The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and dym type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of $N$-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.Comment: 31 pages, no figures, to appear in Commun. Math. Phy

Active waveguides in ferroelectric crystals by ion exchange

Erbium-doped LiNbO3 plates were produced by the ion exchange process. Compositional in-depth analysis of the species involved in the process, performed by secondary ion mass spectrometry, showed that the multivalent Er3+-3Li+ ion exchange occurs. Moreover, active properties of Er3+ ions in LiNbO3 were evidenced by photoluminescence spectroscopy. A systematic study on the role of ion exchange parameters was performed showing that erbium incorporation took place under different regimes. The method was demonstrated to be suitable for Er-doping of LiNbO3 crystals, with the possibility of controlling the optical performances of the material through the preparation parameters

Copper-lithium ion exchange in LiNbO3

Copper-doped LiNbO3 waveguides were prepared by Cu\u2013Li ion-exchange process. Compositional, structural, and optical analyses were performed by secondary ion mass spectrometry, x-ray diffraction, and m-line spectroscopy, respectively. The chemical state of Cu2+ ions was studied by electron paramagnetic resonance, and the results were correlated with structural modification of the LiNbO3 matrix. Copper incorporation in the crystal took place under different regimes, and it induced a lattice rearrangement with the formation of new crystalline phases. Cu2+ ions were surrounded by tetragonally compressed octahedra with rhombic distortions. Cu:LiNbO3 optical waveguides were formed supporting two optical modes

Microanalytical study of Er-doped LiNbO3 crystals obtained by Er-Li ion exchange

Erbium-doped lithium niobate (LiNbO3) crystals were produced by Er\ub1Li ion exchange (IE) process. The role of the process parameters such as exchange time and temperature, crystal cu direction,composition and heating rate of the Er ions liquid source was investigated by means of secondary ion mass spectrometry, X-ray diffraction (XRD) and microluminescence techniques. We demonstrated that Er effective diffusion coefficient is two orders of magnitude greater with respect to the thermal diffusion from Er film. We observed that no Li deficient phase can be achieved and that active Er ions are homogeneously incorporated in the substrate. The method is suitable for Er doping of LiNbO3 crystals at relative low temperature (from 570\ub0C to 650\ub0C)

Characterization of alpha-phase soft proton-exchanged LiNbO3 optical waveguides

Waveguides in LiNbO3 are realized by a soft proton exchange (SPE) process with use of a melt of stearic acid highly diluted by lithium stearate. No phase transitions are formed when alpha-phase waveguides are obtained by SPE. The alpha-phase presents the same crystalline structure as that of pure LiNbO3 crystal, and it maintains the excellent nonlinear and electro-optical properties of the bulk material. The kinetics of the SPE method is studied by the use of secondary-ion mass spectrometry and prism-coupling techniques. The hydrogen effec- tive diffusion coefficient as well as the self-diffusion coefficients of H+ and Li+ ions are determined as a func- tion of the proton-exchange temperature for X-cut LiNbO3

Proton-exchanged waveguides in MgO-doped LiNbO3: Optical and structural properties

We show that MgO:HxLi1-xNbO3 single crystalline solid solutions prepared by proton exchange exhibit very complex structural chemistry which is different from that for undoped congruent material. A correlation between the crystal structure and the refractive index profiles has been experimentally determined that allows us to explain specific peculiarities of the optical properties observed, as well as to predict the characteristics of a variety of waveguides. We have established that the well-known annealing proton exchange process leads to crystal structure disorder in MgO:LiNbO3 within the surface region of waveguides fabricated this way, but the soft proton exchange process preserves the crystal structure of magnesium-doped material

Erbium doping of LiNbO3 by the ion exchange process

Erbium-doped LiNbO3 slides are fabricated by the ion exchange process. Compositional in-depth profiles of the species involved in the exchange are obtained by secondary ion mass spectrometry. Spectroscopic properties of Er3+ ions in the matrix are determined by photoluminescence spectroscopy. Structural and spectroscopic analyses suggest the formation of two different microstructures in the exchanged region. The potential of ion exchange for a controlled doping of LiNbO3 is outlined