Abstract
We consider an integrable nonlinear wave system (anisotropic chiral field model) which exhibits a soliton solution when the Cauchy problem for an infinitely long medium is posed. Whenever the boundary value problem is formulated for the same system but for a medium of finite extension, we reveal that the soliton becomes unstable and the true attractor is a different structure which is called polarization attractor. In contrast to the localized nature of solitons, the polarization attractor occupies the entire length of the medium. By demonstrating the qualitative difference between nonlinear wave propagation in an infinite medium and in a medium of finite extension (with simultaneous change of the initial value problem to the boundary value problem), we would like to point out that solitons may loose their property of being stable attractors. Additionally, our findings show the interest of developing methods of integration for boundary value problems.
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Assémat E., Lagrange S., Picozzi A., Jauslin H.R., Sugny D.: Complete nonlinear polarization control in an optical fiber system. Opt. lett. 35, 2025 (2010)
Degasperis A., Manakov S.V., Santini P.M.: On the initial-boundary value problems for soliton equations. JETP Lett. 74, 481–485 (2001)
Fatome J., Pitois S., Morin P., Millot G.: Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications. Opt. Express 18, 15311–15317 (2010)
Lagrange S., Sugny D., Picozzi A., Jauslin H.R.: Singular tori as attractors of four-wave-interaction systems. Phys. Rev. E 81, 016202 (2010)
Martijn de Sterke C., Jackson K.R., Robert B.D.: Nonlinear coupled-mode equations on a finite interval: a numerical procedure. J. Opt. Soc. Am. 8, 403–412 (1991)
Menyuk C.R., Marks B.S.: Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems. J. Lightwave Technol. 24, 2806–2826 (2006)
Pitois S., Picozzi A., Millot G., Jauslin H.R., Haelterman M.: Polarization and modal attractors in conservative counterpropagating four-wave interaction. Europhys. Lett. 70, 88–94 (2005)
Pitois S., Millot G., Wabnitz S.: Polarization domain wall solitons with counterpropagating laser beams. Phys. Rev. Lett. 81, 1409–1412 (1998)
Pitois S., Millot G., Wabnitz S.: Nonlinear polarization dynamics of counterpropagating waves in an isotropic optical fiber: theory and experiments. J. Opt. Soc. Am. B 18, 432–443 (2001)
Pitois S., Fatome J., Millot G.: Polarization attraction using counter-propagating waves in optical fiber at telecommunication wavelengths. Opt. Express 16, 6646–6651 (2008)
Sugny D., Picozzi A., Lagrange S., Jauslin H.R.: Role of singular tori in the dynamics of spatiotemporal nonlinear wave systems. Phys. Rev. Lett. 103, 034102 (2009)
Wabnitz S.: Chiral polarization solitons in elliptically birefringent spun optical fibers. Opt. Lett. 34, 908–910 (2009)
Wabnitz S.: Cross polarization modulation domain wall solitons for WDM signals in birefringent optical fibers. Photonic Technol. Lett. 21, 875–877 (2009)
Zakharov V.E., Mikhailov A.V.: Polarization domains in nonlinear optics. JETP Lett. 45, 349–352 (1987)
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Kozlov, V.V., Wabnitz, S. Instability of Optical Solitons in the Boundary Value Problem for a Medium of Finite Extension. Lett Math Phys 96, 405–413 (2011). https://doi.org/10.1007/s11005-010-0431-3
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DOI: https://doi.org/10.1007/s11005-010-0431-3