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Journal articles on the topic 'Hamiltonian PDE's'

Author: Grafiati

Published: 4 June 2021

Last updated: 31 January 2023

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1

Rousset, Frederic, and Nikolay Tzvetkov. "Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's." Journal de Mathématiques Pures et Appliquées 90, no. 6 (December 2008): 550–90. http://dx.doi.org/10.1016/j.matpur.2008.07.004.

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2

Kuksin, Sergej B. "Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDE's." Communications in Mathematical Physics 167, no. 3 (February 1995): 531–52. http://dx.doi.org/10.1007/bf02101534.

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3

Dubrovin, B. "Hamiltonian PDEs: deformations, integrability, solutions." Journal of Physics A: Mathematical and Theoretical 43, no. 43 (October 12, 2010): 434002. http://dx.doi.org/10.1088/1751-8113/43/43/434002.

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4

Dubrovin, Boris A. "Hamiltonian PDEs and Frobenius manifolds." Russian Mathematical Surveys 63, no. 6 (December 31, 2008): 999–1010. http://dx.doi.org/10.1070/rm2008v063n06abeh004575.

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5

Bridges, Thomas J., and Sebastian Reich. "Numerical methods for Hamiltonian PDEs." Journal of Physics A: Mathematical and General 39, no. 19 (April 24, 2006): 5287–320. http://dx.doi.org/10.1088/0305-4470/39/19/s02.

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6

Brugnano, Luigi, Gianluca Frasca-Caccia, and Felice Iavernaro. "Line Integral Solution of Hamiltonian PDEs." Mathematics 7, no. 3 (March 18, 2019): 275. http://dx.doi.org/10.3390/math7030275.

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In this paper, we report on recent findings in the numerical solution of Hamiltonian Partial Differential Equations (PDEs) by using energy-conserving line integral methods in the Hamiltonian Boundary Value Methods (HBVMs) class. In particular, we consider the semilinear wave equation, the nonlinear Schrödinger equation, and the Korteweg–de Vries equation, to illustrate the main features of this novel approach.

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7

Oh, Tadahiro, and Jeremy Quastel. "On the Cameron–Martin theorem and almost-sure global existence." Proceedings of the Edinburgh Mathematical Society 59, no. 2 (December 17, 2015): 483–501. http://dx.doi.org/10.1017/s0013091515000218.

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AbstractIn this paper we discuss various aspects of invariant measures for nonlinear Hamiltonian partial differential equations (PDEs). In particular, we show almost-sure global existence for some Hamiltonian PDEs with initial data of the form ‘a smooth deterministic function + a rough random perturbation’ as a corollary to the Cameron–Martin theorem and known almost-sure global existence results with respect to Gaussian measures on spaces of functions.

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8

Bambusi, D., and C. Bardelle. "Invariant tori for commuting Hamiltonian PDEs." Journal of Differential Equations 246, no. 6 (March 2009): 2484–505. http://dx.doi.org/10.1016/j.jde.2008.12.002.

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9

Gong, Yuezheng, and Yushun Wang. "An Energy-Preserving Wavelet Collocation Method for General Multi-Symplectic Formulations of Hamiltonian PDEs." Communications in Computational Physics 20, no. 5 (November 2016): 1313–39. http://dx.doi.org/10.4208/cicp.231014.110416a.

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AbstractIn this paper, we develop a novel energy-preserving wavelet collocation method for solving general multi-symplectic formulations of Hamiltonian PDEs. Based on the autocorrelation functions of Daubechies compactly supported scaling functions, the wavelet collocation method is conducted for spatial discretization. The obtained semi-discrete system is shown to be a finite-dimensional Hamiltonian system, which has an energy conservation law. Then, the average vector field method is used for time integration, which leads to an energy-preserving method for multi-symplectic Hamiltonian PDEs. The proposed method is illustrated by the nonlinear Schrödinger equation and the Camassa-Holm equation. Since differentiation matrix obtained by the wavelet collocation method is a cyclic matrix, we can apply Fast Fourier transform to solve equations in numerical calculation. Numerical experiments show the high accuracy, effectiveness and conservation properties of the proposed method.

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10

Moore, Brian E., and Sebastian Reich. "Multi-symplectic integration methods for Hamiltonian PDEs." Future Generation Computer Systems 19, no. 3 (April 2003): 395–402. http://dx.doi.org/10.1016/s0167-739x(02)00166-8.

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11

Lorenzoni, Paolo. "Flat bidifferential ideals and semi-Hamiltonian PDEs." Journal of Physics A: Mathematical and General 39, no. 44 (October 17, 2006): 13701–15. http://dx.doi.org/10.1088/0305-4470/39/44/006.

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12

Schöberl, M., and K. Schlacher. "Port-Hamiltonian formulation for Higher-order PDEs." IFAC-PapersOnLine 48, no. 13 (2015): 244–49. http://dx.doi.org/10.1016/j.ifacol.2015.10.247.

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13

Bambusi, Dario. "An Averaging Theorem for Quasilinear Hamiltonian PDEs." Annales Henri Poincaré 4, no. 4 (August 2003): 685–712. http://dx.doi.org/10.1007/s00023-003-0144-6.

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14

Song, Mingzhan, Xu Qian, Hong Zhang, and Songhe Song. "Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation." Advances in Applied Mathematics and Mechanics 9, no. 4 (January 18, 2017): 868–86. http://dx.doi.org/10.4208/aamm.2015.m1356.

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AbstractIn this paper, we introduce the Hamiltonian boundary value method (HBVM) to solve nonlinear Hamiltonian PDEs. We use the idea of Fourier pseudospectral method in spatial direction, which leads to the finite-dimensional Hamiltonian system. The HBVM, which can preserve the Hamiltonian effectively, is applied in time direction. Then the nonlinear Schrödinger (NLS) equation and the Korteweg-de Vries (KdV) equation are taken as examples to show the validity of the proposed method. Numerical results confirm that the proposed method can simulate the propagation and collision of different solitons well. Meanwhile the corresponding errors in Hamiltonian and other intrinsic invariants are presented to show the good preservation property of the proposed method during long-time numerical calculation.

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15

Cai, Wenjun, Huai Zhang, and Yushun Wang. "Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs." Communications in Computational Physics 19, no. 5 (May 2016): 1375–96. http://dx.doi.org/10.4208/cicp.scpde14.32s.

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AbstractThis paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, bandlimited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.

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16

Giuliani, Filippo, Marcel Guardia, Pau Martin, and Stefano Pasquali. "Chaotic-Like Transfers of Energy in Hamiltonian PDEs." Communications in Mathematical Physics 384, no. 2 (February 8, 2021): 1227–90. http://dx.doi.org/10.1007/s00220-021-03956-9.

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AbstractWe consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on $${\mathbb {T}}^2$$ T 2 and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered “chaotic-like” since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.

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17

Benzoni-Gavage, Sylvie, Colin Mietka, and L. Miguel Rodrigues. "Modulated equations of Hamiltonian PDEs and dispersive shocks." Nonlinearity 34, no. 1 (January 1, 2021): 578–641. http://dx.doi.org/10.1088/1361-6544/abcb0a.

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18

Arioli, Gianni, and Hans Koch. "Families of Periodic Solutions for Some Hamiltonian PDEs." SIAM Journal on Applied Dynamical Systems 16, no. 1 (January 2017): 1–15. http://dx.doi.org/10.1137/16m1070177.

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19

Eliasson, L. Hakan, Benoît Grébert, and Sergeï B. Kuksin. "A KAM theorem for space-multidimensional Hamiltonian PDEs." Proceedings of the Steklov Institute of Mathematics 295, no. 1 (November 2016): 129–47. http://dx.doi.org/10.1134/s0081543816080071.

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20

Sun, Zhengjie, and Zongmin Wu. "Meshless Conservative Scheme for Multivariate Nonlinear Hamiltonian PDEs." Journal of Scientific Computing 76, no. 2 (March 1, 2018): 1168–87. http://dx.doi.org/10.1007/s10915-018-0658-1.

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21

Faou, Erwan, and Benoît Grébert. "Hamiltonian Interpolation of Splitting Approximations for Nonlinear PDEs." Foundations of Computational Mathematics 11, no. 4 (May 11, 2011): 381–415. http://dx.doi.org/10.1007/s10208-011-9094-4.

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22

Tang, Wensheng, Yajuan Sun, and Wenjun Cai. "Discontinuous Galerkin methods for Hamiltonian ODEs and PDEs." Journal of Computational Physics 330 (February 2017): 340–64. http://dx.doi.org/10.1016/j.jcp.2016.11.023.

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23

Brugnano, Luigi, Felice Iavernaro, Juan I. Montijano, and Luis Rández. "Spectrally accurate space-time solution of Hamiltonian PDEs." Numerical Algorithms 81, no. 4 (August 16, 2018): 1183–202. http://dx.doi.org/10.1007/s11075-018-0586-z.

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24

Ghoussoub, Nassif, and Abbas Moameni. "Hamiltonian systems of PDEs with selfdual boundary conditions." Calculus of Variations and Partial Differential Equations 36, no. 1 (January 31, 2009): 85–118. http://dx.doi.org/10.1007/s00526-009-0224-7.

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25

Bogoyavlenskij, Oleg. "Differential-Geometric Invariants of the Hamiltonian Systems of pde’s." Communications in Mathematical Physics 265, no. 3 (April 25, 2006): 805–17. http://dx.doi.org/10.1007/s00220-006-0016-2.

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26

Kosmas, Odysseas, Pieter Boom, and Andrey P. Jivkov. "On the Derivation of Multisymplectic Variational Integrators for Hyperbolic PDEs Using Exponential Functions." Applied Sciences 11, no. 17 (August 25, 2021): 7837. http://dx.doi.org/10.3390/app11177837.

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We investigated the derivation of numerical methods for solving partial differential equations, focusing on those that preserve physical properties of Hamiltonian systems. The formulation of these properties via symplectic forms gives rise to multisymplectic variational schemes. By using analogy with the smooth case, we defined a discrete Lagrangian density through the use of exponential functions, and derived its Hamiltonian by Legendre transform. This led to a discrete Hamiltonian system, the symplectic forms of which obey the conservation laws. The integration schemes derived in this work were tested on hyperbolic-type PDEs, such as the linear wave equations and the non-linear seismic wave equations, and were assessed for their accuracy and the effectiveness by comparing them with those of standard multisymplectic ones. Our error analysis and the convergence plots show significant improvements over the standard schemes.

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27

Cang, Shijian, Aiguo Wu, Zenghui Wang, and Zengqiang Chen. "Distinguishing Lorenz and Chen Systems Based Upon Hamiltonian Energy Theory." International Journal of Bifurcation and Chaos 27, no. 02 (February 2017): 1750024. http://dx.doi.org/10.1142/s0218127417500249.

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Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lü system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.

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28

Bambusi, Dario, and Massimiliano Berti. "A Birkhoff--Lewis-Type Theorem for Some Hamiltonian PDEs." SIAM Journal on Mathematical Analysis 37, no. 1 (January 2005): 83–102. http://dx.doi.org/10.1137/s0036141003436107.

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29

Islas, A. L., and C. M. Schober. "Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs." Mathematics and Computers in Simulation 69, no. 3-4 (June 2005): 290–303. http://dx.doi.org/10.1016/j.matcom.2005.01.006.

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30

Zhang, Shengliang. "Symplectic Radial Basis Approximation of Multi-variate Hamiltonian PDEs." Iranian Journal of Science and Technology, Transactions A: Science 43, no. 4 (August 28, 2018): 1789–97. http://dx.doi.org/10.1007/s40995-018-0626-5.

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31

Zheng, Yu, and Yong Chen. "Ordered analytic representation of pdes by hamiltonian canonical system." Applied Mathematics-A Journal of Chinese Universities 17, no. 2 (June 2002): 177–82. http://dx.doi.org/10.1007/s11766-002-0042-6.

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32

UDRIŞTE, CONSTANTIN, MARCELA POPESCU, and PAUL POPESCU. "EXTENDED AFFINE CLASSES OF LAGRANGIANS AND HAMILTONIANS RELATED TO CLASSICAL FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 07 (November 2009): 1161–80. http://dx.doi.org/10.1142/s0219887809004156.

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The aim of the paper is to establish a natural affine frame for affine Lagrangians and Hamiltonians, generalizing the well-known classical field theory. Scalar and volume-valued Lagrangians and Hamiltonians can be lifted to the new classes. Using the Hamilton–Jacobi principle, we analyze variational problems corresponding to actions defined by the affine Lagrangians and Hamiltonians. The extremals verify generalizations of the Euler–Lagrange and De Donder–Weyl PDEs. They improve the information about the dynamical solutions of the classical variational problems and refresh the Lagrange–Hamilton theories.

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33

Giuliani, Filippo. "Transfers of energy through fast diffusion channels in some resonant PDEs on the circle." Discrete & Continuous Dynamical Systems 41, no. 11 (2021): 5057. http://dx.doi.org/10.3934/dcds.2021068.

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<p style='text-indent:20px;'>In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely the orbits of a nonlinear resonant model, which is a good approximation of the full equation. The non-convexity of the resonant Hamiltonian allows the existence of <i>fast diffusion channels</i> along which the orbits of the resonant model experience a large drift in the actions in the optimal time. This phenomenon induces a transfer of energy among the Fourier modes of the solutions, which in turn is responsible for the growth of higher order Sobolev norms.</p>

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34

SOLE, ALBERTO DE, MAMUKA JIBLADZE, VICTOR G. KAC, and DANIELE VALERI. "INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS." Transformation Groups 26, no. 2 (April 15, 2021): 479–500. http://dx.doi.org/10.1007/s00031-021-09645-0.

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AbstractWe prove that all classical affine W-algebras 𝒲(𝔤; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.

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35

Udriste, Constantin, and Ionel Tevy. "Geometric Dynamics on Riemannian Manifolds." Mathematics 8, no. 1 (January 3, 2020): 79. http://dx.doi.org/10.3390/math8010079.

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The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail five significant decomposed dynamics: (i) the motion of the four outer planets relative to the sun fixed by a Hamiltonian, (ii) the motion in a closed Newmann economical system fixed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a flow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-flow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into flows and motions transversal to the flows.

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36

Bustillo, Jaime. "Middle dimensional symplectic rigidity and its effect on Hamiltonian PDEs." Commentarii Mathematici Helvetici 94, no. 4 (December 18, 2019): 803–32. http://dx.doi.org/10.4171/cmh/474.

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37

Bernard, Deconinck, and Olga Trichtchenko. "High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs." Discrete & Continuous Dynamical Systems - A 37, no. 3 (2017): 1323–58. http://dx.doi.org/10.3934/dcds.2017055.

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38

Giuliani, Filippo, Marcel Guardia, Pau Martin, and Stefano Pasquali. "Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs." Rendiconti Lincei - Matematica e Applicazioni 32, no. 1 (April 22, 2021): 149–66. http://dx.doi.org/10.4171/rlm/931.

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39

Bernier, Joackim, and Benoît Grébert. "Birkhoff normal forms for Hamiltonian PDEs in their energy space." Journal de l’École polytechnique — Mathématiques 9 (April 4, 2022): 681–745. http://dx.doi.org/10.5802/jep.193.

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40

Benzoni-Gavage, S., C. Mietka, and L. M. Rodrigues. "Co-periodic stability of periodic waves in some Hamiltonian PDEs." Nonlinearity 29, no. 11 (September 9, 2016): 3241–308. http://dx.doi.org/10.1088/0951-7715/29/11/3241.

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41

Zhang, Liying, and Lihai Ji. "Stochastic multi-symplectic Runge–Kutta methods for stochastic Hamiltonian PDEs." Applied Numerical Mathematics 135 (January 2019): 396–406. http://dx.doi.org/10.1016/j.apnum.2018.09.011.

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42

Cai, Jiaxiang, Yushun Wang, and Chaolong Jiang. "Local structure-preserving algorithms for general multi-symplectic Hamiltonian PDEs." Computer Physics Communications 235 (February 2019): 210–20. http://dx.doi.org/10.1016/j.cpc.2018.08.015.

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43

Moore, Brian E., Laura Noreña, and Constance M. Schober. "Conformal conservation laws and geometric integration for damped Hamiltonian PDEs." Journal of Computational Physics 232, no. 1 (January 2013): 214–33. http://dx.doi.org/10.1016/j.jcp.2012.08.010.

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44

Montecchiari, Piero, and Paul H. Rabinowitz. "A Variant of the Mountain Pass Theorem and Variational Gluing." Milan Journal of Mathematics 88, no. 2 (September 1, 2020): 347–72. http://dx.doi.org/10.1007/s00032-020-00318-3.

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AbstractThis paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In particular its application to a system of semilinear elliptic PDEs on $$R^n$$ R n and to a family of Hamiltonian systems involving double well potentials will also be discussed.

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45

Parker, Ross, and Björn Sandstede. "Periodic multi-pulses and spectral stability in Hamiltonian PDEs with symmetry." Journal of Differential Equations 334 (October 2022): 368–450. http://dx.doi.org/10.1016/j.jde.2022.06.019.

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46

Fu, Yayun, Dongdong Hu, and Zhuangzhi Xu. "High-order explicit conservative exponential integrator schemes for fractional Hamiltonian PDEs." Applied Numerical Mathematics 172 (February 2022): 315–31. http://dx.doi.org/10.1016/j.apnum.2021.10.011.

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47

Hong, Jialin, Hongyu Liu, and Geng Sun. "The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs." Mathematics of Computation 75, no. 253 (September 29, 2005): 167–82. http://dx.doi.org/10.1090/s0025-5718-05-01793-x.

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48

Bridges, Thomas J., and Sebastian Reich. "Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity." Physics Letters A 284, no. 4-5 (June 2001): 184–93. http://dx.doi.org/10.1016/s0375-9601(01)00294-8.

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49

Maeda, Masaya. "Stability of bound states of Hamiltonian PDEs in the degenerate cases." Journal of Functional Analysis 263, no. 2 (July 2012): 511–28. http://dx.doi.org/10.1016/j.jfa.2012.04.006.

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50

Li, Yu-Wen, and Xinyuan Wu. "General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs." Journal of Computational Physics 301 (November 2015): 141–66. http://dx.doi.org/10.1016/j.jcp.2015.08.023.

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