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The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell’s equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained. Using a space–time multi-scale asymptotic expansion, a hierarchy of coupled equations describing the spatio-temporal evolution of the perturbed electric and polarization fields are derived. The leading order equation admits a six-wave ansatz satisfying a triad resonance condition. By removing secular terms at next order, a first order in space and time quadratically nonlinear coupled six-wave interaction system is obtained. This resulting system is tied to its integrable counterpart which was postulated by Ablowitz and Haberman in the 1970s. A reduction to a space–time shifted nonlocal three-wave system is presented. The resulting system is solved using the inverse scattering transform, which employs nonlocal symmetries between the associated eigenfunctions and scattering data; soliton solutions are then obtained. Finally, an infinite set of conservation laws for the six-wave system is derived; one is shown to be connected to its Hamiltonian structure. 

We review lattice results related to pion, kaon, $D$ -meson, $B$ -meson, and nucleon physics with the aim of making them easily accessible to the nuclear and particle physics communities. More specifically, we report on the determination of the light-quark masses, the form factor $f_{+}\left(0\right)$ arising in the semileptonic $K\to \pi$ transition at zero momentum transfer, as well as the decay-constant ratio $f_K/f_{\pi }$ and its consequences for the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements $V_{us}$ and $V_{ud}$ . We review the determination of the $B_K$ parameter of neutral kaon mixing as well as the additional four $B$ parameters that arise in theories of physics beyond the Standard Model. For the heavy-quark sector, we provide results for $m_c$ and $m_b$ as well as those for the decay constants, form factors, and mixing parameters of charmed and bottom mesons and baryons. These are the heavy-quark quantities most relevant for the determination of CKM matrix elements and the global CKM unitarity-triangle fit. We review the status of lattice determinations of the strong coupling constant ${\alpha }_s$ . We review the determinations of nucleon charges from the matrix elements of both isovector and flavor-diagonal axial, scalar and tensor local quark bilinears, and momentum fraction, helicity moment and the transversity moment from one-link quark bilinears. We also review determinations of scale-setting quantities. Finally, in this review we have added a new section on the general definition of the low-energy limit of the Standard Model. 

Abstract Supersaturated solid solutions of Al and corrosion-resistant alloying elements (M: V, Mo, Cr, Ti, Nb), produced by non-equilibrium processing techniques, have been reported to exhibit high corrosion resistance and strength. The corrosion mechanism for such improved corrosion performance has not been well understood. We present a fundamental understanding of the role of V in corrosion of an Al-V alloy, which will provide a theoretical background for developing corrosion-resistant Al alloys. High-energy ball milling of the elemental powder of Al and V produced an in situ consolidated Al-V alloy, which exhibited high solid solubility of V. The corrosion resistance of Al-V alloy was significantly higher than that of pure Al, which was attributed to the (1) enrichment of V at the passive film/substrate interface, (2) incorporation of V into the passive film, and (3) deposition of V on the iron-containing cathodic particles and therefore, retardation of cathodic reaction.