Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.
A novel irregular-octagonal-prism-shaped host–guest supramolecular network was synthesized and showed excellent photocatalytic splitting of water performance.
A novel strategy for catalytic oxidation of a variety of benzylic, allylic, propargylic, and aliphatic alcohols to the corresponding aldehydes or ketones has been successfully developed.
Monte Carlo simulations of an unconventional phase transition for a two-dimensional dimerized quantum Heisenberg model