Nonlinear Dynamics of Moving Curves and Surfaces: Applications to Physical Systems

@article{Murugesh2005NonlinearDO,
  title={Nonlinear Dynamics of Moving Curves and Surfaces: Applications to Physical Systems},
  author={S. Murugesh and Muthusamy Lakshmanan},
  journal={Int. J. Bifurc. Chaos},
  year={2005},
  volume={15},
  pages={51-63}
}
The subject of moving curves (and surfaces) in three-dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a variety of physical problems in different disciplines. Making use of the underlying geometry, one can very often relate the associated evolution equations to many interesting nonlinear evolution equations, including soliton possessing nonlinear dynamical systems… Expand
Motion of Space Curves in Three-dimensional Minkowski Space $R_1^{3}$, SO(2,1) Spin Equation and Defocusing Nonlinear Schr\
We consider the dynamics of moving curves in three-dimensional Minkowski space and deduce the evolution equations for the curvature and torsion of the curve. Next by mapping a continuous SO(2,1)Expand
Knot soliton solutions for the one-dimensional non-linear Schr\"{o}dinger equation
We identify that for a broad range of parameters a variant of the soliton solution of the one-dimensional non-linear Schr\"{odinger} equation, the {\it breather}, is distinct when one studies theExpand
An Inhomogeneous Space–Time Patching Model Based on a Nonlocal and Nonlinear Schrödinger Equation
We consider an integrable, nonlocal and nonlinear, Schrödinger equation (NNSE) as a model for building space–time patchings in inhomogeneous loop quantum cosmology (LQC). We briefly review exactExpand
Rogue waves and breathers in Heisenberg spin chain
Abstract Following the connection of the non-linear Schrödinger equation with the continuum Heisenberg spin chain, we find the rogue soliton equivalent in the spin system. The breathers are alsoExpand
Notes on dispersionful and dispersionless vortex filament equations in 1+1 and 2+1 dimensions
The vortex filament equations (VFE) in 1+1 and 2+1 dimensions are considered. Some of these equations are integrable. Also the VFE with potentials and with self-consistent potentials are presented.Expand
Exact and non-exact Fermi–Pasta–Ulam–Tsingou recurrences in a Heisenberg ferromagnet
We visualize the Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in a classical Heisenberg ferromagnetic (HF) spin chain by exploiting its gauge eq uivalence to the nonlinear Schrodinger equation (NLSE).Expand
Anholonomy according to three formulations of non-null curve evolution
In this paper, we present three anholonomy densities and three geometric phases for three formulations of curve evolution in Minkowski 3-space and give applications of three formulation forExpand
Measurement Properties of Simple Biomechanical Measures of Walking Effort
TLDR
Four biomechanical walking effort outcomes are presented that have good theoretical underpinnings, excellent reproducibility and responsiveness, are simple and easy to administer with relatively inexpensive equipment, and can be used in real world environments. Expand
Three classes of non-lightlike curve evolution according to Darboux frame and geometric phase
In this paper, we present three classes of non-lightlike curve evolution according to the Darboux frame and we give three geometric phases according to three classes in Minkowski 3-space.
The Impact of ICTs Diffusion on MDGs and Baroclinic Digital Learning Environments in East and Southern Africa
TLDR
The paper proved the correlation and potential application of Chaos Theory to the design model for digital learning environments and explored the emerging trends in E-learning from ICTs for development. Expand
...
1
2
...

References

SHOWING 1-10 OF 61 REFERENCES
Motion of curves and surfaces and nonlinear evolution equations in (2+1) dimensions
It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable orExpand
Motion of strings, embedding problem and soliton equations
The motion of a flexible string of constant length in E 3 in interaction, corresponding to a variety of physical situations, is considered. It is pointed out that such a system could be studied inExpand
Anholonomy of a moving space curve and applications to classical magnetic chains.
TLDR
This work employs Lamb's formalism for space-curve dynamics to derive an expression for the anholonomy density and the geometric phase for a general time evolution of a space curve. Expand
Pattern formation outside of equilibrium
A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. ExamplesExpand
Rigid body motions, space curves, prolongation structures, fiber bundles, and solitons
The dynamics of a nonlinear string of constant length represented by a helical space curve may be studied through a consideration of the motion of an arbitrary rigid body along it. The resulting setExpand
Three-dimensional vortex dynamics in superfluid 4He: Homogeneous superfluid turbulence.
  • Schwarz
  • Physics, Medicine
  • Physical review. B, Condensed matter
  • 1988
TLDR
The behavior of a tangle of quantized vortex lines subject to uniform superfluid and normal-fluid driving velocities is investigated and the quantitative results obtained are found to be in excellent absolute agreement with a large variety of experiments, including recent studies of the vortex-tangle anisotropy. Expand
Geometric Phases in Physics
During the last few years, considerable interest has been focused on the phase that waves accumulate when the equations governing the waves vary slowly. The recent flurry of activity was set off by aExpand
On the Dynamics of Elastic Strips
TLDR
The dynamics of elastic strips is analyzed by studying the solutions of the appropriate Kirchhoff equations, showing the possibility of both pitchfork and Hopf bifurcations depending on the external and geometric constraints. Expand
An Introduction to Fluid Dynamics
This brief paper derives Euler’s equations for an inviscid fluid, summarizes the Cauchy momentum equation, derives the Navier-Stokes equation from that, and then talks about finite difference methodExpand
VISCOUS NONLINEAR DYNAMICS OF TWIST AND WRITHE
Exploiting the “natural” frame of space curves, we formulate an intrinsic dynamics of a twisted elastic filament in a viscous fluid. Coupled nonlinear equations describing the temporal evolution ofExpand
...
1
2
3
4
5
...