Local ill-posedness of the 1D Zakharov system

Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system for any dimension $d$, in the inhomogeneous Sobolev spaces $(u,n)\in H^k(\mathbb{R}^d)\times H^s(\mathbb{R}^d)$ for a range of exponents $k$, $s$ depending on $d$. Here we restrict to dimension $d=1$ and present a few results establishing local ill-posedness for exponent pairs $(k,s)$ outside of the well-posedness regime. The techniques employed are rooted in the work of Bourgain (1993), Birnir-Kenig-Ponce-Svanstedt-Vega (1996), and Christ-Colliander-Tao (2003) applied to the nonlinear Schroedinger equation

Fast soliton scattering by attractive delta impurities

We study the Gross-Pitaevskii equation with an attractive delta function potential and show that in the high velocity limit an incident soliton is split into reflected and transmitted soliton components plus a small amount of dispersion. We give explicit analytic formulas for the reflected and transmitted portions, while the remainder takes the form of an error. Although the existence of a bound state for this potential introduces difficulties not present in the case of a repulsive potential, we show that the proportion of the soliton which is trapped at the origin vanishes in the limit

Phase-driven interaction of widely separated nonlinear Schr\"odinger solitons

We show that, for the 1d cubic NLS equation, widely separated equal amplitude in-phase solitons attract and opposite-phase solitons repel. Our result gives an exact description of the evolution of the two solitons valid until the solitons have moved a distance comparable to the logarithm of the initial separation. Our method does not use the inverse scattering theory and should be applicable to nonintegrable equations with local nonlinearities that support solitons with exponentially decaying tails. The result is presented as a special case of a general framework which also addresses, for example, the dynamics of single solitons subject to external forces

The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution

We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the $k$-th power of the energy controls the $H^{1}$ Sobolev norm of the solution over $k$-particles. This estimate is new and more difficult for attractive interactions than repulsive interactions. For the proof, we use a version of the finite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging effect is at play in the proof, which is not needed for the corresponding estimate in the repulsive case. This a priori bound allows us to prove that the corresponding BBGKY hierarchy converges to the GP limit as was done in many previous works treating the case of repulsive interactions. As a result, we obtain that the \emph{focusing} nonlinear Schr\"{o}dinger equation is the mean-field limit of the 2D time-dependent quantum many-body system with attractive interatomic interaction and asymptotically factorized initial data. An assumption on the size of the $L^{1}$-norm of the interatomic interaction potential is needed that corresponds to the sharp constant in the 2D Gagliardo-Nirenberg inequality though the inequality is not directly relevant because we are dealing with a trace instead of a power

A class of solutions to the 3d cubic nonlinear Schroedinger equation that blow-up on a circle

We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS) i\partial_t u + \Delta u + |u|^2 u=0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H^1_{axial}(R^3) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Rapha\"el's construction \cite{R} of a family of solutions to the 2d quintic focusing NLS, i\partial_t u + \Delta u + |u|^4 u=0, that blow-up on a circle.Comment: updated introduction, expanded Section 21, added reference

Divergence of infinite-variance nonradial solutions to the 3d NLS equation

We consider solutions $u(t)$ to the 3d NLS equation $i\partial_t u + \Delta u + |u|^2u=0$ such that $\|xu(t)\|_{L^2} = \infty$ and $u(t)$ is nonradial. Denoting by $M[u]$ and $E[u]$, the mass and energy, respectively, of a solution $u$, and by $Q(x)$ the ground state solution to $-Q+\Delta Q+|Q|^2Q=0$, we prove the following: if $M[u]E[u]<M[Q]E[Q]$ and $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}>\|Q\|_{L^2}\|\nabla Q\|_{L^2}$, then either $u(t)$ blows-up in finite positive time or $u(t)$ exists globally for all positive time and there exists a sequence of times $t_n\to +\infty$ such that $\|\nabla u(t_n)\|_{L^2} \to \infty$. Similar statements hold for negative time