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We develop resonance-based low-regularity numerical integrators for stochastic Schr"odinger equations with additive $Q$-Wiener noise, covering both the linear equation with rough potential and the cubic nonlinear case. For the linear…
Let (M,g) be a three-dimensional smooth compact Riemannian manifold such that all geodesics are simple and closed with a common minimal period, such as the 3-sphere S^3 with canonical metric. In this work the global well-posedness problem…
In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schr\"odinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we…
This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) =…
We consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic case, the author proved…
We prove global well-posedness and scattering in $H^1$ for the defocusing nonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases} &(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases} \end{equation*} on the…
We consider the focusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=|u|^\frac{8}{d-4}u$. We prove that if a maximal-lifespan radial solution $u: I\times\Bbb R^d\to\mathbb{C}$ obeys…
Coupled nonlinear Schr\"odinger equations model various physical phenomena, such as wave propagation in nonlinear optics, multi-component Bose-Einstein condensates, and shallow water waves. Despite their extensive applications, analytical…
Consider the initial value problem for systems of cubic derivative nonlinear Schr\"odinger equations in one space dimension with the masses satisfying a suitable resonance relation. We give structural conditions on the nonlinearity under…
We consider the Schr\"odinger equation on a half space in any dimension with a class of nonhomogeneous boundary conditions including Dirichlet, Neuman and the so-called transparent boundary conditions. Building upon recent local in time…
We consider a mass-critical system of nonlinear Sch\"{o}dinger equations \begin{align*} \begin{cases} i\partial_t u +\Delta u =\bar{u}v,\\ i\partial_t v +\kappa \Delta v =u^2, \end{cases} (t,x)\in \mathbb{R}\times \mathbb{R}^4, \end{align*}…
In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schr\"odinger equation (NLS), $$ \textnormal{i} \, \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \R…
In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension $3$. The Hartree type nonlinearity is $[V * |u|^2]u$ and the potential $V$ which generalizes the Yukawa has some growth condition.…
In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schr\"odinger equation \[ i\partial_t u + \Delta u - |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1, \] with $b, \alpha>0$. We firstly study the decaying…
In this paper, we consider the focusing mass-critical nonlinear fourth-order Schr\"odinger equation. We prove that blowup solutions to this equation with initial data in $H^\gamma(\mathbb{R}^d), 5\leq d \leq 7,…
A new four-component nonlinear Schr\"{o}dinger equation is first proposed in this work and studied by Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a $5\times5$ matrix spectral problem for the four-component…
We consider the mass-subcritical nonlinear Schr\"odinger equation in all space dimensions with focusing or defocusing nonlinearity. For such equations with critical regularity $s_c\in(\max\{-1,-\frac{d}{2}\},0)$, we prove that any solution…
We revisit a result by Cuccagna, Kirr and Pelinovsky about the cubic nonlinear Schr\" odinger equation (NLS) with an attractive localized potential and a time-dependent factor in the nonlinearity. We show that, under generic hypotheses on…
This manuscript proves the energy scattering of global solutions to a repulsive fourth-order generalized Hartree equation with non-radial data in the inter-critical regime. This work uses a new approach due to Dodson-Murphy [4] and extends…
Let $\gamma > 0\,$, $\beta > 0\,$, $\alpha > 0$ and $0 < \sigma N < 4$. In the present paper, we study, for $c > 0$ given, the constrained minimization problem \begin{equation*} \label{MinL2fixed} m(c):=\inf_{u\in S (c) }E(u),…