On the role of quadratic oscillations in nonlinear Schroedinger equations II. The $L^2$-critical case

全非线性椭圆方程的Liouville定理：小振荡与x的周期性和右端项的周期性

Liouville theorem for fully nonlinear elliptic equations with the small oscillation and the periodicity in $x$ and the periodic right hand term

本文研究了具有小振荡和周期性条件的全非线性椭圆方程的二次增长解，建立了Liouville型定理，推广了线性及非线性椭圆方程的已有结果。

Multipulse phases in k-mixtures of Bose-Einstein condensates

For a competitive system of k coupled nonlinear Schroedinger equations we prove the existence, when the competition parameter is large, of positive radial solutions on R^N. We show that, when the competition parameter goes to infinity, the profile of each component separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar nonlinear Schroedinger equation. Within an Hartree-Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the…

Nonlinear resonances with a potential: Multilinear estimates and an application to NLS

This paper considers the question of global in time existence and asymptotic behavior of small-data solutions of nonlinear dispersive equations with a real potential $V$. The main concern is treating nonlinearities whose degree is low enough as to preclude the simple use of classical energy methods and decay estimates. In their place, we present a systematic approach that adapts the space-time resonance method to the non-Euclidean setting using the spectral theory of the Schroedinger operator $-Δ+V$. We start by developing tools of independent interest, namely multilinear analysis (Coifman-Mey…

Regularity for minimizers of non-autonomous non-quadratic functionals in the case 1 < p < 2: an a priori estimate

We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

三维二次非线性薛定谔方程组的全局解

Global Solutions For Systems of Quadratic Nonlinear Schrödinger Equations in 3D

本文证明了三维临界二次非线性薛定谔方程组在小局部数据下的全局适定性与散射性，通过对非线性项进行ε正则化并利用分部积分处理时空共振。

Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth

We consider a class of integral functionals with convex integrand with respect to the gradient variable, assuming that the function that measures the oscillation of the integrand with respect to the x variable belongs to a suitable Sobolev space W^{1;q}. We prove a result of higer differentiability for the minimizers. We also infer a result of Lipschitz regularity of minimizers if q > n, and a result of higher integrability for the gradient if q = n. The novelty here is that we deal with integrands satisfying subquadratic growth conditions with respect to gradient variable.

旋转气态恒星绝热振荡的线性化分析

Linearized Analysis of Adiabatic Oscillations of Rotating Gaseous Stars

针对旋转自引力气态恒星的绝热振荡，本文基于恒定角速度 Euler-Poisson 方程建立自由边界问题，并在 Lagrange 坐标下分析线性化解、算子二次束特征值及新稳定性概念。

Lindstedt series for periodic solutions of beam equations with quadratic and velocity dependent nonlinearities

We prove the existence of small amplitude periodic solutions, for a large Lebesgue measure set of frequencies, in the nonlinear beam equation with a weak quadratic and velocity dependent nonlinearity and with Dirichlet boundary conditions. Such nonlinear PDE can be regarded as a simple model describing oscillations of flexible structures like suspension bridges in presence of an uniform wind flow. The periodic solutions are explicitly constructed by means of a perturbative expansion which can be considered the analogue of the Lindstedt series expansion for the invariant tori in classical mecha…

Scattering for the Klein-Gordon equation with quadratic and variable coefficient cubic nonlinearities

We study the 1D Klein-Gordon equation with variable coefficient cubic nonlinearity. This problem exhibits a striking resonant interaction between the spatial frequencies of the nonlinear coefficients and the temporal oscillations of the solutions. In the case where the worst of this resonant behavior is absent, we prove L-Infinity scattering as well as a certain kind of strong smoothness for the solution at time-like infinity with the help of several new normal-forms transformations. Some explicit examples are also given which suggest qualitatively different behavior in the case where the stro…

Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities

We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp de…

非线性薛定谔方程爆破时间的单调性：数值测试

Monotonicity properties of blow-up time for nonlinear Schrödinger equation: numerical tests

通过数值实验研究聚焦非线性薛定谔方程在L^2临界和超临界情形下，爆破时间对耦合常数、初始数据二次振荡强度以及阻尼因子的依赖性，发现大多数情况下单调性不成立，但在临界非线性下初始数据二次振荡情形单调性成立并给出解析证明。