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 ℋ^s for s≥3/2, and the existence of global C_t ℋ^1 solutions.

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(ℝ) and scattering for all such solutions. Moreover, we demonstrate that the same phenomenology holds whenever nonlinear effects are sufficiently concentrated in space.

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 -1/2

Recent well-posedness results have identified the Hardy space L^2_+ as the natural phase space for continuum Calogero-Moser models, both focusing and defocusing, on the line and on the torus. In this paper, we introduce a symplectic form on this phase space and so are able to realize these models as Hamiltonian systems. Moreover, we demonstrate that previously identified conserved quantities are mutually commuting, reinforcing the notion that these models are completely integrable. We further illustrate the utility of these structures by using them to give a new proof of global well-posedness in the critical space L^2_+, under the necessary mass restriction in the focusing case. Our work also brings to light several unforeseen connections: (i) the threshold for well-posedness coincides with that for the nondegeneracy of the symplectic form; (ii) this threshold is connected through Carleman's inequality to the isoperimetric problem in the plane; (iii) the transition from the line to the torus gives rise to a modified dynamical 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 2 and all s∈ℝ , we construct a sequence of solutions u_n whose initial data u_n(0) converges to zero in the Fourier–Lebesgue spaces ℱL^p_s(ℝ) , but whose evolutions at later times t_n diverge to infinity.

We prove the following dichotomy result for L^2(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(ℝ). Specifically, we show that solutions to the Benjamin–Ono equation emanating from small L^2(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(ℝ) for every -12<s≤1/2. This improves the regularity required for stability up to the sharp well-posedness threshold; previous work (even on single solitons) had required s≥1/2. 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.

In this note we present a result from [CKV24] related to a conjecture of Deift from 2008, who posited that almost periodic initial data leads to almost periodic solutions to the Korteweg-de Vries equation (KdV). We show that this is not always the case. Building on the new observation that the conjecture fails for the Airy equation, we construct almost periodic initial data whose KdV evolution remains bounded, but loses almost periodicity at a later time. This text is based on a Laurent Schwartz seminar given by the first author in November 2024, which in turn is based on joint work with Rowan Killip and Monica Vişan.

For slowly-varying initial data, solutions to the Ablowitz-Ladik system have been proven to converge to solutions of the cubic Schr\"odinger 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 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) (p), for all 1 <= p

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.

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 small-data scattering map uniquely determines the nonlinearity for a wide class of gauge-invariant, intercritical nonlinear Schr & ouml;dinger equations. We use the Born approximation to reduce the analysis to a deconvolution problem involving the distribution function for linear Schr & ouml;dinger solutions. We then solve this deconvolution problem using the Beurling-Lax Theorem.

We prove global well-posedness for the cubic nonlinear Schrödinger equation with nonlinearity concentrated on a homogeneous Poisson process.

We prove that the cubic nonlinear Schrodinger 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 [51], 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, 38, 46]. 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 $s\geq \tfrac14$; see [15, 24, 32, 38]. An essential ingredient in our arguments is the demonstration of a local smoothing effect for both equations, with a gain of derivatives matching that of the underlying linear equation. This in turn rests on the discovery of a one-parameter family of microscopic conservation laws that remain meaningful at this low regularity.

We address two pressing questions in the theory of the Korteweg-de Vries (KdV) equation. First, we show the uniqueness of solutions to KdV that are merely bounded, without any further decay, regularity, periodicity, or almost periodicity assumptions. The second question, emphasized by Deift, regards whether almost periodic initial data leads to almost periodic solutions to KdV. Building on the new observation that this is false for the Airy equation, we construct an example of almost periodic initial data whose KdV evolution remains bounded, but fails to be almost periodic at a later time. Our uniqueness result ensures that the solution constructed is the unique development of this initial data.

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 derivative nonlinear Schr\"odinger 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.