• Abstract

This paper is concerned with the time decay estimates of  the fourth order Schrodinger  operator $H=\Delta^{2}+V(x)$ in dimension three, where $V(x)$ is  a real valued decaying potential.   Assume that zero is  a regular point or the first kind resonance of $H$,  and $H$ has no positive eigenvalues, we established the following  time optimal decay estimates of  $e^{-itH}$ with a regular term $H^{\alpha/4}$:

 $$\|H^{\alpha/4}e^{-itH}P_{ac}(H)\|_{L^1-L^\infty}\lesssim |t|^{-\frac{3+\alpha}{4}}, \quad   0 \leq \alpha \leq 3.$$

When zero is the second or  third kind  resonance of $H$,   their decay will be significantly changed. We remark that such improved  time decay estimates with the extra regular term $H^{\alpha/4}$  will be interesting in the well-posedness and scattering of nonlinear fourth order Schrodinger equations with potentials.

• General Summary

• Abstract

The defocusing action ground state of the nonlinear Schr\"odinger equation can be characterized via three different but equivalent minimization formulations. In this work, we propose some deep neural network (DNN) approaches to compute the action ground state through the three  formulations. We first consider the unconstrained formulation, where we propose the DNN with a shift layer and demonstrate its necessity towards finding the correct ground state. The other two  formulations involve the  $L^{p+1}$-normalization or the  Nehari manifold constraint. We enforce them as hard constraints into the networks by  further proposing a normalization layer or a projection layer to the DNN. Our DNNs can then be trained in an unconstrained and unsupervised manner. Systematical numerical experiments are conducted to  demonstrate the effectiveness and superiority of the approaches.

• General Summary

• Abstract

In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators, in two and three spatial dimensions. Indeed, in the multidimensional case very few results are known on the location of the eigenvalues. This leads to solve nonlinear eigenvalue problems. In introduction we begin with a review of theoretical results and numerical results obtained for the one dimensional case. Then we present the numerical methods developed to compute the spectra (finite difference discretization) for the two and three dimensional cases. The numerical results obtained are presented and analyzed. One difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are unstable. This work is in continuity of a previous work in one spatial dimension [3].

• General Summary

• Abstract

This paper studies the $H^p - H^q  $ estimates of a class of oscillatory integrals related to dispersive equations

under the assumption that the level hypersurfaces are convex and of finite type.  As applications, we obtain the decay estimates for the solutions of higher order homogeneous and inhomogeneous Schrodinger equations.

• General Summary

• Abstract

In this work, we modify a conjugate gradient (CG) method  recently proposed in the literature, where a PRP conjugate gradient method is modified using trust region. Particularly, we propose a hybrid CG method that incorporates the parameters $\beta^{PRP}$, $\beta^{FR}$ and $\beta^{CD}$, and this new search direction satisfies both the trust region feature and the sufficient descent conditions. Furthermore, under suitable conditions the developed method is proved to be globally convergent. The method is tested on some benchmark problems from the literature and numerical results show that it is quite efficient in solving large scale problems.

• General Summary