Transcript as For dark soliton For bright soliton

Soliton and related problems in
nonlinear physics
Zhan-Ying Yang , Li-Chen Zhao and Chong Liu
Department of Physics, Northwest University
Outline
Introduction of optical soliton
soliton
Two solitons' interference
Nonautonomous Solitons
Introduction of optical rogue wave
rogue wave
Nonautonomous rogue wave
Rogur wave in two and three mode
nonlinear fiber
Introduction 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 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 soliton
Optical solitons can be subdivided into two broad categories—
spatial and temporal.
Temporal soliton in nonlinear fiber
Spatial soliton in a waveguide
G.P. Agrawal, Nonlinear Fiber Optics, Acdemic
press (2007).
Two solitons' interference
We study continuous wave optical
beams propagating inside a planar
nonlinear waveguide
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 Solitons
Reason: 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）
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 )-Dimensional
Nonlinear Schrodinger Equation.
Wei-Ping Zhong. Phys. Rev. Lett. 101, 123904
(2008).
Nonautonomous Solitons
Engineering integrable nonautonomous
nonlinear Schrödinger equations , Phys. Rev. E.
79, 056610 (2009), Hong-Gang Luo, et.al.)
Bright Solitons solution by Darboux transformation
Under the integrability condition
We get
Dynamics of a nonautonomous soliton in a
generalized nonlinear Schrodinger
equation ,Phys. Rev. E. 83, 066602 (2011) , Z. Y.
Yang, et.al.)
Nonautonomous bright Solitons
under the compatibility condition
We obtain the developing equation.
Nonautonomous bright Solitons
the Darboux transformation
can be presented as
we can derive the evolution equation of
Q as follows:
Nonautonomous bright Solitons
we obtain
Finally, we obtain the solution as
Dynamic description
Dark Solitons solution by Hirota's bilinearization method
Dark 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 method
by Hirota's bilinearization method, we reduce Eq.(6) as
For dark
soliton
For bright
soliton
Dark 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 method
Two dark soliton solution
corresponding to the
different powers of χ
Dark 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 method
Dynamic description of one dark soliton
Nonautonomous 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 fiber
Nonautonomous dark Solitons in optical fiber
Nonautonomous 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 waveguide
Nonautonomous Solitons in a graded-index waveguide
Without the grating , we get
Nonautonomous Solitons in a graded-index waveguide
Nonautonomous Solitons in a graded-index waveguide
Introduction of rogue wave
Mysterious freak wave, killer wave
Oceannography Vol.18，No.3，Sept. 2005。
Introduction 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
Forced and damped nonlinear Schrödinger equation
M. Onorato, D. Proment, Phys. Lett. A 376,
3057-3059(2012).
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)
B. Kibler, J. Fatome, et al., Nature Phys.
6, 790 (2010).
Experimental observation(optical fiber and water tank
B. Kibler, J. Fatome,
et al., Nature Phys.
6, 790 (2010).
Scientific Reports .
2.463(2012) .In
optical fiber
A. Chabchoub, N. P.
Hoffmann, et al., Phys.
Rev. Lett. 106, 204502
(2011).
Optical rogue wave in a graded-index waveguide
Classical rogue wave
Long-life rogue wave
Optical rogue wave in a graded-index waveguide
Rogue 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
Rogue wave of four-petaled flower
Eye-shaped rogue wave
L.C.Zhao, J. Liu, Joun. Opt. Soc. Am. B 29,
3119-3127 (2012)
Two rogue wave
Rogue 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 fiber
Two rogue wave in three-mode fiber
Rogue wave in Three-mode fiber
Three rogue wave in three-mode fiber
Rogue wave in Three-mode fiber
The interaction of three rogue wave
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