arXiv subjects
Publications and source records attributed to W. -M. Wang.
We establish large sets of Anderson localized states for the quasi-periodic nonlinear wave equation on $\mathbb Z^d$, thus extending nonlinear Anderson localization from the random \cite{BW08} to a deterministic setting.
We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$, thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682--706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 10(1):1--45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrarily dimensional phase space, and the application of Bourgain's geometric lemma in [Geom. Funct. Anal., 17(3):682--706, 2007].
We establish Diophantine type estimates on shifts of trigonometric polynomials on the torus $\mathbb{T}^d$, as well as that of their square roots. These estimates arise from the spectral analysis of the quasi-periodic Schrödinger and the quasi-periodic wave operators. They have applications to the nonlinear quasi-periodic Schrödinger equations (NLS) and the nonlinear quasi-periodic wave equations (NLW). One could now, for example, extend the result of Bourgain (Geom. Funct. Anal. 17(3): 682-706, 2007) to the nonlinear setting.
Using a Birkhoff normal form transform to impede mode transfer in a finite "barrier", we prove localization of arbitrary $\ell^2$ data for polynomially long time for the nonlinear quasi-periodic Schrödinger equation on $\mathbb Z^d$.
We prove that Bourgain's separation lemma, Lemma~20.14 [B2] holds at fixed frequencies and their neighborhoods, on sub-lattices, sub-modules of the dual lattice associated with a quasi-periodic Fourier series in two dimensions. And by extension holds on the affine spaces. Previously Bourgain's lemma was not deterministic, and is valid only for a set of frequencies of positive measure. The new separation lemma generalizes classical lattice partition-type results to the hyperbolic Lorentzian setting, with signature $(1, -1, -1)$, and could be of independent interest. Combined with the method in [W2], this should lead to the existence of quasi-periodic solutions to the nonlinear Klein-Gordon equation with the usual polynomial nonlinear term $u^{p+1}$.
It is classical, following Furstenberg's theorem on positive Lyapunov exponent for products of random SL$(2, \mathbb R)$ matrices, that the one dimensional random Schrödinger operator has Anderson localization at arbitrary disorder. This paper proves a nonlinear analogue, thereby establishing a KAM-type persistence result for a non-integrable system.
We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schrödinger equations on $\Bbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous results [B2,1], and their generalizations [W2-4], are quasi-periodic in time, but periodic in space. This paper generalizes Bourgain's semi-algebraic set method [B3] to analyze nonlinear PDEs, in the non-compact space quasi-periodic setting on $\Bbb R$.
We construct space quasi-periodic standing wave solutions to the nonlinear Schrödinger equations on R^d for arbitrary d. This is a type of quasi-periodic nonlinear Bloch-Floquet waves.
Terahertz radiation generation from liquid water has long been considered to be impossible due to strong absorption. A few very recent works reported terahertz generation from water, but the mechanism is not clear and the efficiency demands to be enhanced. We show experimentally that strong single-cycle terahertz radiation with field strength of $\rm 0.2~MV cm^{-1}$ is generated from a water line/column of $\sim 200 μm$ in diameter irradiated by a mJ femtosecond laser beam. This strength is 100-fold higher than that produced from air. We attribute the mechanism to the laser-ponderomotive-force-induced current with the symmetry broken around the water-column interface. This mechanism can explain our following observations: the radiation can be generated only when the laser propagation axis deviates from the column center; the deviation determines its field strength and polarity; it is always p-polarized no matter whether the laser is p- or s-polarized. This study provides a simple and efficient scheme of table-top terahertz sources based on liquid water.
It is shown by three-dimensional QED particle-in-cell simulation that as a laser pulse of 2.5 PW and 20 fs propagates along a sub-wavelength-wide solid wire, directional synchrotron $γ-$rays along the wire surface can be efficiently generated. With 8\% energy conversion from the pulse, the $γ-$rays contains $10^{12}$ photons between 5 and 500 MeV within 10 fs duration, corresponding to peak brilliance of $10^{27}$ photons ${\rm s^{-1}~ mrad^{-2}~ mm^{-2}}$ per 0.1\% bandwidth. The brilliance and photon energy are respectively 2 and 3 orders of magnitude higher than the highest values of synchrotron radiation facilities. The radiation is attributed to the generation of nC, GeV electron beams well guided along the wire surface and their wiggling motion in strong electrostatic and magnetostatic fields induced at the high-density-wire surface. In particular, these quasistatic fields are so strong that QED effects already play a significant role for the $γ-$ray radiation. With the laser power $P_0$ ranging from 0.5 PW to 5 PW available currently, this scheme can robustly produce $γ-$rays peaked at $1^\circ$ with few-mrad divergence and the photon energy and number roughly scales with $P_0$ and $P_0^{3/2}$, respectively. Our scheme embraces both the merits of high directionality comparable to those based upon laser wakefield acceleration and high charge comparable to those based upon laser-solid interaction.
In the widely-studied two-color laser scheme for terahertz (THz) radiation from a gas, the frequency ratio of the two lasers is usually fixed at $ω_2/ω_1=$1:2. We investigate THz generation with uncommon frequency ratios. Our experiments show, for the first time, efficient THz generation with new ratios of $ω_2/ω_1=$1:4 and 2:3. We observe that the THz polarization can be adjusted by rotating the longer-wavelength laser polarization and the polarization adjustment becomes inefficient by rotating the other laser polarization; the THz energy shows similar scaling laws with different frequency ratios. These observations are inconsistent with multi-wave mixing theory, but support the gas-ionization model. This study pushes the development of the two-color scheme and provides a new dimension to explore the long-standing problem of the THz generation mechanism.
We investigate how next-generation laser pulses at 10 PW $-$ 200 PW interact with a solid target in the presence of a relativistically underdense preplasma produced by amplified spontaneous emission (ASE). Laser hole boring and relativistic transparency are strongly restrained due to the generation of electron-positron pairs and $γ$-ray photons via quantum electrodynamics (QED) processes. A pair plasma with a density above the initial preplasma density is formed, counteracting the electron-free channel produced by the hole boring. This pair-dominated plasma can block the laser transport and trigger an avalanche-like QED cascade, efficiently transfering the laser energy to photons. This renders a 1-$\rmμm$-scalelength, underdense preplasma completely opaque to laser pulses at this power level. The QED-induced opacity therefore sets much higher contrast requirements for such pulse in solid-target experiments than expected by classical plasma physics. Our simulations show for example, that proton acceleration from the rear of a solid with a preplasma would be strongly impaired.
We develop the particle-in-cell (PIC) code KLAPS to include the photon generation via the Compton scattering and electron-positron creation via the Breit-Wheeler process due to quantum electrodynamics (QED) effects. We compare two sets of existing formulas for the photon generation and different Monte Carlo algorithms. Then we benchmark the PIC simulation results.
Significant progress has been made towards laser-driven fusion ignition via different schemes, including direct and indirect central ignition, fast ignition, shock ignition, and impact ignition schemes. However, to reach ignition conditions, there are still various technical and physical challenges to be solved for all these schemes. Here, our multi-dimensional integrated simulation shows that the fast-ignition conditions could be achieved when two 2.8 petawatt heating laser pulses counter-propagate along a 3.5 kilotesla external magnetic field. Within a period of 5 picoseconds, the laser pulses heat a nuclear fuel to reach the ignition conditions. Furthermore, we present the parameter windows of lasers and magnetic fields required for ignition for experimental test.
It is shown, by simulation and theory, that circularly or elliptically polarized terahertz radiation can be generated when a static magnetic (B) field is imposed on a gas target along the propagation direction of a two-color laser driver. The radiation frequency is determined by $\sqrt{ω_p^2+{ω_c^2}/{4}} + {ω_c}/{2}$, where $ω_p$ is the plasma frequency and $ω_c$ is the electron cyclotron frequency. With the increase of the B field, the radiation changes from a single-cycle broadband waveform to a continuous narrow-band emission. In high-B-field cases, the radiation strength is proportional to $ω_p^2/ω_c$. The B field provides a tunability in the radiation frequency, spectrum width, and field strength.
We construct time quasi-periodic solutions to the energy supercritical nonlinear Schrödinger equations on the torus in arbitrary dimensions. This introduces a new approach, which could have general applicability.
An integrated simulation approach fully based upon particle-in-cell (PIC) model is proposed, which involves both fast particle generation via laser solid-density plasma interaction and transport and energy deposition of the particles in extremely high density plasma. It is realized by introducing two independent systems in a simulation, where the fast particle generation is simulated by a full PIC system and the transport and energy deposition computed by a second PIC system with a reduced field solver. Data of the fast particles generated in the full PIC system are copied to the reduced PIC system in real time as the fast particle source. Unlike a two-region approach, which takes a single PIC system and two field solvers in two plasma density regions, respectively, the present one need not match the field-solvers since the reduced field solver and the full solver adopted respectively in the two systems are independent. A simulation case is presented, which demonstrates that this approach can be applied to integrated simulation of fast ignition with real target densities.
We construct non-Lipshitz flow in $H^s$ for the cubic nonlinear Schrödinger equation on the 2-torus of revolution with a Lipshitz or smooth metric . The non-Lipshitz property holds for all $s<2/3$ for Lipshitz metric and $s<1/2$ for smooth metric. Both coincide with the Sobolev exponents for uniform local well-posedness.