Schedule for: 25w5419 - Recent Advances in PDE and Mathematical Physics

In this talk we will present a review several results concerning the intermediate long wave (ILW) equation. These will include its deduction, relation with the Korteweg-de Vries and Benjamin-Ono equation, existence and properties of traveling wave solutions, integrability, conservation laws, well-posedness of the associated initial value problems (in classical and weighted Sobolev space) as well as the asymptotic behavior of the solutions.

The concentration-compactness-rigidity method, pioneered by Kenig and Merle, has become standard in the study of global well-posedness and scattering in the context of dispersive and wave equations. Albeit powerful, it requires building some heavy machinery in order to obtain the desired space-time bounds. In this talk, we present a simpler method, based on Tao's scattering criterion and on Dodson-Murphy's Virial/Morawetz inequalities, first proved for the 3d cubic nonlinear Schr\"{o}dinger (NLS) equation. Tao's criterion is, in some sense, universal, and it is expected to work in similar ways for dispersive problems. On the other hand, the Virial/Morawetz inequalities need to be established individually for each problem, as they rely on monotonicity formulae. This approach is versatile, as it was shown to work in the energy-subcritical setting for different nonlinearities, as well as for higher-order equations.

Suppose $u$ is a solution to the equation $\Delta u-Vu=0$, where $V$ is a bounded measurable function in the domain $\Omega_{\rho}=\{x:|x|>\rho>0\}\subset \R^n$, such that $|V(x)|\le k^2$, with $k>0$. Moreover, assume that the solution $u$ in $\Omega_{\rho}$ admits the estimate $$|u(x)|\le Ce^{-(k+\varepsilon)|x|}, \quad \text{ for some } C,\varepsilon>0. $$ Do these conditions imply that $u\equiv 0$ in $\Omega_{\rho}$? This question is known in the literature as Kondrat'ev--Landis' conjecture, or just Landis' conjecture. In more generality, we refer to Landis-type results when we are interested in the maximum vanishing rate of nontrivial solutions to equations with potentials. \\ In the first part of the talk, we will briefly review the main advances concerning the conjecture and formulations of the problem in other contexts. \\ In the second part, we will present recent contributions concerning Landis-type uniqueness results in the context of semidiscrete heat and the stationary discrete Schr\"odinger equations. The results are obtained through quantitative estimates within a spatial lattice which manifest an interpolation phenomenon between continuum and discrete scales. In the case of the elliptic equation, these quantitative estimates exhibit a rate decay which, in the range close to continuum, coincides with the same exponent as in the classical results of Landis' conjecture in the Euclidean setting. \\ Joint work with Aingeru Fern\'andez-Bertolin and Diana Stan.

I will review the recent results on stability of exponential and algebraic solitons in the massive Thirring model and the derivative NLS equation. The exponential solitons correspond to isolated eigenvalues in the Lax spectrum, whereas the algebraic solitons correspond to embedded eigenvalues on the imaginary axis, where the continuous spectrum resides. Stability of exponential solitons can be proven by three methods: constrained energy minimization, Backlund transformations, and inverse scattering. Stability of algebraic solitons is still an open problem, with some explicit results suggesting their orbital stability in both models.

In this talk we consider the existence of traveling wave solutions for a fifth order evolution model with a general class of polynomial-type nonlinearities, which is related to the Kaup Kupershmidt-KdV equation and includes models such as the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, and the Kaup-Kupershmit-Korteweg-de Vries equation. We also include a brief discussion on the nonexistence of solitary wave solutions. The evolution model considered has no Hamiltonian structure, as happens in many water-wave models. Using the Fourier transform, the existence of solitary wave solutions for this model is equivalent to finding a fixed point, for which we use the standard Picard method, choosing appropriately the initial condition.

In this talk we consider the Dirichlet to Neumann map (D-N) for the unit sphere in $R^3$. When we are sufficiently far from the origin, the spectrum of such an operator consists of eigenvalue clusters around the natural numbers. The distribution of the corresponding scaled eigenvalue shifts has an asymptotic expansion when the label of the cluster goes to infinity. The asymptotic expansion consists of distributions called spectral invariants. By using the averaging method, asymptotics of the Berezin symbol of the D-N map and a suitable symbol calculus, we compute the first terms of such an expansion in terms of the Radon transform (averages along geodesis of the unit sphere) of derivatives of a function that encodes conductivity properties of the media in the unit ball in $R^3$.

Taking the nonlinear Klein Gordon equation as a model problem, I will present a result on the qualitative behaviour of solutions of Hamiltonian PDEs on compact boundaryless Riemannian manifolds. Precisely, one has that solutions corresponding to smooth and small initial data remain small and smooth for times of order $\epsilon^{-r}$, $\forall r$. Here $\epsilon$ is the size of the initial datum. The proofs is based on variants of Birkhoff normal form. I will start by reviewing the classical method of Birkhoff normal form for finite dimensional Hamiltonian systems, then I will recall the theory for equations in one space dimensions and finally I will present the ideas leading to the theory in higher space dimension. Joint work with J. Bernier, B. Grebert, R. Imekratz.

In this talk, we present new and optimal estimates for the existence time of maximal solutions to the nonlinear heat equation and a nonlinear parabolic system. By considering a wide range of initial data, we improve and extend existing results. We establish new sufficient conditions for finite-time blow-up and derive both lower and upper bounds for the lifespan of solutions, using local construction methods and necessary conditions for the existence of nonnegative solutions. Furthermore, for the nonlinear heat equation, we construct small initial data that lead to finite-time blow-up and identify sequences of initial data that exhibit distinct blow-up behaviors in terms of lifespan. Part of this talk is a joint work with Fred B. Weissler.

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 Schr\"odinger equation (NLS) on a looping-edge graph (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 with strength $Z<0$ (continuity of the profiles at the vertex is not required), we study the dynamics of standing waves with a periodic-profile on the circle and a 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, $N$ and $Z$. 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 Munoz/IME-USP.

We consider the subcritical nonlinear Schrödinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a new PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided a reasonable integration scheme is available. Applications to traveling waves, breathers and solitons, as well as numerical experiments confirming the validity of the approximation are also presented as well.

This talk focuses on studying how different dispersive effects and nonlinearities influence regularizing phenomena. To this end, we consider a general model that incorporates the Benjamin equation, the Korteweg–de Vries equation, and higher-order extensions of KdV, such as the seventh and ninth-order versions. We confirm that solutions of our model satisfy Kato's smoothing effect and the propagation of the regularity principle, both of which are highly influenced by the highest-order dispersive term in the equation. Joint work with Carlos Garzón, UNAL, Bogotá.

We will discuss recent works on unique continuation for nonlocal nonlinear dispersive equations, including the Benjamin-Ono and water wave equations as well as variable coefficient versions of them. This describes joint works with Pilod, Ponce and Vega.

We consider the problem of transfer of energy to high frequencies in a quasilinear Schrödinger equation with sublinear dispersion, on the one dimensional torus. We exhibit initial data undergoing finite but arbitrary large Sobolev norm explosion: their initial norm is arbitrary small in Sobolev spaces of high regularity, but at a later time becomes arbitrary large. We develop a novel mechanism producing instability, based on Mourre’s positive commutator estimates. This is a joint work with F. Murgante.

We study the stochastic nonlinear Schrödinger (NLS) equation on the half-line $\mathbb{R}^+ = (0, \infty)$ with inhomogeneous Neumann boundary conditions perturbed by Brownian noise. The system considered is \[ \begin{cases} i \, du = (\partial_x^2 u - |u|^{2\sigma} u) \, dt + dW, & x > 0,\, t > 0, \\ u(0, x) = u_0(x), & x > 0, \\ \partial_x u(t, 0) = V(t), & t > 0, \end{cases} \] where $\sigma > 0$, $V(t)$ is a standard Brownian motion (representing boundary noise), and $dW$ is a spatially correlated additive noise. The main objective is to establish the global existence and uniqueness of solutions in the Sobolev space $H^1(\mathbb{R}^+)$. The main difficulty arises from the irregularity of the boundary data, which is not differentiable, and requires a careful analysis of the spatial derivative of the solution near $x = 0$. To overcome this, a method based on the Laplace transform and complex analysis is introduced, which allows obtaining optimal estimates for the stochastic boundary term and proving that $\lim_{x \to 0^+} \partial_x u(t,x) = V(t)$ almost surely. The solution is constructed using the Green operator $U_N(t)$ associated with the linear Schrödinger operator with Neumann condition, and a boundary operator $B_N(t, x)$ that encodes the response to the stochastic noise. Strichartz estimates and Gagliardo-Nirenberg interpolation inequalities are adapted to the half-line case, serving as essential tools to control the nonlinear term in the fixed-point argument. First, local existence is proven using a contraction principle in a suitable function space $X_T = L^\infty(0,T; H^1) \cap L^q(0,T; L^r)$. Global existence is then obtained through energy estimates and by applying Itô's formula to the energy $H(u(t))$, overcoming the lack of conservation laws due to the stochastic noise. This work is pioneering in studying stochastic Neumann boundary conditions for the nonlinear Schrödinger equation. The method used opens new possibilities for analyzing stochastic boundary value problems in other nonlinear dispersive models.

We study the nonhomogeneous Dirichlet initial-boundary value problem (IBV-problem) for the Ginzburg-Landau stochastic equation with fractional Laplacian on a quarter plane \begin{equation} \label{1.1}% \begin{cases} d\mathbf{u}=(\nabla^{\beta}\mathbf{u}+\left\vert \mathbf{u} \right\vert ^{\sigma} \mathbf{u} )dt-d\W, & \text{in } (0, \infty) \times \mathbf{D}, \\ \mathbf{u}(0,\mathbf{x})=\mathbf{u}_{0}, & \text{in } \mathbf{ D},\\ \mathbf{u}(t, \mathbf{x})|_{x_i=0} = \mathbf{V}_{j}(t,x_{j}), & \text{for }x_{j}>0, \ \ t>0 \ \ \text{y } \ j=1,2. \end{cases} \end{equation} where $i \neq j$, $\mathbf{u}=\mathbf{u}(t,\mathbf{x})$, \ $\beta\in\left( \frac{3}{2},2\right)$, \ $\sigma>0$, $$\mathbf{D}=\left\{ \mathbf{x=}\left( x_{1},x_{2}\right) : \ x_{1}>0, \ x_{2}>0\right\},$$ and $\nabla^{\beta}$ is the fractional Laplacian defined by $$ \nabla^{\beta} u= \sum_{j} \dfrac{1}{\Gamma(2-\beta)}\int_{0}^{x_j}\dfrac{u_{y_j y_j}}{(x_j-y_j)^{\beta-1}}d\mathbf{y}.$$ Here, the boundary noise $\V_{j}$ are Wiener processes and the process $\W$ is defined by \begin{equation} \W = \sum_{j} \beta_{j}(t,\omega) \Phi_{e_i}(\x), \ t>0, \ \omega \in \Omega, \ \x \in \D, \end{equation} %donde $\{ e_i(\x)\}$ es una base de Hilbert para $\mathbf{L}^{2}(\D)$, $\{ \beta_{j}(t,\omega) \}$ es una sucesión de movimiento brownianos reales independientes sobre un espacio de probabilidad $(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_{t})$, con $\mathcal{F}_t$ una filtración y $\Phi$ un operador integral definido en términos del Kernel, siempre que, $\| \Phi \|^{2}= \sum_{j} \| \Phi_{e_i} \|_{\W^1(\D)}^{2}$ exista. where $\{ e_j(\x)\}$ is a Hilbert basis for the Sobolev space $\mathbf{W}^{\epsilon}(\D)$, $\{ \beta_{j}(t,\omega) \}$ is a sequence of independent real Brownian motions on a probability space $(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_{t})$, with $\mathcal{F}_t$ a filtration and $\Phi$ an integral operator defined in terms of the Kernel, whenever, $$\| \Phi \|^{2}= \sum_{j} \| \Phi_{e_j} \|_{\W^{\epsilon}}^{2}$$ exists. We study the fundamental aspects of the initial-boundary value (IBV) problem for the stochastic equations: including the existence and uniqueness of local solutions, Itô’s isometry in stochastic calculus, and the influence of initial and boundary conditions on the well-posedness of the problem in $\mathbf{L}^{2}$-based Sobolev spaces for multidimensional domains. Additionally, we derive optimal relationships between the orders of the Sobolev spaces for the initial and boundary data and discuss the lower-order compatibility conditions required between them.

In this talk I will address the stability, under a dynamical viewpoint, of magnetic domain walls in ferromagnetic materials known as Néel walls. The term domain wall refers to a narrow transition region between opposite magnetization vectors inside a ferromagnet. When the thickness of a certain ferromagnet becomes sufficiently small with respect to the exchange characteristic length, it then becomes energetically favorable for the magnetization to rotate in the thin film plane, giving rise to a Néel wall. These domain walls underlie a traveling wave-type dynamics with a constant speed, which depends on the external magnetic field applied. Our result pertains to the nonlinear asymptotic stability of such domain walls under small perturbations in an appropriate energy space. For that purpose we consider the linearized operator around the wave, which is non-local, and prove its spectral stability with a spectral gap. The spectral analysis is the basis to prove, in turn, both the decaying properties of the generated semigroup and the nonlinear asymptotic stability of the moving Néel wall under small perturbations. This is a joint work with A. Capella (IM-UNAM), L. Morales (IIMAS-UNAM) and C. Melcher (RWTH Aachen).

A modified Boussinesq equation is analyzed, incorporating linear damping terms, energy pumping, and convective-type nonlinearity of the form 𝑣 ∂ 𝑥 ( ∣ 𝑈 𝑥 ∣ 𝑈 𝑥 ) v∂ x ​ (∣U x ​ ∣U x ​ ), formulated under periodic boundary conditions. The structure of this equation presents a complex balance between fourth-order dispersion, dissipative terms, and energy sources, distinguishing it from classical models such as KdV, Burgers, and the standard Boussinesq equation. Using the energy method, a priori estimates are obtained that allow the demonstration of the local existence of smooth solutions, and the conditions under which such solutions can be extended globally are discussed. The analysis is complemented by a representation of the Green's operator associated with the linear problem, and its role in the construction of solutions via perturbative methods is examined. Dissipation and nonlinear energy transfer mechanisms in this model are contrasted with those in the Burgers equation, where dissipation dominates, and the KdV equation, where dispersion prevails. Finally, numerical simulations are presented to validate the theoretical predictions and to observe phenomena such as the stabilization of damped waves, formation of persistent structures, and sensitivity to initial conditions. This study provides mathematical tools for a deeper understanding of nonlinear dispersive equations with energy feedback. The dynamic behavior of liquid hydrogen is also explored using a modified version of this equation, which incorporates damping terms, energy pumping, and convective nonlinearities. Theoretical aspects of the model are discussed, and preliminary numerical simulation results are presented, illustrating the generation and propagation of thermal waves in confined tanks. Finally, the relevance of this study to the development of clean and sustainable technologies is highlighted, showing how it can contribute concrete solutions to current energy challenges.