Singular solutions of the elliptic sine-Gordon equation: models of defects

Using analytic and numerical methods, we study a 2d Hamiltonian model of interacting particles carrying ferro-magnetically coupled continuous spins which are also locally coupled to their own velocities. This model has been characterised at the mean field level in a parent paper. Here, we first obtain its finite size ground states, as a function of the spin-velocity coupling intensity and system size, with numerical techniques. These ground states, namely a collectively moving polar state of aligned spins, and two non-moving states embedded with topological defects, are recovered from the analysis of the continuum limit theory and simple energetic arguments that allow us to predict their domains of existence in the space of control parameters. Next, the finite temperature regime is investigated numerically. In some specific range of the control parameters, the magnetisation presents a maximum at a finite temperature. This peculiar behaviour, akin to an order-by-disorder transition, is explained by the examination of the free energy of the system and the metastability of the states of minimal energy. The robustness of our results is checked against the geometry of the boundary conditions and the dimensionality of space.

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“…Mech. (2020) 013209 the addition of the non-linear term in the sine-Gordon equation [24,25]. One-defect solutions can then be written as…”

Section: Continuum Theorymentioning

confidence: 99%

Order by disorder: saving collective motion from topological defects in a conservative model

Casiulis¹,

Tarzia²,

Cugliandolo³

et al. 2020

J. Stat. Mech.

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“…In particular, we only find some theoretical and numerical researches, e.g. in [1,6,7,8,9] and references therein. Additionally, it is worth noting that in principle the Cauchy problem for such equations is ill-posed where the stability of solution fails intrinsically.…”

Section: Introductionmentioning

confidence: 99%

The Cauchy problem of coupled elliptic sine–Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing

Khoa¹,

Nhat²,

Duy³

et al. 2017

Computers & Mathematics with Applications

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Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method for the coupled elliptic sine-Gordon equations along with Cauchy data.The system of equations originates from the static case of the coupled hyperbolic sine-Gordon equations modeling the coupled Josephson junctions in superconductivity, and so far it addresses the Josephson πjunctions. In general, the Cauchy problem is not well-posed, and herein the Hadamard-instability occurs drastically. Generalizing the kernel-based regularization method, we propose a stable approximate solution. Confirmed by the error estimate, this solution strongly converges to the exact solution in L 2 -norm.The main concern of this paper is also with the way to compute the regularized solution formed by an alike integral equation. We employ the proposed techniques that successfully approximated the highly oscillatory integral, and apply the Picard-like iteration to organize an efficient and reliable tool of computations.The results are viewed as the improvement as well as the generalization of many previous works. The paper is also accompanied by a numerical example that demonstrates the potential of this idea.

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“…It has been shown that these solutions are closely related to the solution of the third type Painleve equation with the same logarithmic asymptotic [12,13]. The detailed consideration of topological defects of this type can be found in [14,15]. The interest in these solutions arises in the theory of Josephson junctions [16] and Heisenberg [17] and non-Abelian field models [18].…”

mentioning

confidence: 99%

Numerical study of spirals in a two-dimensionalXYmodel with in-plane magnetic field

Sinitsyn¹,

Bostrem²,

Овчинников³

2004

J. Phys.: Condens. Matter

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The behaviour of topological defects of the spiral type is studied numerically within a two-dimensional XY model with an applied in-plane magnetic field. The energy dependence of the 'spiral-antispiral' pair on the field value and direction is investigated. It turns out to be linear when the field is directed along the pair axis and exhibits a combined linear-logarithmic behaviour for the perpendicular orientation.

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Singular solutions of the elliptic sine-Gordon equation: models of defects

Cited by 12 publications

References 16 publications

Order by disorder: saving collective motion from topological defects in a conservative model

Order by disorder: saving collective motion from topological defects in a conservative model

The Cauchy problem of coupled elliptic sine–Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing

Numerical study of spirals in a two-dimensionalXYmodel with in-plane magnetic field

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