On the Benjamin and Related EquationsKlein, Christian; Linares, Felipe; Pilod, Didier; Saut, Jean-Claude
doi: 10.1007/s00574-024-00428-1pmid: N/A
We consider in this paper various theoretical and numerical issues on classical one dimensional models of internal waves with surface tension. They concern the Cauchy problem, including the long time dynamic, localized solitons or multisolitons, the soliton resolution property. We survey known results, present a few new ones together with open questions and conjectures motivated by numerical simulations. A major issue is to emphasize the differences of the qualitative behavior of solutions with those of the same equations without the capillary term.
A Proof of a Conjecture of W. Hsiang on O(p)×O(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(p)\times O(q)$$\end{document} Invariant CMC Hypersurfaces with a Singularity at the OriginAlencar, Hilário; Garcia, Ronaldo; Silva Neto, Gregório
doi: 10.1007/s00574-024-00437-0pmid: N/A
In 1982, Wu-Yi Hsiang, in an article published in the Journal of Differential Geometry, classified constant mean curvature hypersurfaces in Euclidean space, invariant by the action of the group O(p)×O(q).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(p)\times O(q).$$\end{document} In his work, he conjectured that there is only one of such hypersurfaces in the Euclidean space, invariant by the action of the group O(p)×O(q),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(p)\times O(q),$$\end{document} whose profile curve has a singularity at the origin. In this paper, we prove this conjecture using blowing up techniques for degenerate singularities and invariant manifold theory for a tridimensional system of ordinary differential equations. We remark that the noninvariance of the constant mean curvature equation by homotheties prevents us from using the method developed by E. Bombieri, E. De Giorgi, and E. Giusti, in an article published in the Inventiones Mathematicae in 1969, to classify minimal hypersurfaces in the Euclidean space and transform the constant mean curvature equation in a bidimensional system of ordinary differential equations. This forces us to analyze a tridimensional system of ordinary differential equations with degenerate singularities.
A Gardner-Type Equation: Bore PropagationBona, J. L.; Chen, H.; Panthee, M.; Scialom, M.
doi: 10.1007/s00574-024-00424-5pmid: N/A
Discussed here is a regularized version of the classical Gardner equation ut+ux+uux+Au2ux-uxxt=0,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ u_t + u_x + uu_x + A u^2u_x - u_{xxt} \, = \, 0, $$\end{document}that arises in hydrodynamics and plasma physics. This initial-value problem posed on all of R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}$$\end{document} will be considered with bore-like initial data. That is, the initial wave configuration will consist of a moderately smooth function that asymptotes to zero as the spatial variable x→+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x \rightarrow +\infty $$\end{document}, but converges to r>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r > 0$$\end{document} as x→-∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x \rightarrow -\infty $$\end{document}. Such initial profiles can arise in internal wave propagation, for example. In their idealized versions set on all of R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}$$\end{document}, they possess an infinite amount of potential energy. This makes the analysis of the initial-value problem a slightly more subtle than the common situation where the initial profile is assumed to be localized, so being modelled by Sobolev-class initial data.
Plane Curves Admitting a Non-classical AutomorphismFukasawa, Satoru
doi: 10.1007/s00574-024-00431-6pmid: N/A
An automorphism of a projective curve over an algebraically closed field is said to be non-classical, if the image of any smooth point under the automorphism lies on the tangent line at the point. This paper shows that in characteristic zero, there does not exist a smooth plane curve admitting a non-classical automorphism. This result is a generalization of results according to Levcovitz in 1991. This paper also considers the relation between non-classical automorphisms and Galois points.
Dynamics of the Korteweg–de Vries Equation on a Balanced Metric GraphAngulo, Jaime; Cavalcante, Márcio
doi: 10.1007/s00574-024-00429-0pmid: N/A
In this work, we establish local well-posedness for the Korteweg-de Vries model on a balanced star graph with a structure represented by semi-infinite edges, by considering a boundary condition of δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\delta $$\end{document}-type at the unique graph-vertex. Additionally, we extend the linear instability result in Angulo and Cavalcante (Nonlinearity 34:3373–3410, 2021) to one of nonlinear instability. For the proof of local well posedness theory, the principal new ingredient is the utilization of the special solutions by Faminskii in the context of half-lines. As far as we are aware, this approach is being used for the first time in the context of star graphs and can potentially be applied to other boundary classes. In the case of the nonlinear instability result, the principal ingredients are the linearized instability known result and the fact that data-to-solution map determined by the local theory is at least of class C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^2$$\end{document}.
Partial Groupoid Actions on Smooth ManifoldsMarín, Víctor; Pinedo, Héctor; Rodríguez, José L. Vilca
doi: 10.1007/s00574-025-00441-ypmid: N/A
Given a smooth partial action α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} of a Lie groupoid G on a smooth manifold M, we provide necessary and sufficient conditions for α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} to be globalizable with smooth globalization. As an application, we provide results on the differentiable structure of orbit and stabilizer spaces induced by α,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha ,$$\end{document} which leads to other criteria for its globalization in terms of its orbit maps in the case that α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} is free and transitive. Further, under the assumption that α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha $$\end{document} is free and proper, we prove that there exists exactly one differentiable structure on the quotient structure of the orbit equivalence space M/G such that the quotient map π:M→M/G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\pi :M\rightarrow M/G$$\end{document} is a submersion.
Some zero-sum problems over ⟨x,y∣x2=yn/2,yn=1,yx=xys⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle x,y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle $$\end{document}Ribas, S.
doi: 10.1007/s00574-024-00434-3pmid: N/A
Let n≥8\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n \ge 8$$\end{document} be even, and let G=⟨x,y∣x2=yn/2,yn=1,yx=xys⟩\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G = \langle x, y \mid x^2 = y^{n/2}, y^n = 1, yx = xy^s \rangle $$\end{document}, where s2≡1(modn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s^2 \equiv 1 \pmod n$$\end{document} and s≢±1(modn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s \not \equiv \pm 1 \pmod n$$\end{document}. In this paper, we provide the precise values of some zero-sum constants over G, namely the small Davenport constant, η\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\eta $$\end{document}-constant, Gao constant, and Erdős-Ginzburg-Ziv constant. In particular, the Gao’s and Zhuang-Gao’s Conjectures hold for G. We also solve the associated inverse problems when n≡0(mod4)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n \equiv 0 \pmod 4$$\end{document}.
On the Stability of the s-Nonlocal p-Obstacle Problem and Their Coincidence Sets and Free BoundariesLo, Catharine W. K.; Rodrigues, José Francisco
doi: 10.1007/s00574-025-00439-6pmid: N/A
We show that the solutions to the nonlocal obstacle problems for the nonlocal -Δps\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\Delta _p^s$$\end{document} operator, when the fractional parameter s→σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s\rightarrow \sigma $$\end{document} for 0<σ≤1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<\sigma \le 1$$\end{document}, converge to the solution of the corresponding obstacle problem for -Δpσ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\Delta _p^\sigma $$\end{document}, being σ=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma =1$$\end{document} the classical obstacle problem for the local p-Laplacian. We discuss the weak stability of the quasi-characteristic functions of coincidence sets of the solution with the obstacle, which is a strong convergence of their characteristic functions when s↗1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s\nearrow 1$$\end{document} under a nondegeneracy condition. This stability can be shown also in terms of the convergence of the free boundaries, as well as of the coincidence sets, in Hausdorff distance when s↗1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s\nearrow 1$$\end{document}, under non-degeneracy local assumptions on the external force and a local topological property of the coincidence set of the limit classical obstacle problem for the local p-Laplacian, essentially when the limit coincidence set is the closure of its interior.