The Euler Equations of Spatial Gasdynamics and the Integrable Heisenberg Spin Equation

2011 ◽  
Vol 128 (4) ◽  
pp. 407-419 ◽  
Author(s):  
W. K. Schief ◽  
C. Rogers
Keyword(s):  
Euler Equations ◽  
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.

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

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.

Toward perfectly absorbing boundary conditions for Euler equations

AIAA Journal ◽  
1999 ◽  
Vol 37 ◽  
pp. 912-918
Author(s):  
M. E. Hayder ◽  
Fang Q. Hu ◽  
M. Y. Hussaini
Keyword(s):  
Boundary Conditions ◽  
Euler Equations ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary
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