Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time-symmetric potential

Abstract A two component nonlocal vector nonlinear Schrodinger equation (VNLSE) is considered with a self-induced parity-time-symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and hence integrable. Some of the conserved quantities like number operator, Hamiltonian etc. are found to be real-valued, in spite of the corresponding charge densities being complex. The soliton solution for the same equation is obtained through the method of inverse scattering transformation and the condition of reduction from nonlocal to local case is mentioned.

[1]  N. S. Bergano,et al.  Polarization multiplexing with solitons , 1992 .

[2]  Karsten Trulsen,et al.  A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water , 1996 .

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

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  Jianke Yang,et al.  Multiple Permanent‐wave Trains in Nonlinear Systems , 1998, nlin/0506035.

[6]  P. K. Ghosh.  Deconstructing non-dissipative non-Dirac-Hermitian relativistic quantum systems , 2011, 1105.2495.

[7]  Jianke Yang,et al.  Nonlinear waves in PT -symmetric systems , 2016, 1603.06826.

[8]  P. K. Ghosh,et al.  On the real spectra of the Calogero model with complex coupling , 2003, hep-th/0310276.

[9]  M. Znojil.  symmetric harmonic oscillators , 1999, quant-ph/9905020.

[10]  Pijush K. Ghosh.  Exact results on the dynamics of a multicomponent Bose-Einstein condensate , 2002 .

[11]  A. Levine,et al.  New estimates of the storage permanence and ocean co-benefits of enhanced rock weathering , 2023, PNAS nexus.

[12]  Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum , 2001, math-ph/0110016.

[13]  A. Aceves,et al.  Storage and steering of self-trapped discrete solitons in nonlinear waveguide arrays. , 1994, Optics letters.

[14]  Solvable model of quantum phase transitions and the symbolic-manipulation-based study of its multiply degenerate exceptional points and of their unfolding , 2013, 1305.4822.

[15]  Yuri S. Kivshar,et al.  Nonlinear multicore waveguiding structures with balanced gain and loss , 2015 .

[16]  S. Yomosa,et al.  Soliton excitations in deoxyribonucleic acid (DNA) double helices , 1983 .

[17]  Rodislav Driben,et al.  Stability of solitons in parity-time-symmetric couplers. , 2011, Optics letters.

[18]  Fatkhulla Kh. Abdullaev,et al.  Optical Solitons , 2014 .

[19]  D. N. Christodoulides,et al.  $\mathcal{PT}$-Symmetric Periodic Optical Potentials , 2011 .

[20]  P. K. Ghosh.  A note on the topological insulator phase in non-Hermitian quantum systems , 2011, Journal of physics. Condensed matter : an Institute of Physics journal.

[21]  Mohammad-Ali Miri,et al.  Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Li-Yuan Ma,et al.  Nonlocal nonlinear Schrödinger equation and its discrete version: Soliton solutions and gauge equivalence , 2015, 1503.06909.

[23]  A. Mostafazadeh.  Pseudo-supersymmetric quantum mechanics and isospectral pseudo-Hermitian Hamiltonians , 2002, math-ph/0203041.

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

[25]  D. Christodoulides,et al.  Bragg solitons in nonlinear PT-symmetric periodic potentials , 2012, 1209.0787.

[26]  A. Mostafazadeh.  Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian , 2001, math-ph/0107001.

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

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

[29]  A return to observability near exceptional points in a schematic PT-symmetric model , 2007, quant-ph/0701232.

[30]  Ali Mostafazadeh,et al.  Pseudo-Hermitian Representation of Quantum Mechanics , 2008, 0810.5643.

[31]  Ricardo Carretero-González,et al.  Emergent nonlinear phenomena in Bose-Einstein condensates : theory and experiment , 2008 .

[32]  P. K. Ghosh.  Deconstructing non-Dirac–Hermitian supersymmetric quantum systems , 2011, 1101.2120.

[33]  I. V. Barashenkov,et al.  Breathers in PT-symmetric optical couplers , 2012, 1211.1835.

[34]  Miloslav Znojil.  symmetric square well , 2001 .

[35]  C. Bender.  Introduction to 𝒫𝒯-symmetric quantum theory , 2005, quant-ph/0501052.

[36]  P. K. Ghosh,et al.  The exactly solvable quasi-Hermitian transverse Ising model , 2009, 0904.2852.

[37]  Susumu Takeda,et al.  Modified Nonlinear Schrödinger Equation for Alfvén Waves Propagating along the Magnetic Field in Cold Plasmas , 1975 .

[38]  Min Li,et al.  Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[40]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[41]  Xing Zhu,et al.  Bright spatial solitons in defocusing Kerr media with PT-symmetric potentials , 2011, 1110.5108.

[42]  Y. Kivshar,et al.  Optical solitons in $\mathcal{PT}$-symmetric nonlinear couplers with gain and loss , 2012, 1207.5252.

[43]  Lederer,et al.  Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[44]  Wolfgang Ketterle,et al.  Bose-Einstein Condensation: Identity Crisis for Indistinguishable Particles , 2007 .

[45]  W. Ketterle,et al.  Bose-Einstein condensation , 1997 .

[46]  P. K. Ghosh.  Constructing Exactly Solvable Pseudo-hermitian Many-Particle Quantum Systems by Isospectral Deformation , 2010, 1012.0907.

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

[48]  Ragnar Fleischmann,et al.  Exponentially fragile PT symmetry in lattices with localized eigenmodes. , 2009, Physical review letters.

[49]  Quantum phase transition in a pseudo-Hermitian Dicke model. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[51]  A. Khare,et al.  Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations , 2014, 1405.5267.

[52]  P. K. Ghosh.  On the construction of a pseudo-Hermitian quantum system with a pre-determined metric in the Hilbert space , 2009, 0908.1321.

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

[54]  G. J. Roskes.  Some Nonlinear Multiphase Interactions , 1976 .

[55]  P. K. Ghosh,et al.  Symmetries and exact solutions of a class of nonlocal nonlinear Schrödinger equations with self-induced parity-time-symmetric potential. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

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