Control of orbit and control of chaos in a class of dynamic system

Gen-guo, LI; Zhu, Zhihui

doi:10.1007/s11741-999-0001-z

Cited by 3 publications

(4 citation statements)

References 4 publications

(4 reference statements)

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“…x − = UV VU 0, one can directly obtain equation (2). Next, according to [51] (section 1.4) and [52] (section 4.2), we know that the following transformation,…”

Section: Gauge Transformation and Lax Pairmentioning

confidence: 99%

Higher-order rogue wave solutions of the Kundu–Eckhaus equation

et al. 2014

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In this paper, we investigate higher-order rogue wave solutions of the Kundu-Eckhaus equation, which contains quintic nonlinearity and the Raman effect in nonlinear optics. By means of a gauge transformation, the Kundu-Eckhaus equation is converted to an extended nonlinear Schrödinger equation. We derive the Lax pair, the generalized Darboux transformation, and the Nth-order rogue wave solution for the extended nonlinear Schrödinger equation. Then, by using the gauge transformation between the two equations, a concise unified formula of the Nth-order rogue wave solution with several free parameters for the Kundu-Eckhaus equation is obtained. In particular, based on symbolic computation, explicit rogue wave solutions to the Kundu-Eckhaus equation from the first to the third order are presented. Some figures illustrate dynamic structures of the rogue waves from the first to the fourth order. Moreover, through numerical calculations and plots, we show that the quintic and Raman-effect nonlinear terms affect the spatial distributions of the humps in higher-order rogue waves, although the amplitudes and the time of appearance of the humps are unchanged.

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“…x − = UV VU 0, one can directly obtain equation (2). Next, according to [51] (section 1.4) and [52] (section 4.2), we know that the following transformation,…”

Section: Gauge Transformation and Lax Pairmentioning

confidence: 99%

Higher-order rogue wave solutions of the Kundu–Eckhaus equation

et al. 2014

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“…solutions ( 15)- (20), have been presented and an infinite number of conservation laws, i.e. expressions ( 24)- (30), have also been derived. Figures 1-10 have been plotted via solutions ( 17)- (20) to display the dynamic features of the soliton solutions: (1) figures 1 and 3, respectively, display the head-on and overtaking collisions between two solitons via expressions ( 17)- (18), and the two colliding solitons recover their shapes, amplitudes and velocities after the collisions except for the slight phase shifts, indicating that the collisions are elastic.…”

Section: Discussionmentioning

confidence: 99%

“…The DT, which is comprised of eigenfunction transformation and potential transformation, can be used to construct a series of explicit solutions for the nonlinear evolution equations from the initial ones in a recursive manner [27,28]. Moreover, the iterative algorithm of the DT can be achieved via symbolic computation [29,30].…”

Section: New Lax Pair and Darboux Transformation Of Equations (1) Wit...mentioning

confidence: 99%

Conservation laws and solitons for the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

Shan

Guo

et al. 2011

Phys. Scr.

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Under investigation in this paper are the coupled cubic–quintic nonlinear Schrödinger equations describing the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media. Lax pair of the equations is obtained via the Ablowitz–Kaup–Newell–Segur scheme and the corresponding Darboux transformation is constructed. One-, two- and three-soliton solutions are presented and an infinite number of conservation laws are also derived. The features of solitons are graphically discussed: (i) head-on and overtaking elastic collisions of the two solitons; (ii) periodic attraction and repulsion of the bounded states of two solitons; (iii) energy-exchanging collisions of the three solitons.

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“…(17)(18)(19)(20) can be reduced as a generalized Hirota's lattice equations with variable coefficients:…”

Section: E T S Z R S C T S R Zt D Tmentioning

confidence: 99%

A System of Nonlinear Differential-Difference Equations with Variable Coefficients and Its Reductions

Zhang¹,

Guo²

2015

Proceedings of the 2015 International Industrial Informatics and Computer Engineering Conference

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Abstract. Constructing integrable systems is a significant direction in soliton theory. In this paper, a new system of nonlinear differential-difference equations with variable coefficients is derived by introducing some derivable functions to the corresponding discrete spectral problems. In order to give some special cases of the derived differential-difference equations, three reductions are obtained which include Hirota's lattice equations as special cases. The processes of constructing such a system of variable-coefficient differential-difference equations and obtaining its reductions provide with a necessary help for the beginners.

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Control of orbit and control of chaos in a class of dynamic system

Cited by 3 publications

References 4 publications

Higher-order rogue wave solutions of the Kundu–Eckhaus equation

Higher-order rogue wave solutions of the Kundu–Eckhaus equation

Conservation laws and solitons for the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics

A System of Nonlinear Differential-Difference Equations with Variable Coefficients and Its Reductions

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