Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear PT -symmetric defect |
Optics Express, Vol. 22, Issue 11, pp. 13927-13939 (2014)
http://dx.doi.org/10.1364/OE.22.013927
Acrobat PDF (3387 KB)
Abstract
Discrete fundamental and dipole solitons are constructed, in an exact analytical form, in an array of linear waveguides with an embedded PT -symmetric dimer, which is composed of two nonlinear waveguides carrying equal gain and loss. Fundamental solitons in tightly knit lattices, as well as all dipole modes, exist above a finite threshold value of the total power. However, the threshold vanishes for fundamental solitons in loosely knit lattices. The stability of the discrete solitons is investigated analytically by means of the Vakhitov-Kolokolov (VK) criterion, and, in the full form, via the computation of eigenvalues for perturbation modes. Fundamental and dipole solitons tend to be stable at smaller and larger values of the total power (norm), respectively. The increase of the strength of the coupling between the two defect-forming sites leads to strong expansion of the stability areas. The scattering problem for linear lattice waves impinging upon the defect is considered too.
© 2014 Optical Society of America
1. Introduction
1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]
4. Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012). [CrossRef]
2. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef]
5. D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001). [CrossRef] [PubMed]
6. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002). [CrossRef]
7. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef]
8. A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003). [CrossRef]
10. A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B: At. Mol. Opt. Phys. 43, 163001 (2010). [CrossRef]
11. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007). [CrossRef]
12. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008). [CrossRef] [PubMed]
13. B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006). [CrossRef] [PubMed]
14. B. A. Malomed and P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001). [CrossRef]
18. C. Mejía-Cortés, J. M. Soto-Crespo, M. I. Molina, and R. Vicencio, “Dissipative vortex solitons in two-dimensional lattices,” Phys. Rev. A 82, 063818 (2010). [CrossRef]
19. D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004). [CrossRef] [PubMed]
21. B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Królikowski, F. Kaiser, C. Denz, and Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. 101, 013903 (2008). [CrossRef]
22. U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Nonlinearly induced escape from a defect state in waveguide arrays,” Appl. Phys. Lett. 75, 1348–1350 (1999). [CrossRef]
27. Y. Li, W. Pang, Y. Chen, Z. Yu, J. Zhou, and H. Zhang, “Defect-mediated discrete solitons in optically induced photorefractive lattices,” Phys. Rev. A 80, 043824 (2009). [CrossRef]
28. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005). [CrossRef]
32. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011). [CrossRef]
33. S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011). [CrossRef]
35. A. Regensburger, M. A. Miri, C. Bersch, and J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013). [CrossRef]
36. D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012). [CrossRef]
37. I. V. Barashenkov, G. S. Jackson, and S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013). [CrossRef]
41. T. Mayteevarunyoo, B. A. Malomed, and A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013). [CrossRef]
42. B. Maes, M. Soljačić, J. D. Joannopoulos, P. Bienstman, R. Baets, S.-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006). [CrossRef]
44. E. Bulgakov, A. Sadreev, and K. N. Pichugin, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011). [CrossRef]
45. M. I. Molina and G. P. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330 (1993). [CrossRef]
46. B. C. Gupta and K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997). [CrossRef]
47. V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011). [CrossRef]
28. A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005). [CrossRef]
32. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011). [CrossRef]
2. The model
48. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003). [CrossRef]
1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]
3. Analytical and numerical results
3.1. The analytical consideration
47. V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011). [CrossRef]
49. B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012). [CrossRef]
41. T. Mayteevarunyoo, B. A. Malomed, and A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013). [CrossRef]
50. M. Vakhitov and A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron 16, 783–789 (1973). [CrossRef]
52. E. A. Kuznetsov and F. Dias, “Bifurcations of solitons and their stability,” Phys. Rep. 507, 43–105 (2011). [CrossRef]
3.2. Numerical results for fundamental solitons
53. M. L. Chiofalo, S. Succi, and M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438–7444 (2000). [CrossRef]
54. J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009). [CrossRef]
37. I. V. Barashenkov, G. S. Jackson, and S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013). [CrossRef]
45. M. I. Molina and G. P. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330 (1993). [CrossRef]
46. B. C. Gupta and K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997). [CrossRef]
47. V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011). [CrossRef]
47. V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011). [CrossRef]
3.3. Numerical results for dipole solitons
37. I. V. Barashenkov, G. S. Jackson, and S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013). [CrossRef]
4. Scattering of incident waves on the defect
5. Conclusion
Acknowledgments
References and links
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2. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef] |
3. | I. L. Garanovich, S. Longhi, A. A. Sukhorukova, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012). [CrossRef] |
4. | Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012). [CrossRef] |
5. | D. N. Christodoulides and E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001). [CrossRef] [PubMed] |
6. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002). [CrossRef] |
7. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] |
8. | A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003). [CrossRef] |
9. | A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006). [CrossRef] |
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13. | B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006). [CrossRef] [PubMed] |
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19. | D. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92, 123903 (2004). [CrossRef] [PubMed] |
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28. | A. Ruschhaupt, F. Delgado, and J. G. Muga, “Physical realization of PT-symmetric potential scattering in a planar slab waveguide,” J. Phys. A 38, L171–L176 (2005). [CrossRef] |
29. | K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008). [CrossRef] |
30. | S. Longhi, “Spectral singularities and Bragg scattering in complex crystals,” Phys. Rev. A 81, 022102 (2010). [CrossRef] |
31. | C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192–195 (2010). [CrossRef] |
32. | K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011). [CrossRef] |
33. | S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Y. S. Kivshar, “Scattering of linear and nonlinear waves in a waveguide array with a PT-symmetric defect,” Phys. Rev. A 84, 013833 (2011). [CrossRef] |
34. | S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, “Scattering of the discrete solitons on the PT-symmetric defects,” Europhys. Lett. 100, 54003 (2012). [CrossRef] |
35. | A. Regensburger, M. A. Miri, C. Bersch, and J. Näger, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013). [CrossRef] |
36. | D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in finite-dimensional PT-symmetric systems,” Phys. Rev. Lett. 108, 213906 (2012). [CrossRef] |
37. | I. V. Barashenkov, G. S. Jackson, and S. Flach, “Blow-up regimes in the PT-symmetric coupler and the actively coupled dimer,” Phys. Rev. A 88, 053817 (2013). [CrossRef] |
38. | K. Li, D. A. Zezyulin, P. G. Kevrekidis, V. V. Konotop, and F. K. Abdullaev, “PT-symmetric coupler with χ^{(2)} nonlinearity,” Phys. Rev. A 88, 053820 (2013). [CrossRef] |
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41. | T. Mayteevarunyoo, B. A. Malomed, and A. Roeksabutr, “Solvable model for solitons pinned to a parity-time-symmetric dipole,” Phys. Rev. E 88, 022919 (2013). [CrossRef] |
42. | B. Maes, M. Soljačić, J. D. Joannopoulos, P. Bienstman, R. Baets, S.-P. Gorza, and M. Haelterman, “Switching through symmetry breaking in coupled nonlinear micro-cavities,” Opt. Express 14, 10678–10683 (2006). [CrossRef] |
43. | E. N. Bulgakov and A. F. Sadreev, “Bound states in photonic Fabry-Perot resonator with nonlinear off-channel defects,” Phys. Rev. B 81, 115128 (2010). [CrossRef] |
44. | E. Bulgakov, A. Sadreev, and K. N. Pichugin, “Symmetry breaking for transmission in a photonic waveguide coupled with two off-channel nonlinear defects,” Phys. Rev. B 83, 045109 (2011). [CrossRef] |
45. | M. I. Molina and G. P. Tsironis, “Nonlinear impurities in a linear chain,” Phys. Rev. B 47, 15330 (1993). [CrossRef] |
46. | B. C. Gupta and K. Kundu, “Formation of stationary localized states due to nonlinear impurities using the discrete nonlinear Schrödinger equation,” Phys. Rev. B 55, 894–905 (1997). [CrossRef] |
47. | V. A. Brazhnyi and B. A. Malomed, “Spontaneous symmetry breaking in Schrödinger lattices with two nonlinear sites,” Phys. Rev. A 83, 053844 (2011). [CrossRef] |
48. | Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D 36, R1 (2003). [CrossRef] |
49. | B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, “Pinned modes in lossy lattices with local gain and nonlinearity,” Phys. Rev. E 86, 036608 (2012). [CrossRef] |
50. | M. Vakhitov and A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron 16, 783–789 (1973). [CrossRef] |
51. | L. Bergé, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. 303, 259–370 (1998). [CrossRef] |
52. | E. A. Kuznetsov and F. Dias, “Bifurcations of solitons and their stability,” Phys. Rep. 507, 43–105 (2011). [CrossRef] |
53. | M. L. Chiofalo, S. Succi, and M. P. Tosi, “Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm,” Phys. Rev. E 62, 7438–7444 (2000). [CrossRef] |
54. | J. Yang, “Newton-conjugate-gradient methods for solitary wave computations,” J. Comput. Phys. 228, 7007–7024 (2009). [CrossRef] |
55. | H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures”, Phys. Rev. A 82, 043803 (2010). [CrossRef] |
56. | S. Longhi, “Invisibility in PT-symmetric complex crystals”, J. Phys. A: Math. Theor. 44, 485302 (2011). [CrossRef] |
57. | Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, “Experimental observation of the dual behavior of PT-symmetric scattering,” Phys. Rev. A 85, 050101 (2012). [CrossRef] |
OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.3270) Nonlinear optics : Kerr effect
(190.6135) Nonlinear optics : Spatial solitons
ToC Category:
Nonlinear Optics
History
Original Manuscript: April 8, 2014
Revised Manuscript: May 22, 2014
Manuscript Accepted: May 23, 2014
Published: May 30, 2014
Citation
Xiangyu Zhang, Jinglei Chai, Jiasheng Huang, Zhiqiang Chen, Yongyao Li, and Boris A. Malomed, "Discrete solitons and scattering of lattice waves in guiding arrays with a nonlinear PT -symmetric defect," Opt. Express 22, 13927-13939 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13927
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References
- F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]
- D. N. Christodoulides, F. Lederer, Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003). [CrossRef]
- I. L. Garanovich, S. Longhi, A. A. Sukhorukova, Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518, 1–79 (2012). [CrossRef]
- Z. Chen, M. Segev, D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012). [CrossRef]
- D. N. Christodoulides, E. D. Eugenieva, “Blocking and routing discrete solitons in two-dimensional networks of nonlinear waveguide arrays,” Phys. Rev. Lett. 87, 233901 (2001). [CrossRef] [PubMed]
- N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002). [CrossRef]
- J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef]
- A. A. Sukhorukov, Y. S. Kivshar, H. S. Eisenberg, Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31–50 (2003). [CrossRef]
- A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, “Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica,” Opt. Express 14, 6055–6062 (2006). [CrossRef]
- A. Szameit, S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B: At. Mol. Opt. Phys. 43, 163001 (2010). [CrossRef]
- T. Schwartz, G. Bartal, S. Fishman, M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007). [CrossRef]
- Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008). [CrossRef] [PubMed]
- B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature 440, 1166–1169 (2006). [CrossRef] [PubMed]
- B. A. Malomed, P. G. Kevrekidis, “Discrete vortex solitons,” Phys. Rev. E 64, 026601 (2001). [CrossRef]
- P. G. Kevrekidis, B. A. Malomed, Y. B. Gaididei, “Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity,” Phys. Rev. E 66, 016609 (2002). [CrossRef]
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