Kink scattering from a parity-time-symmetric defect in the ϕ4 model

Abstract In this paper, we study the ϕ4 kink scattering from a spatially localized PT -symmetric defect and the effect of the kink’s internal mode (IM) is discussed. It is demonstrated that if a kink hits the defect from the gain side, a noticeable IM is excited, while for the kink coming from the opposite direction the mode excitation is much weaker. This asymmetry is a principal finding of the present work. Similar to the case of the sine-Gordon kink studied earlier, it is found that the ϕ4 kink approaching the defect from the gain side always passes through the defect, while in the opposite case it must have sufficiently large initial velocity, otherwise it is trapped by the loss region. It is found that for the kink with IM the critical velocity is smaller, meaning that the kink bearing IM can pass more easily through the loss region. This feature, namely the “increased transparency” of the defect as regards the motion of the kink in the presence of IM is the second key finding of the present work. A two degree of freedom collective variable model offered recently by one of the co-authors is shown to be capable of reproducing both principal findings of the present work. A simpler, analytically tractable single degree of freedom collective variable method is used to calculate analytically the kink phase shift and the kink critical velocity sufficient to pass through the defect. Comparison with the numerical results suggests that the collective variable method is able to predict these parameters with a high accuracy.

[1]  H. Yilmaz,et al.  Loss-induced suppression and revival of lasing , 2014, Science.

[2]  Y. Kivshar,et al.  Resonant kink-impurity interactions in the phi4 model. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[3]  Demetrios N. Christodoulides,et al.  PT optical lattices and universality in beam dynamics , 2010 .

[4]  P. Kevrekidis,et al.  Interaction of sine-Gordon kinks and breathers with a parity-time-symmetric defect. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  M. Rice.  Physical dynamics of solitons , 1983 .

[6]  T. Prosen,et al.  PT-symmetric wave Chaos. , 2010, Physical review letters.

[7]  Sergey V. Dmitriev,et al.  Inelastic three-soliton collisions in a weakly discrete sine-Gordon system , 2000 .

[8]  P. Kevrekidis,et al.  The parametrically modified sine-Gordon equation: spectral properties, collisions, resonances and their connection to integrability , 2002 .

[9]  Yuri S. Kivshar,et al.  Nonlinearly PT-symmetric systems: Spontaneous symmetry breaking and transmission resonances , 2011, 1104.0849.

[10]  A. Kudryavtsev,et al.  Solitons and their interactions in classical field theory , 1997 .

[11]  P. Kevrekidis,et al.  Radiationless energy exchange in three-soliton collisions. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Y. Kivshar,et al.  Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect , 2011 .

[13]  I. V. Barashenkov,et al.  Phase sensitivity of light dynamics in PT-symmetric couplers , 2014 .

[14]  P. Kevrekidis,et al.  Effects of parity-time symmetry in nonlinear Klein-Gordon models and their stationary kinks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[16]  Tsampikos Kottos,et al.  Experimental study of active LRC circuits with PT symmetries , 2011, 1109.2913.

[17]  Yuri S. Kivshar,et al.  The Frenkel-Kontorova Model: Concepts, Methods, and Applications , 2004 .

[18]  M. Rice,et al.  Phenomenological theory of soliton formation in lightly-doped polyacetylene , 1980 .

[19]  Yuri S. Kivshar,et al.  The Frenkel-Kontorova Model , 2004 .

[20]  D. Christodoulides,et al.  Observation of asymmetric transport in structures with active nonlinearities. , 2013, Physical review letters.

[21]  Dorje C Brody,et al.  Complex extension of quantum mechanics. , 2002, Physical review letters.

[22]  Usa,et al.  -symmetric electronics , 2012, 1209.2347.

[23]  B. Malomed.  Perturbative analysis of the interaction of a phi 4 kink with inhomogeneities , 1992 .

[24]  David K. Campbell,et al.  Resonance structure in kink-antikink interactions in φ4 theory , 1983 .

[25]  Y. Kivshar,et al.  Nonlocality in PT-symmetric waveguide arrays with gain and loss. , 2012, Optics letters.

[26]  P. Kevrekidis,et al.  On the Spectral Stability of Kinks in Some PT ‐Symmetric Variants of the Classical Klein–Gordon Field Theories , 2014, Studies in Applied Mathematics.

[27]  Yuri S. Kivshar,et al.  Dynamics of Solitons in Nearly Integrable Systems , 1989 .

[28]  Zach DeVito,et al.  Opt , 2017 .

[29]  A. Kudryavtsev,et al.  Quasi-periodical orbits in the scalar classical λφ 4 field theory , 1988 .

[30]  S. Longhi,et al.  Spectral singularities and Bragg scattering in complex crystals , 2010, 1001.0962.

[31]  B. Malomed,et al.  Wave scattering on a domain wall in a chain of PT-symmetric couplers , 2012, 1202.5629.

[32]  C. Bender,et al.  Observation of PT phase transition in a simple mechanical system , 2012, 1206.4972.

[33]  V. Gani,et al.  Kink interactions in the (1+1)-dimensional $\phi^6$ model , 2014, 1402.5903.

[34]  I. V. Barashenkov,et al.  Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer , 2013 .

[35]  P. Kevrekidis,et al.  $\mathcal{P}\mathcal{T}$-symmetric sine-Gordon breathers , 2014 .

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

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

[38]  I. V. Barashenkov,et al.  Actively coupled optical waveguides , 2013, 1308.5862.

[39]  P. Kevrekidis,et al.  The sine-Gordon Model and its Applications , 2014 .

[40]  Integrability of PT-symmetric dimers , 2013, 1307.2788.

[41]  P. Kevrekidis,et al.  Asymmetric wave propagation through nonlinear PT-symmetric oligomers , 2012, 1202.4483.

[42]  Radial sine-Gordon kinks as sources of fast breathers. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Demetrios N. Christodoulides,et al.  Exceptional-point dynamics in photonic honeycomb lattices with PT symmetry , 2011, 1112.4734.

[44]  E. Graefe.  Stationary states of a PT symmetric two-mode Bose–Einstein condensate , 2012, 1206.4806.

[45]  B. Malomed,et al.  $\mathcal{PT}$-symmetric ladders with a scattering core , 2014, 1407.1086.

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

[47]  P. Kevrekidis,et al.  Generalized dimers and their Stokes-variable dynamics , 2014, 1404.4382.

[48]  P. Kevrekidis.  Variational method for nonconservative field theories: Formulation and two PT-symmetric case examples , 2014 .

[50]  T. Dobrowolski,et al.  A numerical test of the collective coordinate method , 2008 .

[51]  W. Zakrzewski,et al.  Scattering of sine-Gordon breathers on a potential well. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  B. Malomed,et al.  Nonlinear -symmetric plaquettes , 2012, 1204.5530.

[53]  Richard Haberman,et al.  Kink-Antikink Collisions in the φ4 Equation: The n-Bounce Resonance and the Separatrix Map , 2005, SIAM J. Appl. Dyn. Syst..

[54]  P. Kevrekidis,et al.  Effect of the ϕ4 kink’s internal mode at scattering on a PT-symmetric defect , 2015 .

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

[56]  Panayotis G. Kevrekidis,et al.  Nonlinear Stationary States in PT-Symmetric Lattices , 2013, SIAM J. Appl. Dyn. Syst..

[57]  Holger Cartarius,et al.  Model of a PT-symmetric Bose-Einstein condensate in a delta-function double-well potential , 2012, 1203.1885.

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

[59]  S. V. Suchkov,et al.  Probability of breaking of the PJ symmetry at collision between breathers with random phases in the model of a PJ symmetric planar coupler , 2014 .

[60]  Yuri S. Kivshar,et al.  Internal Modes of Solitary Waves , 1998 .

[61]  1 Ju l 2 00 9 Wobbling kinks in φ 4 theory , 2008 .

[62]  W. Zakrzewski,et al.  Collective coordinate approximation to the scattering of solitons in the (1+1) dimensional NLS model , 2013, 1308.4072.

[63]  Mohammad Kazem Moravvej-Farshi,et al.  A 2×2 spatial optical switch based on PT-symmetry. , 2011, Optics letters.

[64]  V. Konotop,et al.  Nonlinear modes in a generalized PT?>-symmetric discrete nonlinear Schrödinger equation , 2013, 1310.5651.

[65]  B. Malomed,et al.  Interactions of bright and dark solitons with localized PT-symmetric potentials. , 2014, Chaos.

[66]  P. Kevrekidis,et al.  Nonlinear dynamics in PT-symmetric lattices , 2013, 1307.2973.

[67]  Y. Kivshar,et al.  Resonant kink-impurity interactions in the sine-Gordon model. , 1992, Physical review. A, Atomic, molecular, and optical physics.

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

[69]  K. Javidan,et al.  Collective-Coordinate Analysis of Inhomogeneous Nonlinear Klein–Gordon Field Theory , 2011, 1109.4922.

[70]  Y. Kivshar,et al.  Resonant soliton-impurity interactions. , 1991, Physical review letters.

[71]  Y. Kivshar,et al.  Binary parity-time-symmetric nonlinear lattices with balanced gain and loss. , 2010, Optics letters.

[72]  Y. Kivshar,et al.  Fractal structures and multiparticle effects in soliton scattering. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.