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A soliton ensemble is a particular kind of approximation of the solution of an initial-value problem for an integrable equation by a reflectionless potential that is well adapted to singular asymptotics like the small-dispersion limit. We study soliton ensembles for the Benjamin-Ono equation by using reasonable hypotheses to develop local approximations that capture highly oscillatory features of the solution and hence provide more information than weak convergence results that are easier to obtain. These local approximations are deduced independently from empirically-observed but unproven distributions of eigenvalues of two related matrices, one Hermitian and another non-Hermitian. We perform careful numerical experiments to study the asymptotic behavior of the eigenvalues of these matrices in the small-dispersion limit, and formulate conjectures reflecting our observations. Then we apply the conjectures to construct the local approximations of slowly varying profiles and rapidly oscillating profiles as well. We show that the latter profiles are consistent with the predictions of Whitham modulation theory as originally developed for the Benjamin-Ono equation by Dobrokhotov and Krichever., Comment: 60 pages, 16 figures

We consider the mass-supercritical, defocusing, nonlinear Schr{\"o}dinger equation. We prove loss of regularity in arbitrarily short times for regularized initial data belonging to a dense set of any fixed Sobolev space for which the nonlinearity is supercritical. The proof relies on the construction of initial data as a superposition of disjoint bubbles at different scales. We get an approximate solution with a time of existence bounded from below, provided by the compressible Euler equation, which enjoys zero speed of propagation. Introducing suitable renormalized modulated energy functionals, we prove spatially localized estimates which make it possible to obtain the loss of regularity., Comment: Some typos fixed, more references and explanations. New result in the introduction (Th. 1.1)

We consider the zero-dispersion limit for the Benjamin-Ono equation on the torus for bell shaped initial data. Using the approximation by truncated Fourier series, we transform the eigenvalue equation for the Lax operator into a problem in the complex plane. Then, we use the steepest descent method to get asymptotic expansions of the Lax eigenvalues. As a consequence, we determine the weak limit of solutions as the dispersion parameter goes to zero, as long as the initial data is an even bell shaped potential.

We obtain probabilistic local well-posedness in quasilinear regimes for the Schr\"odinger half-wave equation with a cubic nonlinearity. We need to use a refined ansatz because of the lack of probabilistic smoothing in the Picard's iterations, which is due to the high-low-low frequency interactions. The proof is an adaptation of the method of Bringmann on the derivative nonlinear wave equation to Schr\"odinger-type equations. In addition, we discuss ill-posedness results for this equation.

The purpose of this work is to evidence a pathological set of initial data for which the regularized solutions by convolution experience a norm-inflation mechanism, in arbitrarily short time. The result is in the spirit of the construction from Sun and Tzvetkov, where the pathological set contains superposition of profiles that concentrate at different points. Thanks to finite propagation speed of the wave equation, and given a certain time, at most one profile exhibits significant growth. However, for Schr\"odinger-type equations, we cannot preclude the profiles from interacting between each other. Instead, we propose a method that exploits the regularizing effect of the approximate identity which, at a given scale, rules out the norm inflation of the profiles that are concentrated at smaller scales., Comment: 24 pages

We consider the zero-dispersion limit for the Benjamin-Ono equation on the torus. We prove that when the initial data is a single well, the zero-dispersion limit exists in the weak sense and is uniform on every compact time interval. Moreover, the limit is equal to the signed sum of branches for the multivalued solution of the inviscid Burgers equation obtained by the method of characteristics. This result is similar to the one obtained by Miller and Xu for the Benjamin-Ono equation on the real line for decaying and positive initial data. We also establish some precise asymptotics of the spectral data with the cosine initial data, justifying our approximation method, which is analogous to the work of Miller and Wetzel concerning a family of rational potentials for the Benjamin-Ono equation on the real line.

We study the local well-posedness of the nonlinear Schr\"odinger equation associated to the Grushin operator with random initial data. To the best of our knowledge, no well-posedness result is known in the Sobolev spaces $H^k$ when $k \leq \frac{3}{2}$. In this article, we prove that there exists a large family of initial data such that, with respect to a suitable randomization in $H^k$, $k \in (1,\frac{3}{2}]$, almost-sure local well-posedness holds. The proof relies on bilinear and trilinear estimates.

We consider the Benjamin-Ono equation on the torus with an additional damping term on the smallest Fourier modes (cos and sin). We first prove global well-posedness of this equation in $L^2_{r,0}(\mathbb{T})$. Then, we describe the weak limit points of the trajectories in $L^2_{r,0}(\mathbb{T})$ when time goes to infinity, and show that these weak limit points are strong limit points. Finally, we prove the boundedness of higher-order Sobolev norms for this equation. Our key tool is the Birkhoff map for the Benjamin-Ono equation, that we use as an adapted nonlinear Fourier transform.

We consider the third order Benjamin-Ono equation on the torus $\partial_t u= \partial_x \left( -\partial_{xx}u-\frac{3}{2}u H\partial_x u - \frac{3}{2}H(u\partial_x u) + u^3 \right).$ We prove that for any $t\in\mathbb{R}$, the flow map continuously extends to $H^s_{r,0}(\mathbb{T})$ if $s\geq 0$, but does not admit a continuous extension to $H^{-s}_{r,0}(\mathbb{T})$ if $0

We consider the focusing energy-critical Schr{\"o}dinger equation on the Heisenberg group in the radial case\[i\partial_t u-\Delta_{\mathbb{H}^1} u=|u|^2u,\quad\Delta_{\mathbb{H}^1}=\frac{1}{4}(\partial_x^2+\partial_y^2)+(x^2+y^2)\partial_s^2,\quad(t,x,y,s)\in \mathbb{R}\times\mathbb{H}^1,\]which is a model for non-dispersive evolution equations. For this equation, existence of smooth global solutions and uniqueness of weak solutions in the energy space are open problems. We are interested in a family of ground state traveling waves parametrized by their speed $\beta \in (-1,1)$. We show that the traveling waves of speed close to $1$ present some orbital stability in the following sense. If the initial data is radial and close enough to one traveling wave, then there exists a global weak solution which stays close to the orbit of this traveling wave for all times. A similar result is proven for the limiting system associated to this equation.

We consider radial solutions to the cubic Schr{\"o}dinger equation on the Heisenberg group$$i\partial_t u - \Delta_{\mathbb{H}^1} u = |u|^2u, \quad\Delta_{\mathbb{H}^1} = \frac{1}{4}(\partial_x^2+\partial_y^2) + (x^2+y^2)\partial_s^2, \quad(t,x,y,s) \in \mathbb{R}\times\mathbb{H}^1.$$This equation is a model for totally non-dispersive evolution equations. We show existence of ground state traveling waves with speed $\beta \in (-1,1)$. When the speed $\beta$ is sufficiently close to $1$, we prove their uniqueness up to symmetries and their smoothness along the parameter $\beta$. The main ingredient is the emergence of a limiting system as $\beta$ tends to the limit $1$, for which we establish linear stability of the ground state traveling wave.