The estimates of the ill-posedness index of the (deformed-) continuous Heisenberg spin equation

2021 ◽  
Vol 62 (10) ◽  
pp. 101510
Author(s):  
Penghong Zhong ◽  
Ye Chen ◽  
Ganshan Yang
Keyword(s):  
Heisenberg Spin ◽  
Spin Equation

On relativistic gasdynamics: invariance under a class of reciprocal-type transformations and integrable Heisenberg spin connections

2020 ◽  
Vol 476 (2243) ◽  
pp. 20200487
Author(s):  
C. Rogers ◽  
T. Ruggeri ◽  
W. K. Schief
Keyword(s):  
Conservation Laws ◽  
Classical Limit ◽  
Three Dimensional ◽  
Classical System ◽  
Two Dimensional ◽  
Stationary Case ◽  
Heisenberg Spin ◽  
Lift And Drag ◽  
Spin Equation

A classical system of conservation laws descriptive of relativistic gasdynamics is examined. In the two-dimensional stationary case, the system is shown to be invariant under a novel multi-parameter class of reciprocal transformations. The class of invariant transformations originally obtained by Bateman in non-relativistic gasdynamics in connection with lift and drag phenomena is retrieved as a reduction in the classical limit. In the general 3+1-dimensional case, it is demonstrated that Synge’s geometric characterization of the pressure being constant along streamlines encapsulates a three-dimensional extension of an integrable Heisenberg spin equation.

scholarly journals The ill-posedness of (non-)periodic travelling wave solution for deformed continuous Heisenberg spin equation

2022 ◽  
Author(s):  
Penghong Zhong ◽  
Xingfa Chen ◽  
Ye Chen
Keyword(s):  
Periodic Solution ◽  
Wave Solution ◽  
Fourier Integral ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Heisenberg Spin ◽  
Periodic Travelling Wave ◽  
Ill Posed ◽  
Spin Equation ◽  
Two Parameter

Based on an equivalent derivative nonlinear Schr\”{o}inger equation, some periodic and non-periodic two-parameter solutions of the deformed continuous Heisenberg spin equation are obtained. These solutions are all proved to be ill-posed by the estimates of Fourier integral in ${H}^{s}_{\mathrm{S}^{2}}$ (periodic solution in ${H}^{s}_{\mathrm{S}^{2}}(\mathbb{T})$ and non-periodic solution in ${H}^{s}_{\mathrm{S}^{2}}(\mathbb{R})$ respectively). If $\alpha \neq 0$, the range of the weak ill-posedness index is $1

Nested toroidal flux surfaces in magnetohydrostatics. Generation via soliton theory

2003 ◽  
Vol 69 (6) ◽  
pp. 465-484 ◽  
Author(s):  
W. K. SCHIEF
Keyword(s):  
Magnetic Field ◽  
Total Pressure ◽  
Pressure Profile ◽  
Line Geometry ◽  
Soliton Theory ◽  
Heisenberg Spin ◽  
Rotationally Symmetric ◽  
Field Lines ◽  
Spin Equation ◽  
The Magnetic Field

It is shown that the classical magnetohydrostatic equations of an infinitely conducting fluid reduce to the integrable potential Heisenberg spin equation subject to a Jacobian condition if the magnitude of the magnetic field is constant along individual magnetic field lines. Any solution of the constrained potential Heisenberg spin equation gives rise to a multiplicity of magnetohydrostatic equilibria which share the magnetic field line geometry. The multiplicity of equilibria is reflected by the local arbitrariness of the total pressure profile. A connection with the classical Da Rios equations is exploited to establish the existence of associated helically and rotationally symmetric equilibria. As an illustration, Palumbo's ‘unique’ toroidal isodynamic equilibrium is retrieved.

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