Kink dynamics in a novel discrete sine-Gordon system

We examine various recently proposed discretizations of the well-known φ 4 field theory. We compare and contrast the properties of their fundamental solutions including the nature of their kink-type solitary waves and the spectral properties of the linearization around such waves. We study these features as a function of the lattice spacing h, as one deviates from the continuum limit of h → 0. We then proceed to a more "stringent" comparison of the models, by discussing the scattering properties of a kink-antikink pair for the different discretizations. These collisions are well-known to possess properties that quite sensitively depend on the initial speed even at the continuum limit. We examine how typical model behaviors are modified in the presence (and as a function) of discreteness and attempt to extract qualitative trends and issue pertinent warnings about some of the surprising resulting properties.

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“…Equations (11) and (12) are the discretized first integrals (DFIs) obtained by discretizing Eq. (5) and Eq.…”

Section: φ 4 Field and Its Various Discretizationsmentioning

confidence: 99%

Comparative study of different discretizations of theϕ4model

2007

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“…(2), V ′ (φ) becomes V ′ (φ n ). Another class of Klein-Gordon models which support energy conservation and sustain static kinks but which are free of PNp has been derived by Speight and collaborators [17]. In such models the background potential term of Eq.…”

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confidence: 99%

Standard nearest-neighbour discretizations of Klein–Gordon models cannot preserve both energy and linear momentum

Yoshikawa¹

2006

J. Phys. A: Math. Gen.

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We consider nonlinear Klein-Gordon wave equations and illustrate that standard discretizations thereof (involving nearest neighbors) may preserve either standardly defined linear momentum or total energy but not both. This has a variety of intriguing implications for the "non-potential" discretizations that preserve only the linear momentum, such as the self-accelerating or self-decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices. On the other hand, spatially discrete systems (of coupled nonlinear ordinary differential equations) are also relevant as discretizations and computational implementations of the corresponding continuum field theories that are applicable to a variety of contexts such as statistical mechanics [10], solid state physics [11], fluid mechanics [12] and particle physics [13] (see also references therein). Nonlinear Klein-Gordon type equations are a prototypical example among such wave models and their variants span a variety of applications including Josephson junctions in superconductivity, cosmic domain walls in cosmology, elementary particles in particle physics and denaturation bubbles in the DNA, among others.In this communication, we examine some of the key properties that ensue when discretizing nonlinear KleinGordon (KG) equations, using nearest neighbor approximations (which are the most standard ones implemented in the literature; see e.g., [1]). In particular, we focus on the physically relevant invariances of the continuum equation (more specifically, the conservation of the linear momentum and of the total energy of the system) and illustrate the surprising result that if we demand that the energy be conserved, then the momentum cannot be conserved, while if we demand that the momentum be conserved then the energy cannot be conserved (resulting in a so-called non-potential model [14]). Our presentation will be structured as follows. First, we will provide the general mathematical setting of KG equations and study their discretizations that conserve linear momentum and energy, comparing and constructing the properties of the two. Then, we are going to give an application of our

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“…Представим решеточную версию модели Скирма, применяя указанную нижнюю границу для энергии, т.е., следуя [4], начнем с той же самой функции sin g∂ r (cos g), которая возникает в выражении (15), и реконструируем ограничение…”

Section: Introductionunclassified

Дискретные Двумерные И Трехмерные Скирмионы

Иоанниду¹,

Ioannidou²

2009

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Kink dynamics in a novel discrete sine-Gordon system

Cited by 76 publications

References 8 publications

Comparative study of different discretizations of theϕ4model

Comparative study of different discretizations of theϕ4model

Standard nearest-neighbour discretizations of Klein–Gordon models cannot preserve both energy and linear momentum

Дискретные Двумерные И Трехмерные Скирмионы

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