Related papers: The fourth-order Schr\"{o}dinger equation on latti…
Some solutions for one class of nonlinear fourth-order partial differential equations \[u_{tt} = ({\kappa u + \gamma u^2})_{xx} + \nu uu_{xxxx} + \mu u_{xxtt} + \alpha u_x u_{xxx} + \beta u_{xx}^2 \] where $\alpha ,\;\beta ,\;\gamma ,\;\mu…
This paper studies the inhomogeneous fractional Sch\"odinger equation $$i\dot u-(-\Delta)^s u=\pm(I_\alpha *|\cdot|^b|u|^p)|x|^b|u|^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to…
We prove stability estimates for the problem of recovering the nonlinearity from scattering data. We focus our attention on nonlinear Schr\"odinger equations of the form \[ (i\partial_t+\Delta)u = a(x)|u|^p u \] in three space dimensions,…
In this paper, we introduce a novel first-order derivative for functions on a lattice graph, which extends the discrete Laplacian and generalizes the theory of discrete PDEs on lattices. First, we establish the well-posedness of generalized…
In this work, we study the quasilinear Schr\"{o}dinger equation \begin{equation*} \aligned -\Delta u-\Delta(u^2)u=|u|^{p-2}u+|u|^{q-2}u+\lambda u,\,\, x\in\R^N, \endaligned \end{equation*} under the mass constraint \begin{equation*}…
We consider the focusing mass-critical NLS $iu_t + \Delta u = - |u|^{4/d} u$ in high dimensions $d \geq 4$, with initial data $u(0) = u_0$ having finite mass $M(u_0) = \int_{\R^d} |u_0(x)|^2 dx < \infty$. It is well known that this problem…
Schultz \cite{S98} proved dispersive estimates for the wave equation on lattice graphs $\mathbb{Z}^d$ for $d=2,3,$ which was extended to $d=4$ in \cite{BCH23}. By Newton polyhedra and the algorithm introduced by Karpushkin \cite{K83}, we…
In this paper we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+f(u), \quad \quad \hbox{in }\mathbb{R}^N,\\…
In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schr\"odinger equation, $$ i\partial_{t}u+\Delta u= |u|^p u, \quad (t,x)\in \mathbb R^{d+1}, $$ with $p\in (\frac4d,\frac4{d-2})$. We prove that…
We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation $i \partial_{t}u+ \Delta u=\lambda_{0}u+\lambda_{1}|u|^\alpha u$ in $\mathbb{R}^{N}$, where $\lambda_{0},\lambda_{1}\in\mathbb{C}$, in $H^s$ subcritical and critical…
In this paper we study the multiplicity of normalized solutions to the following nonlinear Schr\"{o}dinger equation with critical growth \begin{align*} \left\{ \begin{aligned} &-\Delta u=\lambda u+\mu |u|^{q-2}u+f(u), \quad \quad \hbox{in…
We obtain global well-posedness, scattering, and global $L_t^4H_{x}^{1,4}$ spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schr\"odinger equation \[iu_t+\Delta u=u(e^{4\pi |u|^2}-1)\] in two spatial…
In this paper, we study uniqueness properties of solutions to the generalized fourth-order Schr\"odinger equations in any dimension $d$ of the following forms, $$i \partial_t u + \sum_{j=1}^d \partial_{x_j}^{\, 4} u = V(t, x) u, \quad…
The finite Laguerre transform is applied to solve Differential Equations Problems of order higher than two and a one-dimensional steady-state Schr\"{o}dinger equation, by using elementary Linear Algebra methods.
Numerical solutions for laser-matter interaction in Schr\"odinger equation has many applications in theoretical chemistry, quantum physics and condensed matter physics. In this paper we introduce a methodology which allows, with a small…
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…
For $n\geq 2$, we establish the smooth effects for the solutions of the linear fourth order Shr\"{o}dinger equation in anisotropic Lebesgue spaces with $\Box_k$-decomposition. Using these estimates, we study the Cauchy problem for the…
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…
This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schr\"{o}dinger equation: % we investigate the fundamental solution of the higher order Schr\"{o}dinger equation…
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…