Programa

Hour	Monday	Tuesday	Wednesday	Thursday	Friday
9:00-9:50	Inscription	7	Free	14	21
9:50-10:40	1	8	Free	15	22
Coffee Break
11:00-11:50	2	9	Free	16	23
11:50-12:40	3	10	Free	17	24
Lunch
14:30-15:20	4	11	Free	18
15:20-16:10	5	12	Free	19
Coffee Break
16:30-17:20	6	13	Free	20
19:30				Event Dinner

 

• • J. Angulo (IME-USP)

    The non-linear Schrödinger equation on a looping-edge graph with \(\delta'\)-interaction at the vertex

  The aim of this lecture is to provide novel results in the mathematical studies associated to the existence and orbital stability of standing wave solutions for the cubic nonlinear Schrödinger equation (NLS) on a looping edge graph \(\mathcal{G}_N\), namely, a graph consisting of a circle and a finite amount N of infinite half-lines attached to a common vertex. By considering interactions of \(\delta'\)-type (where continuity of the profiles at the vertex is not required), we study the dynamics of standing wave solutions with a periodic-profile on the circle and soliton tail-profiles on the half-lines. The existence and (in)stability of these profiles will depend on the relative size of the phase-velocity. The theory developed in this investigation has prospects for the study of other standing wave profiles of the NLS on a looping edge graph.
  This work was done in collaboration with Alexander Muñoz(IME-USP).

• • R. Donninger (Universität Wien) 

    Self-similar blowup for mass supercritical Schrödinger equations

  I present a computer-assisted construction of self-similar blowup for a class of mass supercritical Schrödinger equations in three spatial dimensions. The talk is based on recent joint work with Birgit Schörkhuber and Lorenz Lichtnecker.

• • M. Eugenia Martínez (Universidad de Chile)

    Dynamics of a generalized abcd Boussinesq solitary wave under a slowly variable bottom

  The Boussinesq abcd system is a 4-parameter set of equations, originally derived by Bona, Chen and Saut as first order 2-wave approximations of the incompressible and irrotational, two dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation. Among many particular regimes, depending each of them in terms of the value of the parameters (a, b, c, d) present in the equations, the generic regimeis characterized by the setting b, d > 0 and a, c < 0. If additionally b= d, the abcd system is hamiltonian. In this paper, we investigate existence of generalized solitary waves and the corresponding collision problem in the physically relevant variable bottom regime, introduced by M. Chen in 2003. More precisely, the bottom is represented by a smooth (and small) space-time dependent function, allowing one to describe in detail the weakly, long interaction regime and the evolution of the solitary wave without destroying it. We prove this result by constructing a new approximate solution that represents the interaction of the solitary wave with the slowly varying bottom.
  This is a joint work with André de Laire, Olivier Goubet, Claudio Muñoz and Felipe Poblete.

• • R. Freire (Universidad de Chile)

    On the asymptotic dynamics for the \(L^2\)-supercritical gKDV equation

  We study the \(L^2\)-supercritical generalized Korteweg-de Vries equation (gKdV) with nonlinearities p>5. We develop a unified description of the non-solitonic region for arbitrary \(H^1\) solutions, both global and blowing up. Our analysis shows that the asymptotic \(L^2\) and \(L^p\) dynamics in this region is completely determined by the growth rate of the \(L^2\) norm of the gradient. In particular, we prove sharp far-field decay on both half-lines and establish normalized local vanishing along sequences of times, with improved estimates in the case of even-power nonlinearities. A key ingredient is a new virial method that compensates for the possible unboundedness of the \(H^1\) norm by exploiting the conservation of mass and a careful localization of the nonlinear flux. This yields quantitative versions of decay phenomena previously known only in subcritical settings, and it applies without any smallness or proximity-to-soliton assumptions.

• • I. Glogić (University of Bielefeld)

    Discretely self-similar blowup for a geometric wave equation

  We consider a semilinear wave equation for maps from the Minkowski space $\mathbb{R}^{1+d}$ into $\mathbb{S}^1$, featuring a null-form nonlinearity. For every $d \geq 1$, we construct a countable family of discretely self-similar (DSS) blowup solutions and analyze their nonlinear stability. We show that all DSS solutions are finite co-dimensionally stable, with the precise co-dimension determined by the explicitly computed unstable spectrum. In particular, the ground state profiles are stable. The proof relies on delicate resolvent estimates, which yield the required linear stability. To our knowledge, this is the first result on the existence and stability of DSS blowup for a nonlinear wave equation. This is joint work with David Hilditch (Lisbon) and David Wallauch (EPFL).

• • S. Gustafson (University of British Columbia)

    Logarithmic two-solitons and soliton-potential interactions for the NLS.

  For nonlinear Schrodinger equations with potentials, we identify an effective soliton-potential interaction, computed through a perturbed eigenvalue problem, which can be used to determine if logarithmically separating two-soliton solutions persist in the presence of the potential. Such solutions are known in the absence of a potential -- classically in the integrable case, and more recently due to Nguyen in non-integrable cases. Parts are joint work with Takahisa Inui, and with Mark Choi.
  

• • J. Krieger (EPFL) 

    Stabilisation and control of 1d wave maps into general targets

  I will discuss recent work, joint with J.-M. Coron(Paris) and S. Xiang(Beijing) on stabilisation and control of one dimensional wave maps into general target manifolds. Emphasis is placed on quantitative results; this reveals new geometric and topological features of the problem.

• • A. Lawrie (University of Maryland)

    Scattering for a free Klein Gordon field coupled to a harmonic oscillator

  We study the asymptotic dynamics of solutions to a toy model related to the phi-4 kink stability problem. The analysis involves an internal mode. This is joint work in progress with Gong Chen and Jacek Jendrej.

• • J. Lührmann (University of Cologne) 

    Asymptotic stability of the sine-Gordon kink outside symmetry

  We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the \phi^4 model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and spectral features of the linearized operators such as threshold resonances or internal modes.

  We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

  The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known \(\phi^4\) model.

  This is joint work with Gong Chen (GeorgiaTech).
  

• • N. S. Manton (University of Cambridge)

    Skyrmions — Smooth or Singular?

  Skyrmions are topological solitons in 3+1 dimensions. They are solutions of a nonlinear scalar field theory whose fundamental fields are pions, the particles largely responsible for strong nuclear forces. Mathematically, they are time-dependent maps from \(R^3\) to a 3-sphere, controlled by a Lorentz-invariant Lagrangian. The Skyrmions represent baryons, i.e. protons and neutrons and larger atomic nuclei. Static Skyrmions of minimal energy appear to be smooth, as do slowly spinning and colliding Skyrmions, although complete mathematical proofs are lacking. However, singularities are observed in numerical simulations of higher-energy collisions, and in Skyrmion-antiSkyrmion annihilation, something that is partly understood theoretically. Greater clarity about the singularity formation, and its physical significance, is highly desirable.

• • Y. Martel (Université Paris-Saclay)

    Asymptotic analysis of small energy breathers for the nonlinear Klein-Gordon equation

  For a class of nonlinear Klein-Gordon equations, we prove that in the small energy limit, any sequence of breathers decomposes into a finite sum of decoupled, periodically modulated canonical solitons.
  Work in collaboration with Michal Kowalczyk (U. Chile)

• • C. Maulen (University of Bielefeld)

    On the asymptotic stability problem for soliton solutions to the Boussinesq-type models.

  In this talk, we introduce the Boussinesq family of equations, revisit some known results, and present new findings on the asymptotic stability of solitary wave solutions in the energy space. In particular, we consider the one-dimensional Kaup–Broer–Kuperschmidt (KBK) model with initial data in the energy space \(H^1\times L^2\). This model belongs to the broader family of abcd Boussinesq models introduced by Bona, Chen, and Saut, to describe shallow water waves under the influence of dispersion and large amplitudes. The KBK model admits solitary waves with speeds \(c \in (−1,1)\). Angulo established their orbital stability in \(L^2\times H^1\), assuming local well-posedness in \(H^1\times H^2\). Building on this, we prove that KBK solitary waves are asymptotically stable for initial data in the energy space and for a range of speeds, relying on a new set of virial estimates specifically adapted to the KBK system in a moving frame. This talk is based on joint work with Claudio Muñoz.

• • F. Merle (Université de Cergy-Pontoise and Institut des Hautes Études Scientifiques)

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  TBA

• • A. Moutinho (Georgia Institute of Technology)

    Asymptotic stability of multi-solitons for 1d Supercritical NLS

  In this talk, for any natural m, we show the asymptotic stability of multi-solitons consisting of m solitons of on a subspace of \(L^2\) having finite co-dimension m. More precisely, we show that the remainder converges in the \(H^1\) norm to a scattering state (Modified Wave Operator). In particular, we extend the results of Krieger and Schlag about the stability single soliton to multi-solitons for a large set of supercritical NLS models. This is a joint work with Gong Chen.

• • C. Muñoz (Universidad de Chile)

    Existence and interaction of solitary waves in the Zakharov Water Waves system under a slowly varying bottom

  The Zakharov Water Waves system (ZWW) models the evolution of an inviscid irrotational fluid with free surface in 2 and 3 dimensions. These are characterized by a quasilinear system for the free surface and the fluid potential at the free boundary. In the finite flat bottom case, Amick-Kirchgässner proved the existence of small solitary waves. However, in practical situations, the bottom is always non-constant. In this work, we deal with the generalized solitary wave problem for the ZWW system with surface tension and a non-flat bottom, in one dimension, in the form of a slowly varying (in space) bottom. Our main result establishes that, under suitable conditions on the variation of the bottom, such a generalized nonlinear wave exists and interacts with the bottom in a well-defined fashion, surviving the weak long interaction and exiting the interaction region with well-defined final scaling and shift parameters. The techniques used in the proof of the main result are extensions of the construction of a multi-soliton like solution, and the interaction of solitary waves and different media. However, the ZWW case presents a considerable amount of new challenges, including: shape derivatives of Dirichlet-Neumann and Neumann-Neumann boundary operators, the quasilinear character of the model, and the lack of a suitable asymptotic stability theory for solitary waves.
  Joint work with Frédéric Rousset (U. Paris-Saclay) and María Eugenia Martínez Martini (U. Chile).
  

• • M. Ostermann (University of Bielefeld) 

    Stability of self-similar blowup in wave equations

  We consider a family of wave equations with power-nonlinearities that include spatial inhomogeneities. These equations yield numerous explicit self-similar solutions, making them appealing models from the point of view of blowup dynamics. In the pursuit of understanding their large-data evolution, we deal with the nonlinear asymptotic stability of these solutions in all energy-supercritical dimensions without imposing symmetry restrictions. We outline a general framework for the analysis of the wave flow near a self-similar blowup solution and discuss the underlying spectral-theoretic problems.

• • E. Pacherie (CNRS & Cergy University)

    Stability results for the Gross-Pitaevskii equation

  The Gross-Pitaevskii equation describes the behavior of superfluids and superconductors. Its dispersive estimates are worse than those for Schrödinger, which leads to difficulties when studying stability problems. In this talk, we present recent progress on these questions in dimension 2, in particular concerning the stability of vortices and travelling waves.

• • J. M.  Palácios (EPFL)

    Long finite time bubble trees for two co-rotational wave maps

  We show that the energy critical Wave Maps equation from \( \mathbb{R}^{2+1} \) into \( \mathbb{S}^2 \), restricted to the k=2 co-rotational setting, admits arbitrarily large numbers of concentrating concentric n bubble profiles. For any n \(\in\mathbb{N}\), we construct an n-bubble solution concentrating at scales \(\lambda_1(t)\gg \lambda_2(t)\gg \ldots\gg \lambda_n(t)\), where \(\lambda_n(t)=t^{-1}\vert \log t\vert^\beta\), and \( \lambda_j(t) \geq \exp( \int_t^{t_0} \lambda_{j+1}(s)ds)\), for any \( \beta > \frac{3}{2}\) is a parameter that can be chosen arbitrarily. This shows that, as far as finite time blow-up case is concerned, the entirety of cases postulated in the soliton resolution theorem indeed occur, provided the concentric collapsing bubbles have alternating signs.
  Work in collaboration with Joachim Krieger.

• • W. Schlag (Yale University)

    On the long-term dynamics of nonlinear wave equations on the line with a critical potential.

  We will present recent results with Krieger and Widmayer on a cubic NLS on the line with a repulsive inverse square potential. Some of the context in the wider space-time resonance and wave packet methods will be provided.

• • J. Trespalacios (Universidad Austral de Chile)

    Long time dynamics in Einstein-Belinski-Zakharov soliton spacetimes

  We consider the vacuum Einstein field equations under the Belinski-Zakharov symmetry, which leaves the problem as a 1+1 dimensional quasilinear system of PDEs. Depending on the chosen signature of the metric, these spacetimes contain most of the well-known special solutions in General Relativity. In this talk, we consider the case of cosmological metrics, in the Belinski-Zakharov notation, and prove global existence of small Belinski-Zakharov spacetimes under a natural nondegeneracy condition. We also present new energies and virial functionals to provide a description of the energy decay of smooth global cosmological metrics inside the light cone. Finally, some applications are presented in the case of the particular metrics called generalized Kasner solitons.

• • R. Velozo (Imperial College London) 

    Decay for massless Vlasov fields on Schwarzschild spacetimes — A Hamiltonian approach
    

  In this talk, I will present a Hamiltonian approach to show decay for massless Vlasov fields on the exterior of black hole backgrounds. Vlasov fields are transported along the geodesic flow, and so, the existence of trapped geodesics is an important difficulty. We address this issue by working with a well-chosen defining function of the set of past-trapped geodesics. By using a projection of the associated symplectic gradient we show decay in time of a suitable energy norm. As a corollary, we obtain decay in time of the energy-momentum tensor. This work is motivated by the black hole stability problem with Vlasov matter. This is joint work with Léo Bigorgne (Université de Rennes).
  

Students session  

• • Ignacio Acevedo (Université Paris-Saclay) 

    Kink dynamics for the Yang-Mills field in an extremal Reissner-Nordström black hole

  We consider the spherically symmetric and purely magnetic SU(2) Yang-Mills field in an extremal Reissner-Nordström black hole. The kink is a fundamental, strongly unstable stationary solution in this non-perturbative, variable coefficients model, with a polynomial tail and no explicit form. In this talk, I will discuss the main steps involved in the asymptotic stability of the kink within a suitable stable finite codimensional manifold of the energy space. The analysis relies on extending and adapting several virial techniques to this non-perturbative, variable coefficient model and on a careful construction of the stable manifold. In particular, we handle the emergence of a weak threshold resonance in the description of the stable manifold.
  This is a joint work with C. Muñoz.

• • Javier Monreal (Universidad de Chile) 

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  TBA

• • Nicolás Valenzuela (Universidad de Chile)

    Bounds on the approximation error for deep neural networks applied to dispersive models: Nonlinear waves

  We present a comprehensive framework for deriving rigorous and efficient bounds on the approximation error of deep neural networks in PDE models characterized by branching mechanisms, such as waves, Schrödinger equations, and other dispersive models. This framework utilizes the probabilistic setting established by Henry-Labordère and Touzi. We illustrate this approach by providing rigorous bounds on the approximation error for both linear and nonlinear waves in physical dimensions d=1,2,3, and analyze their respective computational costs starting from time zero. We investigate two key scenarios: one involving a linear perturbative source term, and another focusing on pure nonlinear internal interactions. This is joint work with Claudio Muñoz.