On the multistability of spatial solitons in waveguide and optical lattice

Properties of spatial solitons in channel waveguide and optical lattice are studied with the help of projection operator approach. The nonlinearity is assumed to be of cubic-quintic type. The stability consideration of the fixed point solutions of the ODE’s governing the evolution of soliton parameters indicates to the existence of more than one branch of soliton, giving rise to multistability. Explicit numerical analysis gives more information than the standard Vakhitov–Kolokov criterion. A systematic numerical simulation of the soliton profile gives detailed information about the nature of trapping and structure of the different branches of the pulse. It is observed that even under different launching conditions the solitons do not radiate but get trapped.     

A note on trivial nonmultidimensional superstable theories

Research supported by the NSERC of Canada     

Zakharov-Shabat inverse problem with meromorphic reflection coefficient

We analyze the Zakharov-Shabat-type inverse problem when the reflection coefficient contains poles in the eigenvalue plane, as an extension of the earlier work by Atkinson in the case of the Schrödinger problem. It is demonstrated that due to the mutual influence of such a pole and the usual bound-state pole, a discontinuous solitary wave profile is generated. Furthermore, we also examine the form of the nonlinear field only due to the pole of the reflection coefficient. A different approach is necessary to convert the GLM equation into a purely differential one for its solution.     

Bethe ansatz for an open Heisenberg spin chain with impurity

We have set up the algebraic Bethe ansatz equation for an open Heisenberg spin chain having an impurity of a different type of spin. The chain is considered to be open and hence the QISM approach as modified by Sklyanin is used to set up the equations for the Bethe ansatz.     

Separation of Variables for a Lattice Integrable System and the Inverse Problem

We investigate the relation between the local variables of a discrete integrable lattice system and the corresponding separation variables, derived from the associated spectral curve. In particular, we have shown how the inverse transformation from the separation variables to the discrete lattice variables may be factorized as a sequence of canonical transformations.     

The synthesis of some complexed ketones or sulfones by oxidation of η-arene-η-cyclopentadienyliron cations or η-arenedi-η-cyclopentadienyliron dications

Di-η⁵-cyclopentadienyliron complexed dications of 9,10-dihydroanthracene, xanthene, thioxanthene and diphenylmethane derived from ligand exchange reactions but without their prior isolation were oxidized with KMnO₄ to give, respectively, the dications of anthraquinone, xanthone, thioxanthone and benzophenone, isolated as their dihexafluorophosphate salts. Cyclopentadienyliron complexes of arenes containing a sulfide function were oxidized by m-chloroperbenzoic acid to the corresponding complexed sulfones, and sulfones prepared in this way include the hexafluorophosphate salt of the η⁶-p-tolysulfonylbenzene, thioxanthene-10,10-dioxide, 9,9-dimethylthioxanthene-10,10-dioxide or dibenzothiophene-9,9-dioxide-η⁵-cyclopentadienyliron cation.     

Singular integral equations in elastic dielectrics

Galerkin representations for the displacement vector, polarization vector and the potential field are obtained by elementary matrix inversions of the equations of equilibrium. Matrices of fundamental solutions of an infinite elastic dielectric continuum subjected to a concentrated body force, an electric force, and a charge density, are constructed. Theorems are proved on the discontinuity of double layer potentials and R, M, M operators of single layer potentials. By means of these theorems, the solution of the two basic boundary value problems has been reduced to the solution of a system of seven singular integral equations.     

Some problems in plane elastic dielectrics

Representations of Galerkin type are obtained for the displacement vector, polarization vector and the potential fields in the static plane theory of elastic dielectrics using the method of associated matrices. Fundamental matrix solutions of an infinite elastic dielectric plane subjected to a concentrated body force, electric force and charge density are derived from the singular solutions of harmonic, biharmonic and Helmholtz equations. Using boundary operatorsY, Z, M, the fundamental matrix solutions, and Betti's formulae, a matrix (x, y) is constructed and an integral representation for (u ₁,u ₂,P ₁,P ₂, ) is obtained. Discontinuity theorems are stated for the double layer potential andQ operator of the single layer potential. By means of these theorems, the solutions of interior and exterior boundary value problems are reduced to the solution of a system of five singular integral equations. The index of one of the systems is shown to be zero and it is concluded that Fredholm theorems and its alternatives hold.

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Zusammenfassung Durch Anwendung der Assoziativ-Matrizen-Methode werden Galerkinische Darstellungen für den Verschiebungsvektor, den Polarisierungsvektor und die potentiellen Prüffelder einer statischen ebenen Theorie des elastischen Dielektrikums entwickelt. Von den singulären Lösungen der harmonischen, biharmonischen und Helmholtz-Gleichungen werden grundlegende Matrizenlösungen für das unendliche, ebene und elastische Dielektrikum, das durch konzentrierte Raumkräfte, elektrische Kräfte und Ladungsdichte beansprucht wird, abgeleitet. Unter Verwendung der Randwert-OperatorenY, Z, M, der grundlegenden Matrizen-Lösungen und Betti's Formel, wird eine Spezial-Matrize (x, y) konstruiert und eine Integraldarstellung für (u ₁,u ₂,P ₁,P ₂, ) erhalten. Unstetigkeitssätze werden für das Doppelschicht-Potential und denQ-Operator eines einschichtigen Potentials angeführt. Durch Anwendung dieser Sätze werden die Lösungen der inneren und äusseren Grenzwertprobleme zur Lösung eines Systems von fünf singulären Integralgleichungen reduziert. Der Index eines der Systeme wird als Null bewiesen und es wird die Schlussfolgerung gezogen, dass der Fredholmsche Satz und seine Alternativen fur diese Theorie anwendbar sind.

     

Analysis of peptides and protein digests by reversed phase high performance liquid chromatography–electrospray ionisation mass spectrometry using neutral pH elution conditions

In this study, the advantages of carrying out the analysis of peptides and tryptic digests of proteins under gradient elution conditions at pH 6.5 by reversed-phase liquid chromatography (RP-HPLC) and in-line electrospray ionisation mass spectrometry (ESI-MS) are documented. For these RP separations, a double endcapped, bidentate anchored n-octadecyl wide pore silica adsorbent was employed in a capillary column format. Compared to the corresponding analysis of the same peptides and protein tryptic digests using low pH elution conditions for their RP-HPLC separation, this alternative approach provides improved selectivity and more efficient separation of these analytes, thus allowing a more sensitive identification of proteins at different abundance levels, i.e. more tryptic peptides from the same protein could be confidently identified, enabling higher sequence coverage of the protein to be obtained. This approach was further evaluated with very complex tryptic digests derived from a human plasma protein sample using an online two-dimensional (2D) strong cation-exchange (SCX)-RP-HPLC-ESI-MS/MS system. Again, at pH 6.5, with mobile phases of different compositions, improved chromatographic selectivities were obtained, concomitant with more sensitive on-line electrospray ionisation tandem mass spectrometric (ESI-MS/MS) analysis. As a consequence, more plasma proteins could be confidently identified, highlighting the potential of these RP-HPLC methods with elution at pH 6.5 to extend further the scope of proteomic investigations.