Related papers: Modified scattering for nonlinear Schr\"odinger eq…
Using the Fredholm theory of the linear time-dependent Schr\"odinger equation set up in our previous article arXiv:2201.03140, we solve the final-state problem for the nonlinear Schr\"odinger problem $$ (D_t + \Delta + V) u = N[u], \quad…
We prove a modified scattering and sharp $L^\infty$ decay result for both the Hartree and Schr\"odinger-Bopp-Podolsky equations in dimensions $2$ and $3$ using the testing by wavepackets approach due to Ifrim and Tataru. We show that…
We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein-Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to…
We consider a scattering theory for convolution operators on $\mathcal{H}=\ell^2(\mathbb{Z}^d; \mathbb{C}^n)$ perturbed with a long-range potential $V:\mathbb{Z}^d\to\mathbb{R}^n$. One of the motivating examples is discrete Schr\"odinger…
We investigate scattering properties of a Moyal deformed version of the nonlinear Schr\"odinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general…
A solution of the scattering problem is obtained for the Schr\"odinger equation with the potential of induced dipole interaction, which decreases as the inverse square of the distance. Such a potential arises in the collision of an incident…
The Derivative Nonlinear Schr\"odinger equation is an $L^2$-critical nonlinear dispersive equation model for Alfv\'en waves in space plasmas. Recent numerical studies on an $L^2$-supercritical extension of this equation provide evidence of…
We consider the one-dimensional nonlinear Schr\"odinger equation $$ iu_t + u_{xx} + \mathcal{N}(u)u=0, \quad x,t \in \mathbb R, $$ with the nonlinearity term that is expressed as a sum of powers, possibly infinite: $$ \mathcal{N}(u) = \sum…
We consider a class of one-dimensional nonlinear Schr\"odinger equations of the form \[ (i\partial_t+\Delta)u = [1+a]|u|^2 u. \] For suitable localized functions $a$, such equations admit a small-data modified scattering theory, which…
We prove sharp $L^\infty$ decay and modified scattering for the Hartree nonlinear Schr\"odinger equation in dimensions $2$ and $3$ using the testing by wavepackets method of Ifrim and Tataru. We show that the scattering behavior happens at…
The long-time asymptotic behavior of solutions to the focusing nonlinear Schr\"odinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the…
In this paper, we study one-dimensional linear Schr\"odinger equations with multiple moving potentials, known as transfer charge models. Focusing on the non-self-adjoint setting that arises in the study of solitons, we systematically…
In this paper, we consider a nonlinear Schr\"odinger equation with a repulsive inverse-power potential. It is known that the corresponding stationary problem has a "radial" ground state. Here, the "radial" ground state is a least energy…
We prove the existence of the modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. This is accomplished in three steps. First, we derive a new reduced asymptotic system for the quasilinear wave…
Stationary solutions asymptoting to nonlinear plane waves of the nonlinear Schr\"odinger equation with a PT-symmetric, complex linear potential are characterized. The potential includes both a spatially varying gain-loss profile and a…
In this paper, we consider the existence and the asymptotic completeness of the wave operators for Schrodinger equations with time-dependent potentials which are short-range in space.
Generalizing Dollard's strategy, we investigate the structure of the scattering theory associated to any large time reference dynamics $U_D(t)$ allowing for the existence of M{\o}ller operators. We show that (for each scattering channel)…
We analyse the scattering operator associated with the defocusing nonlinear Schr{\"o}dinger equation which captures the evolution of solutions over an infinite time-interval under the nonlinear flow of this equation. The asymptotic nature…
Recent work by Hintz--Vasy provides a partial asymptotic analysis of the low-energy limit of scattering for Schr\"odinger operators with a short-range potential. Using a slight refinement of Hintz's algorithm, we complete the asymptotic…
Modified scattering phenomena are encountered in the study of global properties for nonlinear dispersive partial differential equations in situations where the decay of solutions at infinity is borderline and scattering fails just barely.…