scholarly journals Binormal Motion of Curves with Constant Torsion in 3-Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Josu Arroyo ◽  
Óscar J. Garay ◽  
Álvaro Pámpano
Keyword(s):  
Soliton Solutions ◽  
Energy Functionals ◽  
Immersed Surfaces ◽  
Constant Torsion ◽  
Curvature Energy ◽  
Curvature And Torsion ◽  
Codazzi Equations

We study curve motion by the binormal flow with curvature and torsion depending velocity and sweeping out immersed surfaces. Using the Gauss-Codazzi equations, we obtain filaments evolving with constant torsion which arise from extremal curves of curvature energy functionals. They are “soliton” solutions in the sense that they evolve without changing shape.

scholarly journals QUADRATIC CURVATURE ENERGIES IN THE 2-SPHERE

2010 ◽  
Vol 81 (3) ◽  
pp. 496-506 ◽  
Author(s):  
JOSU ARROYO ◽  
ÓSCAR J. GARAY ◽  
JOSE MENCÍA
Keyword(s):  
Variational Analysis ◽  
Lagrange Equation ◽  
Numerical Approach ◽  
Real Space ◽  
Closed Curves ◽  
Space Forms ◽  
Energy Functionals ◽  
Critical Curves ◽  
Real Space Forms ◽  
Curvature Energy

AbstractThe classical variational analysis of curvature energy functionals, acting on spaces of curves of a Riemannian manifold, is extremely complicated, and the procedure usually can not be completely developed under such a degree of generality. Sometimes this difficulty may be overcome by focusing on specific actions in real space forms. In this note, we restrict ourselves to quadratic Lagrangian energies acting on the space of closed curves of the 2-sphere. We solve the Euler–Lagrange equation and show that there exists a two-parameter family of closed critical curves. We also discuss the stability of the circular critical points. Since, even for this class of energies, the complete variational analysis is quite involved, we use instead a numerical approach to provide a useful method of visualization of relevant aspects concerning uniqueness, stability and explicit representation of the closed critical curves.

scholarly journals Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space

2020 ◽  
Vol 72 (8) ◽  
pp. 1095-1105
Author(s):  
T. Körpinar ◽  
R. C. Demirkol
Keyword(s):  
Minkowski Space ◽  
Vector Fields ◽  
Energy Functionals ◽  
Curvature And Torsion

UDC 515.1 We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space by using the Bishop orthonormal vector frame and associated Bishop components.  Then we compute the energy of the lightlike elastic and nonelastic Cartan curve in the Minkowski space and investigate its relationship with the energy of the same curve in Bishop vector fields in .  Here, energy functionals are computed in terms of Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space .

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

scholarly journals Dispersive MHD waves and alfvenons in charge non-neutral plasmas

2008 ◽  
Vol 15 (4) ◽  
pp. 681-693 ◽  
Author(s):  
K. Stasiewicz ◽  
J. Ekeberg
Keyword(s):  
Soliton Solutions ◽  
Theoretical Models ◽  
Mhd Waves ◽  
Space Plasmas ◽  
Charge Effects ◽  
Novel Technique ◽  
Solitary Structures ◽  
Spatial Dimensions ◽  
Electric Potentials ◽  
Linear And Nonlinear

Abstract. Dispersive properties of linear and nonlinear MHD waves, including shear, kinetic, electron inertial Alfvén, and slow and fast magnetosonic waves are analyzed using both analytical expansions and a novel technique of dispersion diagrams. The analysis is extended to explicitly include space charge effects in non-neutral plasmas. Nonlinear soliton solutions, here called alfvenons, are found to represent either convergent or divergent electric field structures with electric potentials and spatial dimensions similar to those observed by satellites in auroral regions. Similar solitary structures are postulated to be created in the solar corona, where fast alfvenons can provide acceleration of electrons to hundreds of keV during flares. Slow alfvenons driven by chromospheric convection produce positive potentials that can account for the acceleration of solar wind ions to 300–800 km/s. New results are discussed in the context of observations and other theoretical models for nonlinear Alfvén waves in space plasmas.

scholarly journals Onset of the broad-ranging general stable soliton solutions of nonlinear equations in physics and gas dynamics

2021 ◽  
Vol 20 ◽  
pp. 103762
Author(s):  
Md. Abdul Kayum ◽  
Shamim Ara ◽  
M.S. Osman ◽  
M. Ali Akbar ◽  
Khaled A. Gepreel
Keyword(s):  
Nonlinear Equations ◽  
Gas Dynamics ◽  
Soliton Solutions

scholarly journals N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Common Factors ◽  
Soliton Solutions ◽  
Higher Order ◽  
Kdv Equations ◽  
The Common ◽  
Bilinear Formulation ◽  
Generalized Kdv Equations ◽  
Hirota Bilinear

Abstract We analyze N-soliton solutions and explore the Hirota N-soliton conditions for scalar (1 + 1)-dimensional equations, within the Hirota bilinear formulation. An algorithm to verify the Hirota conditions is proposed by factoring out common factors out of the Hirota function in N wave vectors and comparing degrees of the involved polynomials containing the common factors. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, together with all proofs of the existence of N-soliton solutions to all equations in two classes.

Soliton turbulence in electronegative plasma due to head-on collision of multi solitons

2020 ◽  
Vol 75 (12) ◽  
pp. 999-1007
Author(s):  
Rustam Ali ◽  
Anjali Sharma ◽  
Prasanta Chatterjee
Keyword(s):  
Wave Field ◽  
Soliton Solutions ◽  
Negative Ions ◽  
First Integral ◽  
Statistical Properties ◽  
Charged Dust ◽  
Bilinear Method ◽  
Electronegative Plasma ◽  
Kdv Equations ◽  
First Four Moments

AbstractHead-on interaction of four dust ion acoustic (DIA) solitons and the statistical properties of the wave field due to head-on interaction of solitons moving in opposite direction is studied in the framework of two Korteweg de Vries (KdV) equations. The extended Poincaré–Lighthill–Kuo (PLK) method is applied to obtain two opposite moving KdV equations from an unmagnetized four component plasma model consisting of Maxwellian negative ions, cold mobile positive ions, κ-distributed electrons and positively charged dust grains. Hirota’s bilinear method is adopted to obtain two-soliton solutions of both the KdV equations and accordingly act of soliton turbulence is presented due to head-on collision of four solitons. The amplitude and shape of the resultant wave profile at the point of strongest interaction are obtained. To see the effect of head-on collision on the statistical properties of wave field the first four moments are computed. It is observed that the head-on collision has no effect on the first integral moment while the second, third and fourth moments increase in the dominant interaction region of four solitons, which is a clean indication of soliton turbulence.

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