Soliton and related problems in nonlinear physics
Department of Physics, Northwest University
Zhan-Ying Yang , Li-Chen Zhao and Chong Liu
OutlineOutline
soliton
Introduction of optical soliton
Two solitons' interference
Nonautonomous Solitons
rogue wave
Introduction of optical rogue wave
Nonautonomous rogue wave
Rogur wave in two and three mode nonlinear fiber
Introduction of solitonIntroduction of soliton
Solitons, whose first known description in the scientific literature, in the form of ‘‘a large solitary elevation, a rounded, smooth, and well-defined heap of water,’’ goes back to the historical observation made in a chanal near Edinburgh by John Scott Russell in the 1830s.
Introduction of optical solitonIntroduction of optical soliton
Zabusky and Kruskal introduced for the first time the soliton concept to characterize nonlinear solitary waves that do not disperse and preserve their identity during propagation and after a collision. (Phys. Rev. Lett. 15, 240 (1965) )
Optical solitons. A significant contribution to the experimental and theoretical studies of solitons was the identification of various forms of robust solitary waves in nonlinear optics.
Introduction of optical solitonIntroduction of optical soliton
Optical solitons can be subdivided into two broad categories—spatial and temporal.
G.P. Agrawal, Nonlinear Fiber Optics, Acdemic press (2007).
Temporal soliton in nonlinear fiber
Spatial soliton in a waveguide
Two solitons' interference Two solitons' interference
We study continuous wave optical beams propagating inside a planar nonlinear waveguide
Two solitons' interference Two solitons' interference
Then we can get
The other soliton’s incident angle can be read out, and the nonlinear parameter g will be given
History of Nonautonomous SolitonsHistory of Nonautonomous Solitons
Novel Soliton Solutions of the Nonlinear Schrödinger Equation Model; Vladimir N. Serkin and Akira Hasegawa Phys. Rev. Lett. 85, 4502 (2000) .
Nonautonomous Solitons in External Potentials; V. N. Serkin, Akira Hasegawa,and T. L. Belyaeva Phys. Rev. Lett. 98, 074102 (2007).
Analytical Light Bullet Solutions to the Generalized(3 +1 )-DimensionalNonlinear Schrodinger Equation. Wei-Ping Zhong. Phys. Rev. Lett. 101, 123904 (2008).
A: The test of solitons in nonuniform media with time-dependent density gradients . ( spatial soliton ) B: The test of the core medium of the real fibers, which cannot be homogeneous, fiber loss is inevitable, and dissipation weakens the nonlinearity. ( temporal soliton )
A: The test of solitons in nonuniform media with time-dependent density gradients . ( spatial soliton ) B: The test of the core medium of the real fibers, which cannot be homogeneous, fiber loss is inevitable, and dissipation weakens the nonlinearity. ( temporal soliton )
Reason:
Nonautonomous SolitonsNonautonomous Solitons
Engineering integrable nonautonomous nonlinear Schrödinger equations , Phys. Rev. E. 79, 056610 (2009), Hong-Gang Luo, et.al.)
Bright Solitons solution by Darboux transformation Bright Solitons solution by Darboux transformation
Dynamics of a nonautonomous soliton in a generalized nonlinear Schrodinger equation ,Phys. Rev. E. 83, 066602 (2011) , Z. Y. Yang, et.al.)
Under the integrability condition
We get
Nonautonomous bright SolitonsNonautonomous bright Solitons
under the compatibility condition
We obtain the developing equation.
Nonautonomous bright SolitonsNonautonomous bright Solitons
we can derive the evolution equation of Q as follows:
the Darboux transformation can be presented as
Nonautonomous bright SolitonsNonautonomous bright Solitons
we obtain
Finally, we obtain the solution as
Dynamic description
Dark Solitons solution by Hirota's bilinearization methodDark Solitons solution by Hirota's bilinearization method
Dark Solitons solution by Hirota's bilinearization methodDark Solitons solution by Hirota's bilinearization method
We assume the solution as
Where g(x,t) is a complex function and f(x,t) is a real function
Dark Solitons solution by Hirota's bilinearization methodDark Solitons solution by Hirota's bilinearization method
by Hirota's bilinearization method, we reduce Eq.(6) as
For dark soliton
For bright soliton
Dark Solitons solution by Hirota's bilinearization methodDark Solitons solution by Hirota's bilinearization method
Then we have one dark soliton solution
corresponding to the different powers of χ
Dark Solitons solution by Hirota's bilinearization methodDark Solitons solution by Hirota's bilinearization method
Two dark soliton solution
corresponding to the different powers of χ
Dark Solitons solution by Hirota's bilinearization methodDark Solitons solution by Hirota's bilinearization method
From the above bilinear equations, we obtain the dark soliton soliution as :
Dark Solitons solution by Hirota's bilinearization methodDark Solitons solution by Hirota's bilinearization method
Dynamic description of one dark soliton
Nonautonomous bright Solitons in optical fiberNonautonomous bright Solitons in optical fiber
Dynamics of a nonautonomous soliton in a generalized nonlinear Schrodinger equation ,Phys. Rev. E. 83, 066602 (2011) , J. Opt. Soc. Am. B 28 , 236 (2011) ,Z. Y. Yang, L.C.Zhao et.al.)
Nonautonomous dark Solitons in optical fiberNonautonomous dark Solitons in optical fiber
Nonautonomous dark Solitons in optical fiberNonautonomous dark Solitons in optical fiber
Nonautonomous Solitons in a graded-index waveguideNonautonomous Solitons in a graded-index waveguide
Snakelike nonautonomous solitons in a graded-index grating waveguide , Phys. Rev. A 81 , 043826 (2010), Optic s Commu nications 283 (2010) 3768 . Z. Y. Yang, L.C.Zhao et.al.)
Nonautonomous Solitons in a graded-index waveguideNonautonomous Solitons in a graded-index waveguide
Nonautonomous Solitons in a graded-index waveguideNonautonomous Solitons in a graded-index waveguide
Without the grating , we get
Nonautonomous Solitons in a graded-index waveguideNonautonomous Solitons in a graded-index waveguide
Nonautonomous Solitons in a graded-index waveguideNonautonomous Solitons in a graded-index waveguide
Introduction of rogue waveIntroduction of rogue wave
Oceannography Vol.18 , No.3 , Sept. 2005 。
Mysterious freak wave, killer wave
Introduction of rogue waveIntroduction of rogue wave
Observe “New year” wave in 1995, North sea
D.H.Peregrine, Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B25,1643 (1983);Wave appears from nowhere and disappears without a trace,N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373 (2009) 675
M. Onorato, D. Proment, Phys. Lett. A 376, 3057-3059(2012).
Forced and damped nonlinear Schrödinger equation
B. Kibler, J. Fatome, et al., Nature Phys. 6, 790 (2010).
Experimental observation(optical fiber)
As rogue waves are exceedingly difficult to study directly, the relationship between rogue waves and solitons has not yet been definitively established, but it is believed that they are connected. Optical rogue waves.Nature 450,1054-1057 (2007)
A. Chabchoub, N. P. Hoffmann, et al., Phys. Rev. Lett. 106, 204502 (2011).
B. Kibler, J. Fatome, et al., Nature Phys. 6, 790 (2010). Scientific Reports . 2.463(2012) .In optical fiber
Experimental observation(optical fiber and water tank)
Optical rogue wave in a graded-index waveguide Optical rogue wave in a graded-index waveguide
Long-life rogue wave Long-life rogue wave Classical rogue wave Classical rogue wave
Optical rogue wave in a graded-index waveguide Optical rogue wave in a graded-index waveguide
Rogue wave in Two-mode fiberRogue wave in Two-mode fiber
F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, Phys. Rev. Lett. 109, 044102 (2012).
B.L. Guo, L.M. Ling, Chin. Phys. Lett. 28, 110202 (2011).
Bright rogue wave and dark rogue wave
Two rogue wave
L.C.Zhao, J. Liu, Joun. Opt. Soc. Am. B 29, 3119-3127 (2012)
Rogue wave of four-petaled flower
Eye-shaped rogue wave
Rogue wave in Three-mode fiberRogue wave in Three-mode fiber
One rogue wave in three-mode fiber
Rogue wave of four-petaled flower
Eye-shaped rogue wave
Rogue wave in Three-mode fiberRogue wave in Three-mode fiber
Two rogue wave in three-mode fiber
Rogue wave in Three-mode fiberRogue wave in Three-mode fiber
Three rogue wave in three-mode fiber
Rogue wave in Three-mode fiberRogue wave in Three-mode fiber
The interaction of three rogue wave
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