arXiv subjects
Publications and source records attributed to Monica Visan.
We prove pointwise decay and scattering for small-mass solutions to the focusing and defocusing continuum Calogero--Moser models under suitable decay assumptions on the initial data. Our proof is based on an explicit formula for solutions and the exact conservation of the Galilean vector field associated to the equation.
It was recently discovered by Boiti--Manfrin that a variant of the Kirchhoff wave equation introduced by Pohozaev admits infinitely many conserved quantities. In this paper, we obtain explicit formulae for such conserved quantities, as well as introduce a generating function for them that is coercive. These tools are then employed to establish a priori bounds, the propagation of equicontinuity, global well-posedness in $\mathcal H^s$ for $s\geq \frac32$, and the existence of global $C_t \mathcal H^1$ solutions.
We prove the following dichotomy result for $L^2(\mathbb R)$ solutions to the Benjamin--Ono equation: On windows traveling at any speed, the solution either converges to zero or to a soliton dictated by the spectral properties of the Lax operator associated to the initial data. As an application of this result, we prove asymptotic stability of Benjamin--Ono multisolitons in $L^2(\mathbb R)$. Specifically, we show that solutions to the Benjamin--Ono equation emanating from small $L^2(\mathbb R)$ perturbations of multisolitons evolve towards a series of separating one-solitons when viewed in windows traveling with these solitons.
We show that multisoliton solutions to the Benjamin--Ono equation are uniformly orbitally stable in $H^s(\mathbb{R})$ for every $-\tfrac12<s\leq \frac12$. This improves the regularity required for stability up to the sharp well-posedness threshold; previous work (even on single solitons) had required $s\geq \frac12$. One key ingredient in our argument is a new variational characterization of multisolitons. A second ingredient is the extension to low-regularity slowly-decaying solutions of the Wu identity on eigenfunctions of the Lax operator. This extension also allows us to clarify the spectral type of the Lax operator for such potentials by precluding embedded eigenvalues.
We show that the small-data scattering map uniquely determines the nonlinearity for a class of nonlinear Schrödinger equations with radial, Hartree-type nonlinearities. Our assumptions on the convolution kernel require only a mild decay condition at infinity and permit a locally integrable singularity at the origin.
We prove uniform-in-time a priori $H^s$ bounds for solutions to the intermediate long wave equation posed both on the line and on the circle, covering the range $-\frac12<s\leq0$. Additionally, we prove that the set of orbits emanating from a bounded and equicontinuous set in $H^s$ is also bounded and equicontinuous in $H^s$. Our proof is based on the identification of a suitable Lax pair formulation for the intermediate long wave equation.
We demonstrate inflation of Fourier--Lebesgue norms for solutions to the focusing modified Korteweg--de Vries equation posed on the real line. For $p\neq 2$ and all $s\in \mathbb{R}$, we construct a sequence of solutions $u_n$ whose initial data $u_n(0)$ converges to zero in the Fourier--Lebesgue spaces $\mathcal F L^p_s(\mathbb{R})$, but whose evolutions at later times $t_n$ diverge to infinity.
We prove global well-posedness for the cubic nonlinear Schrödinger equation with nonlinearity concentrated on a homogeneous Poisson process.
We consider the cubic defocusing nonlinear Schrödinger equation in one dimension with the nonlinearity concentrated at a single point. We prove global well-posedness in the scaling-critical space $L^2(\mathbb{R})$ and scattering for all such solutions. Moreover, we demonstrate that the same phenomenology holds whenever nonlinear effects are sufficiently concentrated in space.
We establish global well-posedness for both the defocusing and focusing complex-valued modified Korteweg--de Vries equations on the real line in modulation spaces $M_p^{s,2}(\mathbb{R})$, for all $1\leq p<\infty$ and $0\leq s<3/2-1/p$. We will also show that such solutions admit global-in-time bounds in these spaces and that equicontinuous sets of initial data lead to equicontinuous ensembles of orbits. Indeed, such information forms a crucial part of our well-posedness argument.
For slowly-varying initial data, solutions to the Ablowitz-Ladik system have been proven to converge to solutions of the cubic Schrödinger equation. In this paper we show that in the continuum limit, solutions to the Ablowitz-Ladik system with $H^1$ initial data may also converge to solutions of the modified Korteweg--de Vries equation. To exhibit this new limiting behavior, it suffices that the initial data is supported near the inflection points of the dispersion relation associated with the Ablowitz-Ladik system. Our arguments employ harmonic analysis tools, Strichartz estimates, and the conservation of mass and energy. Correspondingly, they are applicable beyond the completely integrable models of greatest interest to us.
We prove dispersive decay, pointwise in time, for solutions to the mass-critical nonlinear Schrödinger equation in spatial dimensions $d=1,2,3$.
We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s(\mathbb R)$ for any regularity $s>-\frac12$. Well-posedness has long been known for $s\geq 0$, see [55], but not previously for any $s<0$. The scaling-critical value $s=-\frac12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg-de Vries equations in $H^s(\mathbb R)$ for any $s>-\frac12$. The best regularity achieved previously was $s\geq \frac14$; see [15, 24, 33, 39]. To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the non-perturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity.
We prove that the small-data scattering map uniquely determines the nonlinearity for a wide class of gauge-invariant, intercritical nonlinear Schrödinger equations. We use the Born approximation to reduce the analysis to a deconvolution problem involving the distribution function for linear Schrödinger solutions. We then solve this deconvolution problem using the Beurling--Lax Theorem.
We construct dynamics for the defocusing real-valued (Miura) mKdV equation on the real line with initial data distributed according to Gibbs measure. We also prove that Gibbs measure is invariant under these dynamics. On the way, we provide a new proof of the invariance of the Gibbs measure under mKdV on the torus. Building on these results, we construct new measure-preserving dynamics for the KdV equation on the whole real line. Samples from this family of measures exhibit the same local regularity as white-noise, but completely different statistics!
We prove that the focusing and defocusing continuum Calogero--Moser models are well-posed in the scaling-critical space $L^2_+(\mathbb R)$. In the focusing case, this requires solutions to have mass less than that of the soliton.
We prove that the derivative nonlinear Schrödinger equation in one space dimension is globally well-posed on the line in $L^2(\mathbb{R})$, which is the scaling-critical space for this equation.
We demonstrate that in three space dimensions, the scattering behaviour of semilinear wave equations with quintic-type nonlinearities uniquely determines the nonlinearity. The nonlinearity is permitted to depend on both space and time.