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Conclusion

In our investigation into Hall MHD near a 2D X-type neutral point, we have tested the accuracy and validity of the asymptotic solutions found in [15] by checking to see if they satisfy exact equations. We have found that these solutions satisfy the incompressibility condition. The asymptotic solution for long time trivially satisfies the exact invariant. However, the small time asymptotic solution only satisfies the invariant up to O(t). Further investigation into the nature of the invariant revealed that it could be used to recursively refine the small time solution to higher degree polynomials. Despite having an arbitrary order of precision, the refined solution is only valid for a short time interval—a length which is dependent on the externally determined parameters of the system. Unlike the ideal MHD model investigated in [14], the Hall MHD model has the ability to not exhibit a finite-time singularity. This allows the magnetic field lines to stay in a more open hyperbolic configuration and prevents the current-density blow-up and plasma collapse at long times in Hall MHD.

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