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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