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Article Citation - WoS: 3Citation - Scopus: 5Global Energy Preserving Model Reduction for Multi-Symplectic Pdes(Elsevier Science inc, 2023) Uzunca, Murat; Karasozen, Bulent; Aydin, AyhanMany Hamiltonian systems can be recast in multi-symplectic form. We develop a reduced -order model (ROM) for multi-symplectic Hamiltonian partial differential equations (PDEs) that preserves the global energy. The full-order solutions are obtained by finite difference discretization in space and the global energy preserving average vector field (AVF) method. The ROM is constructed in the same way as the full-order model (FOM) applying proper orthogonal decomposition (POD) with the Galerkin projection. The reduced-order system has the same structure as the FOM, and preserves the discrete reduced global energy. Ap-plying the discrete empirical interpolation method (DEIM), the reduced-order solutions are computed efficiently in the online stage. A priori error bound is derived for the DEIM ap-proximation to the nonlinear Hamiltonian. The accuracy and computational efficiency of the ROMs are demonstrated for the Korteweg de Vries (KdV) equation, Zakharov-Kuznetzov (ZK) equation, and nonlinear Schrodinger (NLS) equation in multi-symplectic form. Preser-vation of the reduced energies shows that the reduced-order solutions ensure the long-term stability of the solutions.(c) 2022 Elsevier Inc. All rights reserved.Article Citation - WoS: 23Citation - Scopus: 25Multisymplectic Integration of n-coupled Nonlinear Schrodinger Equation With Destabilized Periodic Wave Solutions(Pergamon-elsevier Science Ltd, 2009) Aydin, AyhanN-coupled nonlinear Schrodinger equation (N-CNLS) is shown to be in multisymplectic form. 3-CNLS equation is studied for analytical and numerical purposes. A new six-point scheme which is equivalent to the multisymplectic Preissman scheme is derived for 3-CNLS equation. A new periodic wave solution is obtained and its stability analysis is discussed. 3-CNLS equation is integrated for destabilized periodic solutions both for integrable and non-integrable cases by multisymplectic six-point scheme. Different kinds of evolutions are observed for different parameters and coefficients of the system. Numerical results show that, the multisymplectic six-point scheme has excellent local and global conservation properties in long-time computation. (C) 2008 Elsevier Ltd. All rights reserved.Article Citation - WoS: 12Citation - Scopus: 12Conservative Finite Difference Schemes for the Chiral Nonlinear Schrodinger Equation(Springer international Publishing Ag, 2015) Ismail, Mohammad S.; Al-Basyouni, Khalil S.; Aydin, AyhanIn this paper, we derive three finite difference schemes for the chiral nonlinear Schrodinger equation (CNLS). The CNLS equation has two kinds of progressive wave solutions: bright and dark soliton. The proposed methods are implicit, unconditionally stable and of second order in space and time directions. The exact solutions and the conserved quantities are used to assess the efficiency of these methods. Numerical simulations of single bright and dark solitons are given. The interactions of two bright solitons are also displayed.
