Schedule for: 22w5018 - New trends in Mathematics of Dispersive, Integrable and Nonintegrable Models in Fluids, Waves and Quantum Physics

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We give upper and lower bounds on the growth of Euler flows, in three and higher dimensions, of ``vortex-tube-pair-type” (axisymmetric, swirl-free, vorticity satisfying certain sign, oddness, and decay properties), generalizing and improving bounds of Choi-Jeong in three dimensions. Joint work with Evan Miller and Tai-Peng Tsai.

In this talk I will discuss on a recent derivation of the mean-field limit for multi-agent systems on a large class of sparse graphs. More specifically, the case of non-exchangeable multi-agent systems consisting of non-identical agents is addressed, where the heterogeneous distribution of connectivities in the network is known to have critical effects on the collective dynamics. Our method of proof does not only involve PDEs and stochastic analysis, but also graph theory through a novel concept of limits of sparse graphs (extended graphons) for the structure of the network, which can be regarded as a new non-trivial extension of the seminal works by L. Lovasz and B. Szegedy for dense graph limits. Our proof allows removing some of the main restrictive hypotheses in the previous literature on the connectivities between agents (dense graphs) and the cooperation between them (symmetric interactions). This is a joint work with Pierre-Emmanuel Jabin (Penn State University) and Juan Soler (University of Granada).

Zakharov water waves arises as a free surface model for an irrotational and incompressible fluid under the influence of gravity. Such fluid is considered in a domain with rigid bottom (described as ha(x)) and a free surface. When considering the pressure over the surface, Amick-Kirchgässner proved the existence of solitary waves Qc (solutions that maintain its shape as they travel in time) of speed c for the flat-bottom case (a=1). In this talk, we are interested in the analysis of the behavior of the solitary wave solution of the flat-bottom problem when the bottom actually presents a (slight) change at some point. We construct a solution to the Zakharov water waves system with non-flat bottom that is time assympotic (as time t tends to - infinity) to the Amick-Kirchgässner soliton Q_c.

The Peskin problem models the dynamics of a closed elastic membrane immersed in an incompressible Stokes fluid. This set of equations was proposed as a simplified model for the motion of red blood cells, and it serves as a canonical test problem for numerical methods. Mathematically, the problem can be seen as a generalization of the Stokes two-phase interface with surface tension, and it shares the linear structure with the Muskat problem. We will review some of the latest well-posedness results, and in particular we will focus on the global regularity issue for 2D Peskin with viscosity jump and the local well-posedness for 3D membranes.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk, we study two types of the generalized Korteweg-De Vries equation, where the nonlinearity is given with or without absolute value. We will focus more on the low power nonlinearity case, an example of which is the Schamel equation. We begin studying the local well-posedness of both equations in polynomial weighted Sobolev spaces. We then investigate the large time behavior of solutions numerically, we compare solutions to both types of equations, in particular, we confirm soliton resolution for the positive and negative data. This talk is based on joint work with S. Roudenko (FIU), K. Yang (FIU), I. Friedman, Otterbein University, and D. Son, University of Tennessee Knoxville.

In this talk, I shall consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space $H^1\times L^2$. I will present in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Mu\~noz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.

The fractional modified Korteweg-de Vries equation : \begin{align*} \partial_t u + \partial_x \left( \vert D \vert^\alpha u + u^3 \right)=0, \end{align*} for $\alpha\in (1,2)$ enjoys the existence of solitary waves : those solutions keep their form along the time and move with a constant velocity in one direction. Since the existence of multi-solitary waves with different velocities has been established (see [Eychenne 2021]), we are interested in constructing solutions behaving at large time as a sum of two solitary waves with the same velocity. I will first introduce the equation and the asymptotic behaviour of solitary waves, and state the existence of solutions whose asymptotic behaviour is a sum of two strongly interacting solitary waves with almost the same velocity. This is a joint work with Arnaud Eychenne.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk focuses on the well-posedness of the derivative nonlinear Schr\"odinger equation on the line. This model is known to be completely integrable and $L^2$-critical with respect to scaling. However, until recently not much was known regarding the well-posendess of the equation below $H^{\frac 1 2}$. In this talk we prove that the problem is well-posed in the critical space $L^2$ on the line, highlighting several recent results that led to this resolution. This is joint work with Benjamin Harrop-Griffiths, Rowan Killip, and Monica Visan.

I will report my recent joint work with Jacek Jendrej on muti-solitons to the Klein-Gordon equations including their asymptotic stability and classification.

A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions' existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices, and extension of these results for the 2d generalized SQG. This is research in collaboration with J. Dávila, A. Fernández, M. Musso and J. Wei.

Consider the Klein-Gordon equation with the focusing cubic power on R^3. Cote and Munoz (2014) constructed solutions asymptotic to sum of solitons generated by the Lorentz transform from the ground state. In view of their instability and the soliton resolution conjecture, it is natural to study global behavior of solutions initially close to those multi-solitons. Assuming that the solitons are far enough and getting away from each other, we may classify the initial data in a small neighborhood into open sets of scattering and blow-up, and a connected union of manifolds consisting of multi-solitons of various numbers. To treat the remainder globally, a key ingredient is a local energy estimate for the Klein-Gordon equation along multi-soliton trajectories, together with uniform decay for their mutual distance. It may be regarded as an extension of the classical estimate by Morawetz, including the proof, but it becomes tricky for the uniform decay, since we may not simply localize the Morawetz multiplier, which requires a space-time vector field with non-negative Jacobian matrix.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation \[ i u_t +\Delta u+|x|^{-b}|u|^{2\sigma} u = 0, \,\,\, x\in \mathbb{R}^N, \] with $N\geq 1$ and $b\in (0, \min \{ \frac{N}{2} , 2\})$. The above model is a generalization of the classical nonlinear Schr\"odinger equation (NLS), obtained when $b = 0$. We focus on the intercritical case, where the scaling invariant Sobolev index $s_c=\frac{N}{2}-\frac{2-b}{2\sigma}$ satisfies $s_c\in (0,1)$. In this talk we discuss well-posedness, scattering and blow-up results for the INLS equation in the radial and non-radial settings. These results were obtained in collaboration with Mykael Cardoso (UFPI-Brazil), Carlos G\'uzman (UFF-Brazil), Luccas Campos (UFMG-Brazil), Jason Murphy (Missouri S\&T-USA) and Sim\~ao Correa (IST-Portugal). This work is partially supported by CNPq, CAPES and FAPEMIG-Brazil.

In this talk, we consider the inhomogeneous Dirichlet initial boundary value problem for the Benjamin-Ono equation formulated on the half line. We study the global in time existence of solutions to this equation. The novelty of the present work is that we combine two different approaches between the real and the complex analysis. First, we start our work by applying the analytic continuation method by Hayashi and Kaikina (in the aforementioned references) related to the Riemann-Hilbert problem. Indeed, the construction of the Green operator is based on the introduction of a suitable necessary condition at the singularity points of the symbol, the integral representation for the sectionally analytic function, and the theory of singular integrodifferential equations with Hilbert kernels and with discontinuous coefficients. Later on, via the contraction principle, we deduce of global existence of a solutions $u\in \mathbf H^1(\mathbf R^+)$ to \eqref{nolineal}. Finally, using the Calder\'on commutator technique as developed by Ponce and Fonseca, we prove that $u\in \mathbf{L}^{2,1}(\mathbf R^+)$, in the case where the initial data $u_0$ belongs to $ \textbf H^{1+\epsilon}(\mathbf R^+)\cap \mathbf L^{1,2}(\mathbf R^+)$ and the boundary condition $h(t)\in \mathbf H^1(\mathbf R^+)\cap \mathbf{L}^1(\mathbf R^+). $

In this talk we consider the Kadomtsev-Petviashvili equations posed on R^2. For both models, we provide sequential in time asymptotic descriptions of solutions obtained from arbitrarily large initial data, inside regions of the plane not containing lumps or line solitons, and under minimal regularity assumptions. The results that we will present are consequence of two new virial identities adapted to the KP dynamics and do not require the use of the integrability of KP. Joint work with: Argenis J. Mendez, Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile. Claudio Muñoz, Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Santiago, Chile. Juan C. Pozo, Departamento de Matemáticas, Universidad de Chile, Santiago, Chile.

In this talk we will consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials. This is a joint work with V. Lotoreichik, A. Mas and M.Tušek.

We present a generalization of the famous Kuramoto Velarde equation that also, at the same time, under certain particularities, represents others interesting dispersive-dissipative equations in the field of fluid mechanics. We show that the initial value problem is well posed locally (and globally in some cases) in certain Sobolev spaces. We will also study the optimal decay in spatial variable of the associated solution

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we discuss new techniques to construct global-in-time critical Muskat solutions. The Muskat problem models the evolution of an incompressible fluid filtered in porous media driven by gravity. We show that initial Lipschitz graphs of arbitrary size provide global-in-time well-posedness for the stable scenario.

I will overview NLS models with intensity-dependent dispersion where bright and dark solitons may exist. For bright solitons, a continuous family of singular solitons exists with a cusped soliton as the limiting lowest energy state. We show that this family can be obtained variationally by minimization of mass at fixed energy and fixed length between two singularities. For dark solitons, we show that the spectral stability problem possesses only isolated eigenvalues on the imaginary axis and the energetic stability argument holds in Sobolev spaces with exponential weights.

The aim of this talk is to introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension~$2$ together with a partial result in dimension~$3$. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted $L^p$ norms of the marginals or observables of the system, uniformly in the number of particles. This allows to treat very singular interaction kernels between the particles, including repulsive Poisson interactions.

The KP-II equation is a 2-dimensional generalization of the KdV equation which takes slow variations in the transversal direction into account. The KP-II equation has explicit multi-line soliton solutions. In this talk, I will talk on linear stability of 2-line soliton solutions of P-type whose line solitons interact elastically.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will present some recent results obtained in collaboration with L. Roncal and N. Schiavone, concerning with the spectral properties of the Sublaplcian on the Heisenberg Group with a potential. The main ingredients are a priori estimates similar to the so called “Kato-Yajima” estimate for the Euclidean setting. We found a way to prove the result in this subriemannian geometry by using horizontal weights and direct methods. A connection with local smoothing properties of the associated Schrödinger equation will be given.

We consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability.

This talk concerns the nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space-dependent mass, known as the (generalized) Soler model. The equation has standing wave solutions for frequencies w in (0,m), where m is the mass in the Dirac operator. These standing waves are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations, but there are very few results in this direction. The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-self-adjoint operator with a particular block structure. I will present some partial results for the one-dimensional case, highlight the differences and similarities with the Schrödinger case, and discuss (a lot of) open problems.

Every multi-soliton manifold of the Benjamin–Ono equation on the line is invariant under the Benjamin–Ono flow. Its generalized action–angle coordinates allow to solve this equation by quadrature and we have the explicit expression of every multi-solitary wave. References [1] Gérard, P., Kappeler, T. On the integrability of the Benjamin–Ono equation on the torus, Commun. Pure Appl. Math. 74 (2021), no.8, 1685-1747, https://doi.org/10.1002/cpa.21896, 2021. [2] Sun, R. Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds, Com- mun. Math. Phys. 383, 1051–1092 (2021). https://doi.org/10.1007/s00220-021-03996-1

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this lecture I will present recent results regarding the asymptotic behavior of solutions to the initial value problem associated with the Benjamin-Ono equation. We use new techniques in order to show that solutions of this system decay to zero in the energy space in an appropriate domain. The result is independent of the integrability of the equation involved and it does not require any size assumptions. We also consider the asymptotic behavior of the solution in a domain moving in time in the right direction. Finally, we discuss the decay of the solution in the far left region. This is a joint work in collaboration with R. Freire (IMPA), C. Mu\~noz (UChile) and G. Ponce (UCSB).

We study a non-equilibrium Gross-Pitaevskii type system recently proposed to model exciton-polariton condensates. The coupled dispersive-dissipative equations present numerous mathematical challenges, and the known previous methods do not seem to apply in a standard way to study the global dynamics and singularity formation. We consider initial data in Sobolev spaces defined on euclidean and periodic domains and we prove global in-time existence results for small data (in all dimensions) with regularity above the algebra structure under some extra hypotheses. By using Strichartz estimates, we obtain global well-posedness in the one-dimensional case in the space L2xL2 (with exponential decay in some physical cases), which can not be applied to higher dimensions. Furthermore, under some physical assumptions, we show the existence of initial data, in both cases (euclidean and periodic), such that the corresponding solutions blow-up in finite or infinite time, with exponential rate. We also present an interesting result about the existence of initial data with higher regularity, in periodic domains, such that the corresponding solutions either blow-up in finite time or have unboundedness Sobolev norms with vanishing dissipation parameter.

In this talk I will discuss the stability problem for mKdV breathers on the left half-line. We are able to show that leftwards moving breathers, initially located far away from the origin, are strongly stable for the problem posed on the left half-line, when assuming homogeneous boundary conditions. The proof involves a Lyapunov functional which is almost conserved by the mKdV flow once we control some boundary terms which naturally arise. Also, recent results about orbital and asymptotic stability of solitons on the positive half-line will be discussed. This is a joint work with Miguel Alejo and Adán Corcho.

In this talk, we consider the heat equation with the natural polynomial non-linear term; and with two different cases in the diffusion term. The first case (anomalous diffusion) concerns to the fractional Laplacian operator with parameter $1<\alpha <2$, while, the second case (classical diffusion) involves the classical Laplacian operator. When $\alpha \to 2$, we prove the uniform convergence of the solutions of the anomalous diffusion case to a solution of the classical diffusion case. Moreover, we rigorously derive a convergence rate, which was experimentally exhibited in previous related works.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we will construct a large class of non-trivial (non radial) self-similar solutions of the generalized surface quasi-geostrophic equation. To the best of our knowledge, this is the first rigorous construction of any self-similar solution for these equations. Moreover, the solutions are of spiral type, locally integrable and may have a change of sign. This is a joint work with Javier Gómez-Serrano.

We examine the stability of vortex-like configuration of magnetization in magnetic materials, so-called the magnetic skyrmion. These correspond to critical points of the Landau-Lifshitz energy with the Dzyaloshinskii-Moriya (DM) interactions. From an earlier work of Doring and Melcher, it is known that the skyrmion is a ground state when the coefficient of the DM term is small. In this paper, we prove that there is an explicit critical value of the coefficient above which the skyrmion is unstable, while stable below this threshold. Moreover, we show that in the unstable regime, the infimum of energy is not bounded from below, by giving an explicit counterexample with a sort of helical configuration. This mathematically explains the occurrence of phase transition observed in some experiments. This is a joint work with I. Shimizu (Osaka, U.)

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

In this talk we will look at some recent results concerning NLS equations with slowly decaying and non-decaying initial data. We show global wellposedness for the tooth problem (that is initial data in $H^{s_1} (\mathbb R) + H^{s_2} (\mathbb T)$), based on work of the speaker with Peer Kunstmann. Moreover we show local and global wellposedness results in modulation spaces, which include slowly decaying functions with a finite number of slowly decaying derivatives, based on recent work of the speaker.