Propagation of ultra-short solitons in stochastic Maxwell's equations

Kurt, Levent; Schäfer, Tobias

doi:10.1063/1.4859815

Cited by 13 publications

(5 citation statements)

References 29 publications

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“…3D stochastic Maxwell equations with multiplicative noise play an important role in many scientific fields, especially in stochastic electromagnetism and statistical radiophysics [2,7,12,17]. We refer interested readers to [14] for the well-posedness of equations (1.1).…”

Section: Introductionmentioning

confidence: 99%

An energy-conserving method for stochastic Maxwell equations with multiplicative noise

Hong

Zhang

et al. 2017

Journal of Computational Physics

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In this paper, it is shown that three-dimensional stochastic Maxwell equations with multiplicative noise are stochastic Hamiltonian partial differential equations possessing a geometric structure (i.e. stochastic mutli-symplectic conservation law), and the energy of system is a conservative quantity almost surely. We propose a stochastic multi-symplectic energy-conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time. Numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy. The mean square convergence result of the method in temporal direction is tested numerically, and numerical comparisons with finite difference method are also investigated.

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Section: Introductionmentioning

confidence: 99%

An energy-conserving method for stochastic Maxwell equations with multiplicative noise

Hong

Zhang

et al. 2017

Journal of Computational Physics

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“…Based on this model, [14] proposed a method based on Wiener chaos expansion to determine the near field thermal radiation, and [10] described the fluctuation of the electromagnetic field using spectral representation. Without modeling the precise origins of randomness, rather assume that they lead to small stochastic variations of the coefficients of the equations, [7] studied the propagation of ultra-short solitons in a cubic nonlinear media, which is modeled by nonlinear Maxwell equations with stochastic variations of media; and assume that the externally imposed source is a random field, which is expressed by a Q-Wiener process, [4,9,11] dealt with the mathematical analysis of stochastic problems arising in the theory of electromagnetic in complex media, including well posedness, controllability and homogenisation. The stochastic model considered in this paper is based on [11,Chapter 12] for the isotropic homogeneous medium with an external source.…”

Section: Introductionmentioning

confidence: 99%

Preservation of physical properties of stochastic Maxwell equations with additive noise via stochastic multi-symplectic methods

Chen

Hong

Zhang

2016

Journal of Computational Physics

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Stochastic Maxwell equations with additive noise are a system of stochastic Hamiltonian partial differential equations intrinsically, possessing the stochastic multi-symplectic conservation law. It is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic Maxwell equations with additive noise preserves the divergence in the sense of expectation. Moreover, we propose three novel stochastic multi-symplectic methods to discretize stochastic Maxwell equations in order to investigate the preservation of these properties numerically. We make theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence. Meanwhile, we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method. Especially, the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case. Numerical experiments are performed to verify our theoretical results.

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“…On the other hand, Liu-Pelinovsky-Sakovich [33] found the conditions of the wave breaking in the SP equation (see also Sakovich [46]). Some generalizations of the SP equation have been considered as well, including vector SP equation [45], regularized SP equation [8], integrable coupled SP equation [9], higher-order SP equation [29], stochastic SP equation [30], and (coupled) complex SP equation [10]. There are also a few results for the circle (periodic) case.…”

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confidence: 99%

A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation

Sato,

Oguma,

Matsuo

et al. 2016

Preprint

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The short pulse equation was introduced by Schäfer- Wayne (2004) for modeling the propagation of ultrashort optical pulses. While it can describe a wide range of solutions, its ultrashort pulse solutions with a few cycles, which the conventional nonlinear Schrödinger equation does not possess, have drawn much attention. In such a region, existing numerical methods turn out to require very fine numerical mesh, and accordingly are computationally expensive. In this paper, we establish a new efficient numerical method by combining the idea of the hodograph transformation and the structure-preserving numerical methods. The resulting scheme is a self-adaptive moving mesh scheme that can successfully capture not only the ultrashort pulses but also exotic solutions such as loop soliton solutions.

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Propagation of ultra-short solitons in stochastic Maxwell's equations

Cited by 13 publications

References 29 publications

An energy-conserving method for stochastic Maxwell equations with multiplicative noise

An energy-conserving method for stochastic Maxwell equations with multiplicative noise

Preservation of physical properties of stochastic Maxwell equations with additive noise via stochastic multi-symplectic methods

A self-adaptive moving mesh method for the short pulse equation via its hodograph link to the sine-Gordon equation

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