Related papers: Asymptotic behavior for the long-range nonlinear S…
We consider the Schr\"{o}dinger equation with power type long-range nonlinearity on star graph. Under a general boundary condition at the vertex, including Kirchhoff, Dirichlet, $\delta$, or $\delta'$ boundary condition, we show that the…
We consider the cubic nonlinear Schr\"odinger equation with long-range linear potentials in one space dimension, and prove the modified scattering in the energy space for the associated final state problem with a prescribed small asymptotic…
In this paper we consider the long time behavior of solutions to the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. For sufficiently small, smooth, decaying data we prove…
We consider the nonlinear Schr\"{o}dinger (NLS) equation with the subcritical power nonlinearity on a star graph consisting of $N$ edges and a single vertex under generalized Kirchhoff boundary conditions. The stationary NLS equation may…
We consider the nonlinear Schr\"odinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy…
We define the Schr\"odinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff…
In this work, we consider the long-time asymptotics for the Cauchy problem of a fourth-order dispersive nonlinear Schr\"{o}dinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic…
The long-time asymptotic behavior of the focusing nonlinear Schr\"odinger (NLS) equation on the line with symmetric nonzero boundary conditions at infinity is characterized by using the recently developed inverse scattering transform (IST)…
In this paper, we prove the asymptotic stability of nonlinear Schrodiger equations on star graphs, which partially solves an open problem in D. Noja \cite{DN}. The essential ingredient of our proof is the dispersive estimate for the…
In the present paper an introduction to the new subject of nonlinear dispersive hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of nonlinear Schr\"odinger equation. Special…
In this paper, we extend $\overline\partial$ steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schr\"odinger (NNLS) equation with weighted Sobolev initial data %and finite density initial data \begin{align*}…
We study a nonlinear Schr\"odinger equation with logarithmic nonlinearity on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We…
We consider the cubic nonlinear Schr\"odinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of…
We consider the cubic nonlinear Schr\"odinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leq…
We undertake a comprehensive study of the nonlinear Schr\"odinger equation $$ i u_t +\Delta u = \lambda_1|u|^{p_1} u+ \lambda_2 |u|^{p_2} u, $$ where $u(t,x)$ is a complex-valued function in spacetime $\R_t\times\R^n_x$, $\lambda_1$ and…
We consider the cubic Schrodinger equation on the line, for which the scattering theory requires modifications due to long range effects. We revisit the construction of the modified wave operator, and recall the construction of its inverse,…
We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schr"odinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we…
We study the scattering problem for the nonlinear Schr\"odinger equation $i\partial_t u + \Delta u = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that…
In this paper, we consider the Cauchy problem for the modified NLS equation. Using nonlinear steepest descent method and combining the Dbar-analysis, we show that inside any fixed cone, the long time asymptotic behavior of the solution for…
We study the nonlinear Schr\"odinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate an orbital…