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1. The Benjamin–Ono initial-value problem for rational data with application to long-time asymptotics and scattering

   https://doi.org/10.4171/aihpc/169  

   Blackstone, Elliot; Gassot, Louise; Gérard, Patrick; Miller, Peter D (October 2025, Annales de l'Institut Henri Poincaré C, Analyse non linéaire)

   We show that the initial-value problem for the Benjamin–Ono equation on\mathbb{R}withL^{2}(\mathbb{R})rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The dimension of the determinant depends on the number of simple poles of the rational initial data only and the matrix elements depend explicitly on the independent variables(t,x)and the dispersion coefficient\epsilon. This allows for various interesting asymptotic limits to be resolved quite efficiently. As an example, and as a first step towards establishing the soliton resolution conjecture, we prove that the solution with initial datum equal to minus a soliton exhibits scattering. 

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2. On the Maxwell‐Bloch system in the sharp‐line limit without solitons

   https://doi.org/10.1002/cpa.22136  

   Li, Sitai; Miller, Peter D (January 2024, Communications on Pure and Applied Mathematics)

   Abstract We study the (characteristic) Cauchy problem for the Maxwell‐Bloch equations of light‐matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically‐motivated initial‐boundary conditions are satisfied. In particular, we present a proper Riemann‐Hilbert problem that generates the uniquecausalsolution to the Cauchy problem, that is, the solution vanishes outside of the light cone. Inside the light cone, we relate the leading‐order asymptotics to self‐similar solutions that satisfy a system of ordinary differential equations related to the Painlevé‐III (PIII) equation. We identify these solutions and show that they are related to a family of PIII solutions recently discovered in connection with several limiting processes involving the focusing nonlinear Schrödinger equation. We fully explain a resulting boundary layer phenomenon in which, even for smooth initial data (an incident pulse), the solution makes a sudden transition over an infinitesimally small propagation distance. At a formal level, this phenomenon has been described by other authors in terms of the PIII self‐similar solutions. We make this observation precise and for the first time we relate the PIII self‐similar solutions to the Cauchy problem. Our analysis of the asymptotic behavior satisfied by the optical field and medium density matrix reveals slow decay of the optical field in one direction that is actually inconsistent with the simplest version of scattering theory. Our results identify a precise generic condition on an optical pulse incident on an initially‐unstable medium sufficient for the pulse to stimulate the decay of the medium to its stable state. 

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3. Global Bifurcation for Corotating and Counter-Rotating Vortex Pairs

   https://doi.org/10.1007/s00220-023-04741-6  

   García, Claudia; Haziot, Susanna V (September 2023, Communications in Mathematical Physics)

   Abstract The existence of a local curve of corotating and counter-rotating vortex pairs was proven by Hmidi and Mateu (in Commun Math Phys 350(2):699–747, 2017) via a desingularization of a pair of point vortices. In this paper, we construct a global continuation of these local curves. That is, we consider solutions which are more than a mere perturbation of a trivial solution. Indeed, while the local analysis relies on the study of the linear equation at the trivial solution, the global analysis requires on a deeper understanding of topological properties of the nonlinear problem. For our proof, we adapt the powerful analytic global bifurcation theorem due to Buffoni and Toland to allow for the singularity at the bifurcation point. For both the corotating and the counter-rotating pairs, along the global curve of solutions either the angular fluid velocity vanishes or the two patches self-intersect. 

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4. The Cauchy Problem for Boltzmann Bi-linear Systems: The Mixing of Monatomic and Polyatomic Gases

   https://doi.org/10.1007/s10955-023-03221-4  

   Alonso, Ricardo J; Čolić, Milana; Gamba, Irene M (January 2024, Journal of Statistical Physics)

   Abstract From a unified vision of vector valued solutions in weighted Banach spaces, this paper establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unified approach for vector valued solutions in weighted Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities, angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate forp-binomial forms producing sharper estimates for thek-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only a$$2^+$$ $2^{+}$moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation. 

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5. Global well-posedness and exponential decay for the inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation

   https://doi.org/10.1090/qam/1644  

   Wang, Dehua; Ye, Zhuan (June 2023, Quarterly of Applied Mathematics)

   We consider the Cauchy problem for the inhomogeneous incompressible logarithmical hyper-dissipative Navier-Stokes equations in higher dimensions. By means of the Littlewood-Paley techniques and new ideas, we establish the existence and uniqueness of the global strong solution with vacuum over the whole space ${\mathbb{R}}^n$. Moreover, we also obtain the exponential decay-in-time of the strong solution. Our result holds without any smallness on the initial data and the initial density is allowed to have vacuum. 

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