Exact Solutions to the Sine-Gordon Equation

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable $x$ and the temporal variable $t,$ and they are exponentially asymptotic to integer multiples of $2\pi$ as $x\to\pm\infty.$ The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation as a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation.Comment: 43 page

3-D crustal structure along the North Anatolian Fault Zone in north-central Anatolia revealed by local earthquake tomography

3-D P-wave velocity structure and Vp/Vs variations in the crust along the North Anatolian Fault Zone (NAFZ) in north-central Anatolia were investigated by the inversion of local P- and S-wave traveltimes, to gain a better understanding of the seismological characteristics of the region. The 3-D local earthquake tomography inversions included 5444 P- and 3200 S-wave readings obtained from 168 well-located earthquakes between 2006 January and 2008 May. Dense ray coverage yields good resolution, particularly in the central part of the study area. The 3-D Vp and Vp/Vs tomographic images reveal clear correlations with both the surface geology and significant tectonic units in the region. We observed the lower limit of the seismogenic zone for north-central Anatolia at 15 km depth. Final earthquake locations display a distributed pattern throughout the study area, with most of the earthquakes occurring on the major splays of the NAFZ, rather than its master strand. We identify three major high-velocity blocks in the mid-crust separated by the Izmir-Ankara-Erzincan Suture and interpret these blocks to be continental basement fragments that were accreted onto the margin following the closure of Neo-Tethyan Ocean. These basement blocks may have in part influenced the rupture propagations of the historical 1939, 1942 and 1943 earthquakes. In addition, large variations in the Vp/Vs ratio in the mid-crust were observed and have been correlated with the varying fluid contents of the existing lithologies and related tectonic structures

Una explicación para el desamargado natural de aceitunas Hurma durante su maduración en el árbol

Harvested olives require further processing to make them edible due to their content in the bitter substance oleuropein. However, some olives of the Erkence cultivar naturally de-bitter on the tree giving rise to the so-called Hurma olives. In this study, the evolution of the chemical characteristics of Erkence and Hurma olives harvested from the northeast and southwest area of trees located in the Karaburun Peninsula was assayed. It was confirmed that the oleuropein content in Hurma olives was much lower (< 2000 mg/kg fresh weight) than Erkence, which reached 35.000 mg/kg fresh weight at the beginning of the season. In addition, no free or polymerized anthocyanins were found in Hurma fruit in contrast to ripened Erkence fruit. The concentration of glucose was also lower in Hurma than Erkence olives. These results suggest that the enzymatic oxidation of oleuropein could be responsible for the natural de-bittering of Hurma olives during their ripening on the tree.Las aceitunas recién cogidas del árbol necesitan ser procesadas para hacerlas comestibles, debido a su contenido en el compuesto amargo oleuropeína. Sin embargo, algunas aceitunas de la variedad Erkence desamargan de forma natural en el árbol dando lugar a las aceitunas conocidas como Hurma. En este trabajo se han analizado las características químicas de aceitunas Erkence y Hurma recolectadas de la zona noreste y suroeste de árboles situados en la provincia de Karaburun. Se ha confirmado que el contenido en oleuropeína de aceitunas Hurma es muy inferior (< 2000 mg/kg) que Erkence, las cuales alcanzaron una concentración en dicha sustancia hasta de 35.000 mg/kg al principio del periodo de maduración. Además, no se encontraron en aceitunas Hurma antocianinas ni libres ni polimerizadas, a diferencia de Erkence. Estos resultados indican que la oxidación enzimática de la oleuropeína podría ser la responsable de la eliminación del amargor de forma natural en aceitunas Hurma durante su maduración en el árbol

Explicit solutions to the Korteweg-de Vries equation on the half line

Certain explicit solutions to the Korteweg-de Vries equation in the first quadrant of the $xt$-plane are presented. Such solutions involve algebraic combinations of truly elementary functions, and their initial values correspond to rational reflection coefficients in the associated Schr\"odinger equation. In the reflectionless case such solutions reduce to pure $N$-soliton solutions. An illustrative example is provided.Comment: 17 pages, no figure

Exact solutions to the focusing nonlinear Schrodinger equation

A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates.Comment: 60 pages, 18 figure

Properties of Solutions in 2+1 Dimensions

We solve the Einstein equations for the 2+1 dimensions with and without scalar fields. We calculate the entropy, Hawking temperature and the emission probabilities for these cases. We also compute the Newman-Penrose coefficients for different solutions and compare them.Comment: 16 pages, 1 figures, PlainTeX, Dedicated to Prof. Yavuz Nutku on his 60th birthday. References adde