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In this paper we develop a new approximation method valid for a wide family of nonlinear wave equations of Nonlinear Schr\"odinger type. The result is a reduced set of ordinary differential equations for a finite set of parameters measuring…
This paper is devoted to the study of the nonlinear Schr\"odinger-Poisson system with a doping profile. We are interested in the existence of stable standing waves by considering the associated $L^2$-minimization problem. The presence of a…
The present paper is a numerical study of the dynamics of solitary wave solutions of the fractional nonlinear Schr\"{o}dinger equation, whose existence was analyzed by the authors in the first part of the project. The computational study…
We consider the focusing fractional periodic Korteweg-deVries (fKdV) and fractional periodic nonlinear Schr\"odinger equations (fNLS) equations, with $L^2$ sub-critical dispersion. In particular, this covers the case of the periodic KdV and…
We establish the full asymptotic stability of solitary wave solutions for the 1D focusing cubic Schr\"odinger equation on the line under small perturbations in weighted Sobolev spaces, building upon our results in [58]. The proof integrates…
In this study, we consider the nonlinear Sch\"odinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean…
For the one-dimensional mass-critical/supercritical pseudo-relativistic nonlinear Schrodinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy…
We consider the nonlinear Schroedinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we…
We introduce two multiscale numerical schemes for the time integration of weakly nonlinear Schr\"odinger equations, built upon the discretization of Picard iterates of the solution. These high-order schemes are designed to achieve high…
In this paper we study dynamical properties of blowup solutions to the focusing mass-critical nonlinear fractional Schr\"odinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we…
We study a defocusing quasilinear Schr\"odinger equation with nonzero conditions at infinity in dimension one. This quasilinear model corresponds to a weakly nonlocal approximation of the nonlocal Gross--Pitaevskii equation, and can also be…
This manuscript investigates the existence and spectral stability of multiple periodic standing wave solutions for a nonlinear Schr\"odinger system. By considering both cnoidal and snoidal profiles, we provide a comprehensive spectral…
We consider the subcritical nonlinear Schr\"odinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical…
A concept of semiclassically concentrated solutions is formulated for the multidimensional nonlinear Schr\"odinger equation (NLSE) with an external field. These solutions are considered as multidimensional solitary waves. The center of mass…
The nonlinear Schr\"odinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that…
We review some recent results on the minimization of the energy associated to the nonlinear Schr\"odinger Equation on non-compact graphs. Starting from seminal results given by the author together with C. Cacciapuoti, D. Finco, and D. Noja…
We consider travelling periodic and quasiperiodic wave solutions of a set of coupled nonlinear Schr\"odimger equations. In fibre optics these equations can be used to model single mode fibers with strong birefringence and two-mode optical…
We study localization, pinning, and mobility in the fractional discrete nonlinear Schr\"odinger equation (fDNLS) with generalized power-law coupling. A finite-dimensional spatial-dynamics reduction of the nonlocal recurrence yields onsite…
We consider a semilinear Schr\"odinger equation, driven by the power degenerate second order differential operator $\nabla\cdot (|x|^{2a} \nabla), a\in (0,1)$. We construct the solitary waves, in the sharp range of parameters, as minimizers…
In this work we deal with a following nonlinear Schrodinger equation in dimension greater or equal to 3, with a subcritical power-type nonlinearity and a positive potential satisfying a local condition. We prove the existence and…