Exact solutions and dynamics of the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity

This paper investigate the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity by using the method of dynamical systems. The functions $q(x,t)=\phi(\xi)\exp(i(-kx+\omega t))$ are solutions of the equation (1.1) that governs the propagation of Raman solitons through optical metamaterials, where $\xi=x-vt$ and $\phi(\xi)$ in the solutions satisfy a singular planar dynamical system (1.5) which has two singular straight lines. By using the bifurcation theory method of dynamical systems to the equation of $\phi(\xi)$, bifurcations of phase portraits for this dynamical system are obtained under 28 different parameter conditions. Based on those phase portraits, 62 exact solutions of system (1.5) including periodic solutions, heteroclinic and homoclinic solutions, periodic peakons and peakons as well as compacton solutions are derived.     

Besov Functions and Tangent Space to the Integrable Teichm\"uller Space

The authors identify the function space which is the tangent space to the integrable Teichmüller space. By means of quasiconformal deformation and an operator induced by a Zygmund function, several characterizations of this function space are obtained.     

Discerning the degenerate transitions of scalar coupled H NMR spectra: Correlation and resolved techniques at higher quantum

The blend of spin topological filtering and the spin state selective detection of single quantum transitions by the two dimensional multiple quantum-single quantum correlation and higher quantum resolved techniques have been employed for simplifying the complexity of scalar coupled ¹H NMR spectra. The conventional two dimensional COSY and TOCSY experiments, though identify the coupled spin networks, fail to differentiate them due to severe overlap of transitions. Non-selective excitation of homonuclear higher quantum of protons results in filtering of spin systems irrespective of their spin topologies. The spin state selection by passive ¹⁹F spins provides fewer transitions in each cross section of the single quantum dimension simplifying the analyses of the complex spectra. The degenerate single quantum transitions are further discerned by spin selective double and/or triple quantum resolved experiments that mimic simultaneous heteronuclear and selective homonuclear decoupling in the higher quantum dimension. The techniques aided the determination of precise values of spectral parameters and relative signs of the couplings.     

Q.E.COSY: determining sign and size of small deuterium residual quadrupolar couplings using an extended E.COSY principle

Residual quadrupolar couplings contain important structural information comparable with residual dipolar couplings. However, the measurement of sign and size of especially small residual quadrupolar couplings is difficult. Here, we present an extension of the E.COSY principle to spin systems consisting of a Spin 1 coupled to a spin ½ nucleus, which allows the determination of the sign of the quadrupolar coupling of the Spin 1 nucleus relative to the heteronuclear coupling between the spins. The so‐called Q.E.COSY approach is demonstrated with its sign‐sensitivity using variable angle NMR, stretched gels and liquid crystalline phases applied to various CD and CD₃ groups. Especially the sign‐sensitive measurement of residual quadrupolar couplings that remain unresolved in conventional deuterium 1D spectra is shown. Copyright © 2016 John Wiley & Sons, Ltd.     

NMR “Crystallography” for Uniformly (13C, 15N)‐Labeled Oriented Membrane Proteins

In oriented‐sample (OS) solid‐state NMR of membrane proteins, the angular‐dependent dipolar couplings and chemical shifts provide a direct input for structure calculations. However, so far only ¹H–¹⁵N dipolar couplings and ¹⁵N chemical shifts have been routinely assessed in oriented ¹⁵N‐labeled samples. The main obstacle for extending this technique to membrane proteins of arbitrary topology has remained in the lack of additional experimental restraints. We have developed a new experimental triple‐resonance NMR technique, which was applied to uniformly doubly (¹⁵N, ¹³C)‐labeled Pf1 coat protein in magnetically aligned DMPC/DHPC bicelles. The previously inaccessible ¹H_α–¹³C_α dipolar couplings have been measured, which make it possible to determine the torsion angles between the peptide planes without assuming α‐helical structure a priori. The fitting of three angular restraints per peptide plane and filtering by Rosetta scoring functions has yielded a consensus α‐helical transmembrane structure for Pf1 protein.     

On generalized Lax equation of the Lax triple of KP Hierarchy

In terms of the operator Nambu 3-bracket and the Lax pair (L, B_n) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, B_n, B_m). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, B_n, B_m). Furthermore we derive some no integrable evolution equations and present their single soliton solutions.     

NEW INTEGRABLE MULTICOMPONENT NONLINEAR PARTIAL DIFFERENTIAL-DIFFERENCE EQUATIONS

We report a new three and four coupled nonlinear partial differential-difference equations each admits Lax representation, possess infinitely many generalized (nonpoint) symmetries, conserved quantities and a recursion operator. Hence they are completely integrable both in the sense of Lax and Liouville.     

On Neumann eigenfunctions in lip domains

A ``lip domain' is a planar set lying between graphs of two Lipschitz functions with constant 1. We show that the second Neumann eigenvalue is simple in every lip domain except the square. The corresponding eigenfunction attains its maximum and minimum at the boundary points at the extreme left and right. This settles the ``hot spots' conjecture for lip domains as well as two conjectures of Jerison and Nadirashvili. Our techniques are probabilistic in nature and may have independent interest.

     

Temperature dependence of proton–proton residual dipolar couplings in adenosine dissolved in charged phospholipid bicelles. Application of a two‐dimensional selective refocusing method

¹H NMR spectra of nucleoside adenosine were recorded in 7.5% (w/v) DMPC/DHPC bicelles doped with CTAB. Despite the small size of adenosine in comparison with the distances between discoidal bicelles, intramolecular ¹H,¹H residual dipolar couplings D were observed at 29.5–38°C. The absolute values of ¹H,¹H spin interactions Δ (sum of J‐ and D‐couplings) were measured selectively by a series of 2D SERF experiments as a function of temperature. At lower temperatures dipolar couplings are scaled down, which leads to simplification of the ¹H NMR spectra. For some pairs of protons, which are simultaneously J‐ and D‐coupled, the exact values of D can be determined from temperature dependence of Δ. Copyright © 2002 John Wiley & Sons, Ltd.