Model order reduction for nonlinear Schrödinger equation

Karasözen, Bülent; Akkoyunlu, Canan; Uzunca, Murat

doi:10.1016/j.amc.2015.02.001

Cited by 8 publications

(4 citation statements)

References 10 publications

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“…In the following, we introduce the SIPG method [8,52] for the spatial discretization of NLSE. We remark that among the three common interior penalty Galerkin discretizations, nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin (IIPG), only SIPG leads to a Hamiltonian system of ODEs [36].…”

Section: Full Order Modelmentioning

confidence: 99%

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Energy preserving model order reduction of the nonlinear Schrödinger equation

Karasözen

Uzunca

2018

Adv Comput Math

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An energy preserving reduced order model is developed for two dimensional nonlinear Schrödinger equation (NLSE) with plane wave solutions and with an external potential. The NLSE is discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of Hamiltonian ordinary differential equations are integrated in time by the energy preserving average vector field (AVF) method. The mass and energy preserving reduced order model (ROM) is constructed by proper orthogonal decomposition (POD) Galerkin projection. The nonlinearities are computed for the ROM efficiently by discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD). Preservation of the semi-discrete energy and mass are shown for the full order model (FOM) and for the ROM which ensures the long term stability of the solutions. Numerical simulations illustrate the preservation of the energy and mass in the reduced order model for the two dimensional NLSE with and without the external potential. The POD-DMD makes a remarkable improvement in computational speed-up over the POD-DEIM. Both methods approximate accurately the FOM, whereas POD-DEIM is more accurate than the POD-DMD.

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Section: Full Order Modelmentioning

confidence: 99%

“…With increasing number of the modes, the singular values decay rather slowly. We choose in the ROM the number of POD modes k = 10 which satisfies the energy criterion (36) with ε 10 > 0.9999. In Fig.…”

Section: Defocusing Nlse With Progressive Wave Solutionsmentioning

confidence: 99%

Energy preserving model order reduction of the nonlinear Schrödinger equation

Karasözen

Uzunca

2018

Adv Comput Math

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“…Es decir, se observa que el comportamiento de los solitones a propagarse a lo largo de fibras ópticas es importante para los sistemas de transmisión de información, ya que al contrarrestarse los efectos de dispersión de primer orden es posible transmitir pulsos más cortos a mayores distancias, permitiendo reducir los costos de implementación y funcionamiento de un sistema comunicaciones óptico. Los resultados obtenidos esta en concordancia con los reportados en [20], donde solucionan numéri-camente la ecuación de Schrödinger no lineal por el método de descomposición ortogonal evidenciando la no distorsión de perfil a lo largo de una guía de propagación del pulso.…”

Section: Conclusionesunclassified

Solución De La Ecuación No Lineal De Schrodinger (1+1) en Un Medio Kerr

Segovia

Cabrera

2015

Redes Ing.

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Se presenta un marco teórico y se muestra una simulación numérica de la propagación de solitones. Con especial atención a los solitones ópticos espaciales, se calcula analíticamente el perfil de solitón correspondiente a la ecuación Schrodinger no-lineal para un medio Kerr. Los resultados muestran que los solitones ópticos son pulsos estables cuya forma y espectro son preservados en grandes distancias.Solution of the nonlinear Schrodinger equation (1+1) in a Kerr mediumABSTRACTThis document presents a theoretical framework and shows a numerical simulation for the propagation of solitons. With special attention to the spatial optical solitons, we calculates analytically the profile of solitón corresponding to the non-linear Schrodinger equation for a Kerr medium. The results show that the optical solitons are stable pulses whose shape and spectrum are preserved at great distances.Keywords: nonlinear optics, nonlinear Schrodinger equation, solitons.

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“…In order to solve the problem of modal truncation, many scholars have made in-depth research and achieved certain results [25,26,27,28,29]. Among them, the modal acceleration method [26] is put forward on the basis of combining the special solution of the motion equation with the existing modal when the excitation frequency is zero, which can improve the vibration response analysis to some extent, especially in the lower frequency range.…”

Section: Introductionmentioning

confidence: 99%

Modal truncation method for continuum structures based on matrix norm: modal perturbation method

Kang,

Yuan,

et al. 2024

Nonlinear Dyn

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Modal analysis is a widely applied method to study the vibration phenomenon of continuum structures, but there is no clear method to solve the modal truncation problem at present. In order to determine the contribution of different modes to the whole system, a new mode truncation method based on perturbation theory is proposed in this paper. In the process of discretization, perturbation parameters are introduced into the modes, and the modal number in the continuum structure system is confirmed according to the norm error of stiffness matrix in different degrees of freedom (DOFs) systems. The results show that the DOF identified by the modal perturbation method is related to the perturbation parameters, and the smaller the perturbation parameters are, the fewer modes need to be considered. When the perturbation parameter is large enough, the response of the system can only be accurately explained by truncation to higher-order modes. Finally, the perturbation parameter is fixed to 1, and the modal perturbation is connected with the traditional Galerkin method, which means that the traditional discretization is a special case of the modal perturbation method, and the number of modes can still be determined by the induced norm. This method can significantly reduce the modal truncation error, which is of great significance to the dynamic analysis of engineering applications.

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Model order reduction for nonlinear Schrödinger equation

Cited by 8 publications

References 10 publications

Energy preserving model order reduction of the nonlinear Schrödinger equation

Energy preserving model order reduction of the nonlinear Schrödinger equation

Solución De La Ecuación No Lineal De Schrodinger (1+1) en Un Medio Kerr

Modal truncation method for continuum structures based on matrix norm: modal perturbation method

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