Finite-Gap Solutions of the Vortex Filament Equation: Genus One Solutions and Symmetric Solutions

Calini, Annalisa; Ivey, Thomas

doi:10.1007/s00332-004-0679-9

Cited by 53 publications

(62 citation statements)

References 28 publications

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“…Butγ ∈ L 2 (T 1 ) and Parseval imply (1 − cos(u)) 2 + (sin u − u) 2 u 4+p du as well. Collecting constants and extending the region of integration then yields the claim.…”

Section: Appendixmentioning

confidence: 99%

Dynamics of embedded curves by doubly-nonlocal reaction–diffusion systems

Brecht¹,

Blair²

2017

J. Phys. A: Math. Theor.

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We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of complexity for embedded curves. This line of investigation culminates in a family of complexity bounds that relate a rather broad class of models to a generalized, or weighted, variant of the crossing number. Our dynamic results include global well-posedness of the associated partial differential equations, regularity of equilibria for these flows as well as a more detailed investigation of dynamics near such equilibria. Finally, we explore a few global dynamical properties of these models numerically.

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“…Butγ ∈ L 2 (T 1 ) and Parseval imply (1 − cos(u)) 2 + (sin u − u) 2 u 4+p du as well. Collecting constants and extending the region of integration then yields the claim.…”

Section: Appendixmentioning

confidence: 99%

Dynamics of embedded curves by doubly-nonlocal reaction–diffusion systems

Brecht¹,

Blair²

2017

J. Phys. A: Math. Theor.

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“…This can also be seen by analytically continuing the time-evolved Jost solution ψ(λ, x; t) to the entire x-axis, by using (2.3), (2.9), and 15) by evaluating the integral with the help of (4.10) and by observing that the limit in (5.15) vanishes.…”

Section: Further Properties Of Our Explicit Solutionsmentioning

confidence: 99%

“…There are also other methods to obtain solutions to (1.1). Such methods include the use of a Darboux transformation [16], the use of a Bäcklund transformation [12,14], the bilinear method of Hirota [28], the use of various other transformations such as the Hasimoto transformation [15,27] and various other techniques [6] based on guessing the form of a solution and adjusting various parameters. The main idea behind using the transformations of Darboux and Bäcklund is to produce new solutions to (1.1) from previously known solutions, and other transformations are used to produce solutions to the NLS equation from solutions to other integrable PDEs.…”

Section: Introductionmentioning

confidence: 99%

Marchenko Equations and Norming Constants of the Matrix Zakharov-Shabat System

Demontis¹,

Mee²

2008

Operators and Matrices

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A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrödinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric and polynomial functions of the spatial and temporal coordinates.

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“…24) This approach, based on the standard finite-band integration method, has the merit of giving solutions of multiphase case in terms of the Riemann theta function. However, these solutions have a complicated form, requiring numerical integrations to explicitly enumerate parameters appearing in the solution.…”

Section: §1 Introductionmentioning

confidence: 99%

Three-Dimensional Curve Motions Induced by Modified Korteweg-de Vries Equation

Shin

2008

Progress of Theoretical Physics

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We have constructed the one-phase quasi-periodic solution of the curve equation induced using the modified Korteweg-de Vries equation. The solution is expressed in terms of the elliptic functions of Weierstrass. This solution can describe curve dynamics such as a vortex filament with axial velocity embedded in an incompressible inviscid fluid. There exist two types of curve (type A, type B) according to the form of the main spectra of the finite-band integrated solution. Our solution includes various filament shapes such as the Kelvin-type wave, the rigid vortex, plane curves, closed curves, and the Hasimoto one-solitonic filament. §1. IntroductionSpace curves are of interest in mathematics and physics. This is particularly true with how these curves' mathematical structures are related to integrable equations. The first appearance of such a relationship was between the sine-Gordon equation and the differential geometry for surfaces of constant negative curvature. 1) Another interesting development was from Hasimito, who showed that the vortex filament motion is related to the nonlinear Schrödinger equation. 2) Motivated by this development, Lamb studied various helical curves that were described using the well-known integrable equations. 3) He showed that there exists a curve that has a constant torsion and its curvature can be described using the modified Korteweg-de Vries (mKdV) equation. The mKdV equation is one of the important integrable equations, which can be solved exactly by the inverse scattering method. 4) A special class of zero torsion curves is the planar curves, which already appeared in the famous old problem of elastica. Typical solutions of the elastica problem, found by a numerical method, have been published. 5) Interestingly, this problem has occurred repeatedly in diverse forms. For example, Ishimori 6) found that the soliton solution of the mKdV equation is essentially the same as the loop soliton solution found in 7). The loop soliton describes the loop solitary wave propagating along a stretched rope. Mumford showed that computer vision problems are related to the mKdV equation. 8) A new form of the solution was given by Mumford, which was expressed using the theta functions of the genus-one Riemann surface. 8) More general solutions, corresponding to the genus-N Riemann surfaces, were given by Matsutani in terms of the Weierstrass' sigma functions and their generalizations. 9) On the other hand, 10) and 11) related the mKdV equation to the motions of curves on a plane. Some curves are explicitly constructed in 12) including those described * )

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Finite-Gap Solutions of the Vortex Filament Equation: Genus One Solutions and Symmetric Solutions

Cited by 53 publications

References 28 publications

Dynamics of embedded curves by doubly-nonlocal reaction–diffusion systems

Dynamics of embedded curves by doubly-nonlocal reaction–diffusion systems

Marchenko Equations and Norming Constants of the Matrix Zakharov-Shabat System

Three-Dimensional Curve Motions Induced by Modified Korteweg-de Vries Equation

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