Three-dimensional Curve Motions Induced by the Modified Korteweg-de Vries Equation

We have constructed one-phase quasi-periodic solutions of the curve equation induced by the mKdV 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 curves (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.Comment: 26 pages, 4 figure

Sine-Gordon Soliton on a Cnoidal Wave Background

The method of Darboux transformation, which is applied on cnoidal wave solutions of the sine-Gordon equation, gives solitons moving on a cnoidal wave background. Interesting characteristics of the solution, i.e., the velocity of solitons and the shift of crests of cnoidal waves along a soliton, are calculated. Solutions are classified into three types (Type-1A, Type-1B, Type-2) according to their apparent distinct properties.Comment: 11 pages, 5 figures, Contents change

Non-local Wess-Zumino Model on Nilpotent Noncommutative Superspace

We investigate the theory of the bosonic-fermionic noncommutativity, $[x^{\mu},\theta^{\alpha}] = i \lambda^{\mu \alpha}$, and the Wess-Zumino model deformed by the noncommutativity. Such noncommutativity links well-known space-time noncommutativity to superspace non-anticommutativity. The deformation has the nilpotency. We can explicitly evaluate noncommutative effect in terms of new interactions between component fields. The interaction terms that have Grassmann couplings are induced. The noncommutativity does completely break full $\mathcal{N}=1$ supersymmetry to $\mathcal{N} = 0$ theory in Minkowski signature. Similar to the space-time noncommutativity, this theory has higher derivative terms and becomes non-local theory. However this non-locality is milder than the space-time noncommutative field theory. Due to the nilpotent feature of the coupling constants, we find that there are only finite number of Feynman diagrams that give noncommutative corrections at each loop order.Comment: Latex, 16 pages, 2 figures, typos corrected, some references and comments on auxiliary field added, a figure replaced, English refine