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We study the existence and strong instability of standing wave solutions for the fractional nonlinear Schrödinger equation
\begin{equation*}\begin{cases}\mathrm{i} \psi_{t}=\left(-\Delta\right)^s \psi-\left(|\psi|^{p-2}\psi+\mu|\psi|^{q-2}\psi\right), & \quad(t,x)\in(0,\infty)\times\mathbb{R}^N,\\\psi(0,x)=\psi_0(x),&\quad x\in\mathbb{R}^N,\end{cases}\end{equation*}
where 
$N\geq2$, 
$0 \lt s \lt 1$, 
$2 \lt q \lt p \lt 2_s^*=2N/(N-2s)$, and 
$\mu\in\mathbb{R}$. The primary challenge lies in the inhomogeneity of the nonlinearity.We deal with the following three cases: (i) for 
$2 \lt q \lt p \lt 2+4s/N$ and µ < 0, there exists a threshold mass a0 for the existence of the least energy normalized solution; (ii) for 
$2+4s/N \lt q \lt p \lt 2_s^*$ and µ > 0, we reveal the existence of the ground state solution, explore the strong instability of standing waves, and provide a blow-up criterion; (iii) for 
$2 \lt q\leq2+4s/N \lt p \lt 2_s^*$ and µ < 0, the strong instability of standing wave solutions is demonstrated. These findings are illuminated through variational characterizations, the profile decomposition, and the virial estimate.
In this paper, we investigate Kolmogorov-type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter ξ as 
$ H(y,x,\xi) = \langle\omega(\xi),y\rangle {+ \bar h(y,\xi)}+\varepsilon P(y,x,\xi,\varepsilon) $, where ɛ > 0. We assume that the frequency mapping 
$\omega(\cdot)$, 
$\bar h(y,\cdot)=O(|y|^2)$ and the perturbation 
$\varepsilon P(y,x,\cdot, \varepsilon)$ maintain Hölder continuity about ξ. We prove that the persistent invariant tori retain the same frequency as those of the unperturbed tori, under a certain topological degree condition and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions with only assumption of Hölder continuous dependence of frequency mapping ω on the parameter.
The research on elasto-inertial turbulence (EIT), a new type of turbulent flow, has reached the stage of identifying the minimal flow unit (MFU). On this issue, direct numerical simulations of FENE-P fluid flow in two-dimensional channels with variable sizes are conducted in this study. We demonstrate with the increase of channel length that the simulated flow experiences several different flow patterns, and there exists an MFU for EIT to be self-sustained. At Weissenberg number (
$Wi$) higher than the one required to excite EIT, when the channel length is relatively small, a steady arrowhead regime (SAR) flow structure and a laminar-like friction coefficient is achieved. However, as the channel length increases, the flow can fully develop into EIT characterized with high flow drag. Close to the size of the MFU, the simulated flow behaves intermittently between the SAR state with low drag and EIT state with high drag. The flow falling back to ‘laminar flow’ is caused by the insufficient channel size below the MFU. Furthermore, we give the relationship between the value of the MFU and the effective 
$Wi$, and explain its physical reasons. Moreover, the intermittent flow regime obtained based on the MFU gives us an opportunity to look into the origin and exciting process of EIT. Through capturing the onset process of EIT, we observed that EIT originates from the sheet-like extension structure located near the wall, which is maybe related to the wall mode rather than the centre mode. The fracture and regeneration of this sheet-like structure is the key mechanism for the self-sustaining of EIT.
A high-energy picosecond 355 nm ultraviolet (UV) laser operating at 100 Hz was demonstrated. A 352 mJ, 69 ps, 1064 nm laser at 100 Hz was realized firstly by cascaded regenerative, laser diode end-pumped single-pass and side-pumped main amplifiers. The stimulated Raman scattering-based beam shaping technique, thermally induced birefringence compensation and 4f spatial filter-image relaying systems were used to maintain a relatively homogeneous beam intensity distribution during the amplification process. By using lithium triborate crystals for second- and third-harmonic generation (THG), a 172 mJ, approximately 56 ps, 355 nm UV laser was achieved with a THG conversion efficiency of 49%. To the best of our knowledge, it is the highest pulse energy of a picosecond 355 nm UV laser so far. The beam quality factor 
${M}^2$ and pulse energy stability were 
${M}_x^2$=3.92, 
${M}_y^2$=3.71 and root mean square of 1.48%@3 hours. This laser system could play significant roles in applications including photoconductive switch excitation, laser drilling and laser micro-fabrication.
