Related papers: Weakly nonlinear Schr\"odinger equation with rando…

We consider a class of nonlinear Schr\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…

We derive a scheme by which to solve the Liouville equation perturbatively in the nonlinearity, which we apply to weakly nonlinear classical field theories. Our solution is a variant of the Prigogine diagrammatic method, and is based on an…

We consider the cubic defocusing nonlinear Schr\"odinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This weakly turbulent behavior is…

We study the nonlinear Schr\"odinger equation with a competing cubic-quintic power law nonlinearity on the waveguide domain $\mathbb R_x \times \mathbb T_{L_y}$. This model is globally well-posed and admits line solitary wave solutions,…

In this paper a nonlinear coupled Schrodinger system in the presence of mixed cubic and superlinear power laws is considered. A non standard numerical method is developed to approximate the solutions in higher dimensional case. The idea…

In this paper we consider the Schr\"odinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $\H^{s}$ if $s$ is large enough and…

We consider a generic Hamiltonian system of nonlinear interacting waves with 3-wave interactions. In the kinetic regime of wave turbulence, which assumes weak nonlinearity and large system size, the relevant observable associated with the…

In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a…

Starting from a stochastic Zakharov-Kuznetsov (ZK) equation on a lattice, the previous work [ST21] by the last two authors gave a derivation of the homogeneous 3-wave kinetic equation at the kinetic limit under very general assumptions: the…

In this Series, we study the weakly nonlinear dynamics of chemically active particles near the threshold for spontaneous motion. In this part, we focus on steady solutions and develop an `adjoint method' for deriving the nonlinear amplitude…

We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schr\"odinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the…

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…

We show that the Schr\"{o}dinger-Newton equation, which describes the nonlinear time evolution of self-gravitating quantum matter, can be made compatible with the no-signaling requirement by elevating it to a stochastic differential…

We study a system of nonlinear Schr\"odinger equations with cubic interactions in one space dimension. The orbital stability and instability of semitrivial standing wave solutions are studied for both non-degenerate and degenerate cases.

We study the asymptotic behavior of the Schr\"odinger equation in the presence of a nonlinearity of Hartree type in the semi-classical regime. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading…

Consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations in one space dimension. We provide a detailed lower bound estimate for the lifespan of the solution, which can be computed explicitly from the initial…

We consider the semiclassical limit of nonlinear Schr\"odinger equations with wavepacket initial data. We recover the Wigner measure of the problem, a macroscopic phase-space density which controls the propagation of the physical…

We propose the suppression of dispersive spreading of wave packets governed by the free-space Schr\"odinger equation with a periodically pulsed nonlinear term. Using asymptotic analysis, we construct stroboscopically-dispersionless quantum…

We consider a linear Schr\"odinger equation with a small nonlinear perturbation in $R^3$. Assume that the linear Hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition. We prove that if the initial…

We consider random Schr\"odinger equations on $\bR^d$ for $d\ge 3$ with a homogeneous Anderson-Poisson type random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. The space and time…