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1. The algebraic and analytic compactifications of the Hitchin moduli space

   https://doi.org/10.1112/mod.2024.6  

   He, Siqi; Mazzeo, Rafe; Na, Xuesen; Wentworth, Richard (January 2024, Moduli)

   Abstract Following the work of Mazzeo–Swoboda–Weiß–Witt [Duke Math. J. 165 (2016), 2227–2271] and Mochizuki [J. Topol. 9 (2016), 1021–1073], there is a map$$\overline{\Xi }$$between the algebraic compactification of the Dolbeault moduli space of$${\rm SL}(2,\mathbb{C})$$Higgs bundles on a smooth projective curve coming from the$$\mathbb{C}^\ast$$action and the analytic compactification of Hitchin’s moduli space of solutions to the$$\mathsf{SU}(2)$$self-duality equations on a Riemann surface obtained by adding solutions to the decoupled equations, known as ‘limiting configurations’. This map extends the classical Kobayashi–Hitchin correspondence. The main result that this article will show is that$$\overline{\Xi }$$fails to be continuous at the boundary over a certain subset of the discriminant locus of the Hitchin fibration. 

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2. Non-BPS floating branes and bubbling geometries

   https://doi.org/10.1007/JHEP02(2022)162  

   Heidmann, Pierre (February 2022, Journal of High Energy Physics)

   A bstract We derive a non-BPS linear ansatz using the charged Weyl formalism in string and M-theory backgrounds. Generic solutions are static and axially-symmetric with an arbitrary number of non-BPS sources corresponding to various brane, momentum and KKm charges. Regular sources are either four-charge non-extremal black holes or smooth non-BPS bubbles. We construct several families such as chains of non-extremal black holes or smooth non-BPS bubbling geometries and study their physics. The smooth horizonless geometries can have the same mass and charges as non-extremal black holes. Furthermore, we find examples that scale towards the four-charge BPS black hole when the non-BPS parameters are taken to be small, but the horizon is smoothly resolved by adding a small amount of non-extremality. 

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3. Semiexplicit symplectic integrators for non-separable Hamiltonian systems

   https://doi.org/10.1090/mcom/3778  

   Jayawardana, Buddhika; Ohsawa, Tomoki (January 2023, Mathematics of Computation)

   We construct a symplectic integrator for non-separable Hamiltonian systems combining an extended phase space approach of Pihajoki and the symmetric projection method. The resulting method is semiexplicit in the sense that the main time evolution step is explicit whereas the symmetric projection step is implicit. The symmetric projection binds potentially diverging copies of solutions, thereby remedying the main drawback of the extended phase space approach. Moreover, our semiexplicit method is symplectic in the original phase space. This is in contrast to existing extended phase space integrators, which are symplectic only in the extended phase space. We demonstrate that our method exhibits an excellent long-time preservation of invariants, and also that it tends to be as fast as and can be faster than Tao’s explicit modified extended phase space integrator particularly for small enough time steps and with higher-order implementations and for higher-dimensional problems. 

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4. Analytical construction of soliton families in one‐ and two‐dimensional nonlinear Schrödinger equations with nonparity‐time‐symmetric complex potentials

   https://doi.org/10.1111/sapm.12383  

   Yang, Jianke (April 2021, Studies in Applied Mathematics)

   Abstract The existence of soliton families in nonparity‐time‐symmetric complex potentials remains poorly understood, especially in two spatial dimensions. In this article, we analytically investigate the bifurcation of soliton families from linear modes in one‐ and two‐dimensional nonlinear Schrödinger equations with localized Wadati‐type nonparity‐time‐symmetric complex potentials. By utilizing the conservation law of the underlying non‐Hamiltonian wave system, we convert the complex soliton equation into a new real system. For this new real system, we perturbatively construct a continuous family of low‐amplitude solitons bifurcating from a linear eigenmode to all orders of the small soliton amplitude. Hence, the emergence of soliton families in these nonparity‐time‐symmetric complex potentials is analytically explained. We also compare these analytically constructed soliton solutions with high‐accuracy numerical solutions in both one and two dimensions, and the asymptotic accuracy of these perturbation solutions is confirmed. 

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5. Numerical Investigations of Non-uniqueness for the Navier–Stokes Initial Value Problem in Borderline Spaces

   https://doi.org/10.1007/s00021-023-00789-5  

   Guillod, Julien; Šverák, Vladimír (August 2023, Journal of Mathematical Fluid Mechanics)

   We consider the Cauchy problem for the incompressible Navier–Stokes equations in R3 for a one-parameter family of explicit scale-invariant axi-symmetric initial data, which is smooth away from the origin and invariant under the reflection with respect to the xy-plane. Working in the class of axi-symmetric fields, we calculate numerically scale-invariant solutions of the Cauchy problem in terms of their profile functions, which are smooth. The solutions are necessarily unique for small data, but for large data we observe a breaking of the reflection symmetry of the initial data through a pitchfork-type bifurcation. By a variation of previous results by Jia and ˇSver´ak (Invent Math 196(1):233–265, 2013, https://doi.org/10. 1007/s00222-013-0468-x) it is known rigorously that if the behavior seen here numerically can be proved, optimal nonuniqueness examples for the Cauchy problem can be established, and two different solutions can exists for the same initial datum which is divergence-free, smooth away from the origin, compactly supported, and locally (−1)-homogeneous near the origin. In particular, assuming our (finite-dimensional) numerics represents faithfully the behavior of the full (infinitedimensional) system, the problem of uniqueness of the Leray–Hopf solutions (with non-smooth initial data) has a negative answer and, in addition, the perturbative arguments such those by Kato (Math Z 187(4):471–480, 1984, https://doi.org/ 10.1007/BF01174182) and Koch and Tataru (Adv Math 157(1):22–35, 2001, https://doi.org/10.1006/aima.2000.1937), or the weak-strong uniqueness results by Leray, Prodi, Serrin, Ladyzhenskaya and others, already give essentially optimal results. There are no singularities involved in the numerics, as we work only with smooth profile functions. It is conceivable that our calculations could be upgraded to a computer-assisted proof, although this would involve a substantial amount of additional work and calculations, including a much more detailed analysis of the asymptotic expansions of the solutions at large distances. 

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