Nonlinear patches

Figure 4. Open circles, angular dependence of the second-order scattered light intensity (hyper-Rayleigh scattering from 1064 to 532 nm) for a suspension of purple membrane patches of bacteriorhodopsin before any solubilization has taken place. Solid curve best fit to the data by combining the nonlinear hyperpolarizability of retinal with the linear refractive index of the upo-protein.

Therefore, (nonlinear) photonic crystal properties can be attributed to purple membrane patches of bacteriorhodopsin. This is an important finding, since this natural material exhibits photonic bandgap properties without careful and costly engineering, but through self-assembly. In... 

It is illuminating to study the time evolution of a river plume as an initial value problem. It can be shown that the current pattern is governed by a geostrophically adjusted eddy confined to the buoyancy patch (near field) and a coastally trapped flow that develops in the wake of a Kelvin wave (far field). Behind the front of the first Kelvin wave mode, undercurrents are set up. Although the velocities of the flow forced by the momenrnm of the river mnoff are small enough to justify a linear treatment, there are important nonlinear effects owing to the advection of density, which limits the validity of the linear analytical models. In particular, the structure of the near field in front of the river mouth is dominated by the response to the buoyancy flux associated with the river discharge. 

The method of the normal solution expansion provides a means to obtain the critical patch size if F (0) > 0, the saddle-node bifurcation point if p iff) < 0, and the stability of the steady states. An analytical expression for the steady state can also be obtained, but it is an approximate solution since we truncate the expansion. The normal solution expansion is a well-known method to obtain solutions of the nonlinear Boltzmann equation [180, 359]. It assumes that the distribution function f(jc, v, t), describing the density of atoms or structureless molecules at position x with velocity V at time t depends on time only through the velocity moments /o(jc, t), u(x, t), Pij x, t), i.e., f(jc, V, t) f(jc, V, p(x, t), u(x, t), Pjjix, t)), where p, u, Pij are found by integrating f, fw, and fVjVj, respectively, over the full velocity space. We express the solution of (9.1) in terms of an appropriate complete set of orthogonal spatial basis functions ... 

Terekhova LP, Skoryi lA (1973) Stresses in bonded joints of thin cylindrical shells. Strength Mater 4 1271 Tong L, Sun X (2003) Nonlinear stress analysis for bonded patch to curved thin-walled structures. Int J Adhes Adhes 23 349... 

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