On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space

Abstract

In this paper, we introduce Bäcklund transformation of a pseudo null curve in Minkowski 3-space as a transformation mapping a pseudo null helix to another pseudo null helix congruent to the given one. We also give the sufficient conditions for a transformation between two pseudo null curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation (LIA), we derive the vortex filament equation for a pseudo null curve and prove that the evolution equation for the torsion is the viscous Burger’s equation. As an application, we show that pseudo null curves and their Frenet frames generate solutions of the Da Rios vortex filament equation.

On Killing Magnetic Fields of Magnetic Pseudo Null Curves in R_1^3

“…On the other hand, there are different approaches in framing a pseudo null curve, see [8][9][10][11]. Let be a -dimensional Riemannian manifold and be an affine connection defined on .…”

Integrability aspects of the vortex filament equation for pseudo-null curves

“…In particular, if we take a = 0 and b = 1, we get the pseudo-null vortex filament equation (see [10]) given by γ t = N = γ s × γ ss , whose pseudo-torsion flow is…”

“…It has been shown in [8,9] that Burgers' hierarchy naturally arises from inextensible motions of plane curves in similarity geometry. More recently in [10], the authors derived the vortex filament equation for pseudo-null curves in the Minkowski space, and proved that the evolution equation for its invariant (pseudo-torsion) is the Burgers' equation. Let us remind that Burgers' equation is the nonlinear parabolic equation given by…”

Differential geometric aspects of nonlinear Schrödinger equation

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