International Journal of Geometric Methods in Modern Physics Vol. 13, No. 6 (2016) 1650077 (14 pages) c  World Scientific Publishing Company DOI: 10.1142/S0219887816500778 On B¨ acklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space Milica Grbovi´ c ∗ and Emilija Neˇ sovi´ c † Faculty of Science, University of Kragujevac Department of mathematics and informatics Kragujevac, Serbia ∗ milica.grbovic@kg.ac.rs † nesovickg@sbb.rs Received 6 January 2016 Accepted 11 March 2016 Published 20 April 2016 In thispaper, we introduce B¨acklund transformation of a pseudo nullcurve 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 local- ized 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. Keywords : B¨acklund transformation; vortex filament flow; pseudo null curve. Mathematics Subject Classification 2010: 53C50, 53Z05 1. Introduction In classical differential geometry, the B¨ acklund transformation f :Σ → Σ ′ maps a pseudospherical surface Σ to a new pseudospherical surface Σ ′ , such that the mentioned surfaces are connected by the tangent line segments of fixed lengths and the angle between the normal vector fields of the surfaces at the corresponding points is constant. Moreover, f takes asymptotic lines on Σ to asymptotic lines on Σ ′ , having equal constant torsions. Hence B¨ acklund transformation can be restricted to a transformation mapping constant torsion curve α in E 3 to new curve ¯ α in E 3 , having equal constant torsion ¯ τ = τ , whereby the curvatures of α and ¯ α are related by ¯ κ = κ - 2C sin β,[1]. In Minkowski 3-space, B¨acklund transformations of non-null curves with non-null Frenet vectors are obtained in [2]. For B¨ acklund transformations of curves in hyperbolic 3-space and higher dimensional spaces, we refer to [3–6]. It is known that constant torsion curves have some applications in 1650077-1