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

Abstract

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 * )

Soliton dynamics in phase-modulated lattices

“…(It thus avoids the 'reality problem'.) Details of this procedure are now appearing in various literature [21][22][23][24][25], and we summarize only the results required in the following. First, we introduce the constants of motion s i , i = 2, 4, which are defined (related to the main spectrum variables λ i , shown in the following) as…”

“…In some cases, we need an explicit enumeration of the integration. We perform this integration using Weierstrass' functions and their identities [19,22,24]. For this, we first express ν i , i = 0, 1, 2, in equations ( 2) and (36) in terms of Weierstrass' P (u, g 2 , g 3 ) function as follows:…”

“…The MSW approach simplifies the imposition of the reality condition, and has the merit that the obtained auxiliary variables are directly related to the experimental parameters [21]. It has been used widely to find periodic solutions of nonlinear problems including vortex filament motion [22], the Rund-Regge equation [23], the thin rod dynamics [24], as well as the NLS equation [25].…”