NLS-MKdV Hierarchy and Its Hamiltonian Structures

A new simple loop algebra is constructed, which is devote to establishing an isospectral problem. By making use of Tu scheme, NLS-MKdV hierarchy is obtained. Again via expanding the loop algebra above, another higher-dimensional loop algebra is presented. It follows that an integrable coupling of NLS-MkdV hierarchy is given. Also, the trace identity is extended to the quadratic-form identity and the Hamiltonian structures of the NLS-MKdV hierarchy and integrable coupling of NLS-MkdV hierarchy are obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.     

Schonland ambiguity in the electron nuclear double resonance analysis of hyperfine interactions: principles and practice

For the analysis of the angular dependence of electron paramagnetic resonance (EPR) spectra of low-symmetry centres with S=1/2 in three independent planes, it is well-established-but often overlooked-that an ambiguity may arise in the best-fit g<--> tensor result. We investigate here whether a corresponding ambiguity also arises when determining the hyperfine coupling (HFC) A<--> tensor for nuclei with I=1/2 from angular dependent electron nuclear double resonance (ENDOR) measurements. It is shown via a perturbation treatment that for each set of M(S) ENDOR branches two best-fit A<--> tensors can be derived, but in general only one unique solution simultaneously fits both. The ambiguity thus only arises when experimental data of only one M(S) multiplet are used in analysis or in certain limiting cases. It is important to realise that the ambiguity occurs in the ENDOR frequencies and therefore the other best-fit result for an ENDOR determined A<--> tensor depends on various details of the ENDOR experiment: the M(S) state of the fitted transitions, the microwave frequency (or static magnetic field) in the ENDOR measurements and the rotation planes in which data have been collected. The results are of particular importance in the identification of radicals based on comparison of theoretical predictions of HFCs with published literature data. A procedure for obtaining the other best-fit result for an ENDOR determined A<--> tensor is outlined.     

N-fold Darboux Transformation for Integrable Couplings of AKNS Equations

For the integrable couplings of Ablowitz-Kaup-Newell-Segur(ICAKNS) equations, N-fold Darboux transformation(DT) TN, which is a 4 × 4 matrix, is constructed in this paper. Each element of this matrix is expressed by a ratio of the(4N + 1)-order determinant and 4N-order determinant of eigenfunctions. By making use of these formulae,the determinant expressions of N-transformed new solutions p~([N ]), q~([N ]), r~([N ])and s~([N ])are generated by this N-fold DT.Furthermore, when the reduced conditions q =-p*and s =-r*are chosen, we obtain determinant representations of N-fold DT and N-transformed solutions for the integrable couplings of nonlinear Schr?dinger(ICNLS) equations.Starting from the zero seed solutions, one-soliton solutions are explicitly given as an example.     

超广义Burgers方程族的非线性可积耦合及其Bargmann对称约束

基于李超代数,构造了超广义Burgers方程族的非线性可积耦合,并且利用超级恒等式得到了它的超Hamilton结构.此外,该文计算出超广义Burgers方程族的非线性可积耦合的Bargmann对称约束.     

Influence of the Thiophene Ring on the Molecular Order of Structurally Simple π-Conjugated Smectogens: 13C NMR Study

Structurally simple rod-like π-conjugated mesogens with thiophene directly connected to phenyl, biphenyl, and fluorenone rings with terminal chains are synthesized respectively. The occurrence of smectic A/smectic C phases is concurred by a hot-stage optical polarising microscope (HOPM), differential scanning calorimetry (DSC), and X-ray diffraction (XRD). The static 1D and 2D ¹³C nuclear magnetic resonance (NMR) studies in the liquid crystalline phase are carried out to find the alignment-induced chemical shifts (AIS) and ¹³C−¹H dipolar couplings. The orientational order parameters of the mesogens determined from ¹³C−¹H dipolar couplings disclose that the long axis is not only collinear to the C3−C4 bond of the thiophene ring but also for the local axes of phenyl and biphenyl rings. For fluorenone-based mesogen, the molecular biaxiality is found to be high owing to the increased breadth of the molecule. The study unveils that the orientation of thiophene and the phenyl rings is similar in the current mesogens in stark contrast to mesogens, where thiophene is connected to phenyl rings through linking groups.