Schrödinger equation — A new completely integrable system , 1981 .

[17]  D. Christodoulides,et al.  Parity-time–symmetric microring lasers , 2014, Science.

[18]  P. Kevrekidis,et al.  PT-symmetric oligomers: analytical solutions, linear stability, and nonlinear dynamics. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  M. Segev,et al.  Theory of Self-Trapped Spatially Incoherent Light Beams , 1997 .

[20]  Z. Ahmed.  Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex -invariant potential , 2001 .

[21]  B. Malomed,et al.  Stable dark solitons in PT-symmetric dual-core waveguides , 2012, 1211.3746.

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

[23]  V. Konotop,et al.  Nonlinear modes in finite-dimensional PT-symmetric systems. , 2012, Physical review letters.

[24]  Franco Nori,et al.  PT-symmetric phonon laser. , 2014, Physical review letters.

[25]  Bikashkali Midya,et al.  Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells , 2013, 1304.2105.

[26]  V. Konotop,et al.  Fundamental, multipole, and half-vortex gap solitons in spin-orbit coupled Bose-Einstein condensates. , 2014, Physical review letters.

[27]  A. Mostafazadeh.  Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries , 2002, math-ph/0203005.

[28]  Shanhui Fan,et al.  Parity–time-symmetric whispering-gallery microcavities , 2013, Nature Physics.

[29]  H. Harney,et al.  PT symmetry and spontaneous symmetry breaking in a microwave billiard. , 2011, Physical review letters.

[30]  M. Wadati,et al.  Relationships among Inverse Method, Bäcklund Transformation and an Infinite Number of Conservation Laws , 1975 .

[31]  Carl M. Bender,et al.  Making sense of non-Hermitian Hamiltonians , 2007, hep-th/0703096.

[32]  Boris A. Malomed,et al.  Solitons in nonlinear lattices , 2011 .

[33]  Zhenya Yan.  Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

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

[36]  M. Kruskal,et al.  New similarity reductions of the Boussinesq equation , 1989 .

[37]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

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

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

[40]  G. Bluman,et al.  Symmetries and differential equations , 1989 .

[41]  S. Longhi,et al.  Bloch oscillations in complex crystals with PT symmetry. , 2009, Physical review letters.

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

[43]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[44]  D. H. Peregrine,et al.  Water waves, nonlinear Schrödinger equations and their solutions , 1983, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

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

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

[47]  V. Pérez-García,et al.  Dissipation-induced coherent structures in Bose-Einstein condensates. , 2009, Physical review letters.

[48]  B. M. Fulk.  MATH , 1992 .

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

[50]  J. Soto-Crespo,et al.  Rogue waves and rational solutions of the nonlinear Schrödinger equation. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  Yasuhiro Ohta,et al.  General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.