Localization of the sine-Gordon equation solutions

Порубов, А. В.; Fradkov, Alexander L.; Bondarenkov, R. S.; Andrievsky, Boris

doi:10.1016/j.cnsns.2016.02.043

Cited by 10 publications

(5 citation statements)

References 14 publications

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“…In contrast, distributive control has infinite dimensions and may be different for every spatial coordinate in the system [1,2,7,8]. Our non-distributive algorithm does not allow us to localize the nonlinear wave solution to the sine-Gordon equation [14,17]. Therefore, it seems that localized waves may be efficiently controlled by a distributive algorithm.…”

Section: Choice Of Control Methodsmentioning

confidence: 93%

“…Let us illustrate this with the example of the sine-Gordon equation. Our previous results [15,17] show that the control of the coefficient allows us to provide stable propagation of those waves whose shape is described by analytical travelling wave solutions to the equation. However, waves with another shape are not generated.…”

Section: Choice Of Control Methodsmentioning

confidence: 99%

“…However, waves with another shape are not generated. This is why modification of the algorithm from [15,17] Then, one obtains…”

Section: Choice Of Control Methodsmentioning

confidence: 99%

“…As noted before, the control of solutions to the sine-Gordon equation has been studied in our previous papers [15,17]. However, it was found that only particular localized waves may be supported by the algorithm used in [15,17]. Now, we first consider the improved algorithm developed in §3 to achieve the target function U * (x, t),…”

Section: Control Of Localized Nonlinear Waves (A) Single Equationmentioning

confidence: 99%

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Control methods for localization of nonlinear waves

Порубов

Andrievsky

2017

Phil. Trans. R. Soc. A.

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A general form of a distributed feedback control algorithm based on the speed-gradient method is developed. The goal of the control is to achieve nonlinear wave localization. It is shown by example of the sine-Gordon equation that the generation and further stable propagation of a localized wave solution of a single nonlinear partial differential equation may be obtained independently of the initial conditions. The developed algorithm is extended to coupled nonlinear partial differential equations to obtain consistent localized wave solutions at rather arbitrary initial conditions.This article is part of the themed issue 'Horizons of cybernetical physics'.

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Section: Choice Of Control Methodsmentioning

confidence: 93%

Section: Choice Of Control Methodsmentioning

confidence: 99%

“…However, waves with another shape are not generated. This is why modification of the algorithm from [15,17] Then, one obtains…”

Section: Choice Of Control Methodsmentioning

confidence: 99%

Section: Control Of Localized Nonlinear Waves (A) Single Equationmentioning

confidence: 99%

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Control methods for localization of nonlinear waves

Порубов

Andrievsky

2017

Phil. Trans. R. Soc. A.

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“…Recently, the SG algorithms were successfully applied to control of energy in the distributed systems described by a nonlinear PDE (sine-Gordon equation) [38]. Extensions to localization of nonlinear waves problem are presented in [39,40]. The first rigorous results justifying the SG energy control for PDE are obtained in [41].…”

Section: (B) Control Of Wavesmentioning

confidence: 99%

Horizons of cybernetical physics

Fradkov

2017

Phil. Trans. R. Soc. A.

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The subject and main areas of a new research field—cybernetical physics—are discussed. A brief history of cybernetical physics is outlined. The main areas of activity in cybernetical physics are briefly surveyed, such as control of oscillatory and chaotic behaviour, control of resonance and synchronization, control in thermodynamics, control of distributed systems and networks, quantum control. This article is part of the themed issue ‘Horizons of cybernetical physics’.

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Smooth positon solutions of the focusing modified Korteweg–de Vries equation

Xing

Mihalache

et al. 2017

Nonlinear Dyn

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The n-fold Darboux transformation T n of the focusing real modified Korteweg-de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the n-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues λ j and the corresponding eigenfunctions of the associated Lax equation. The nonsingular n-positon solutions of the focusing mKdV equation are obtained in the special limit λ j → λ 1 , from the corresponding n-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the n-positon solution into n single-soliton solutions, the trajectories, and the corresponding "phase shifts" of the multipositons are also investigated.

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Localization of the sine-Gordon equation solutions

Cited by 10 publications

References 14 publications

Control methods for localization of nonlinear waves

Control methods for localization of nonlinear waves

Horizons of cybernetical physics

Smooth positon solutions of the focusing modified Korteweg–de Vries equation

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