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The purpose of this paper is to illustrate the I-method by studying low-regularity solutions of the nonlinear Schr\'[o]dinger equation in two space dimensions. By applying this method, together with the interaction Morawetz estimate, (see…
In this paper, we analyse a new exponential-type integrator for the nonlinear cubic Schr\"odinger equation on the $d$ dimensional torus $\mathbb T^d$. The scheme has recently also been derived in a wider context of decorated trees in [Y.…
In this paper, we prove new multilinear Strichartz estimates, which are obtained by using techniques of Bourgain. These estimates lead to new critical well-posedness results for the nonlinear Schr\"odinger equation on irrational tori in two…
We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on…
We prove large-data scattering in $H^1$ for inhomogeneous nonlinear Schr\"odinger equations in two space dimensions for all powers $p>0$. We assume the inhomogeneity is nonnegative and repulsive; we additionally require decay at infinity in…
We prove global well-posedness for low regularity data for the $L^2-critical$ defocusing nonlinear Schr\"odinger equation (NLS) in 2d. More precisely we show that a global solution exists for initial data in the Sobolev space $H^{s}(\mathbb…
The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schr{\"o}dinger equation posed on $\mathbb{R}^d$, in a supercritical regime. Firstly, we establish Bourgain type bilinear estimates…
In this paper, we establish an optimal dual version of trace estimate involving angular regularity. Based on this estimate, we get the generalized Morawetz estimates and weighted Strichartz estimates for the solutions to a large class of…
In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating…
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem…
We prove large-data scattering in $H^1$ for inhomogeneous nonlinear Schr\"odinger equations in one space dimension for powers $p>2$. We assume the inhomogeneity is nonnegative and repulsive; we additionally require decay at infinity in the…
We prove large-data scattering and existence of wave operators in the energy space for the systems of $N$ defocusing fourth-order Schr\"odinger equations with mass-supercritical and energy-subcritical power-type nonlinearity. In addition,…
We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schr\"odinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation…
We give a new proof of the scattering below the ground state energy level for a class of nonlinear Schr\"odinger equations (NLS) with mass-energy intercritical competing nonlinearities. Specifically, the NLS has a focusing leading order…
We investigate the global existence and scattering for the cubic fourth-order Schr\"{o}dinger equation $iu_t+\Delta^2u+|u|^2u=0$ in the low regularity space $H^s(\R^n)$ with $s<2$. We provide an alternative approach to obtain a new…
We introduce a perturbative formulation for a nonlinear extension of the J-matrix method of scattering in two dimensions. That is, we obtain the scattering matrix for the time-independent nonlinear Schr\"odinger equation in two dimensions…
We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator…
In the framework of multidimensional Compressed Sensing (CS), we introduce an analytical reconstruction formula that allows one to recover an $N$th-order $(I_1\times I_2\times \cdots \times I_N)$ data tensor $\underline{\mathbf{X}}$ from a…
We develop the existence, uniqueness, continuity, stability, and scattering theory for energy-critical nonlinear Schr\"odinger equations in dimensions $n \geq 3$, for solutions which have large, but finite, energy and large, but finite,…
In this paper, we present a microscopic derivation of the two-dimensional focusing cubic nonlinear Schr\"odinger equation starting from an interacting $N$-particle system of Bosons. The interaction potential we consider is given by…