The Cahn-Hilliard equation is related with a number of interesting physical phenomena like the spinodal decomposition, phase separation and phase ordering dynamics. On the other hand this equation is very stiff an the difficulty to solve it numerically increases with the dimensionality and therefore, there are several published numerical studies in one dimension (1D), dealing with different approaches, and much fewer in two dimensions (2D). In three dimensions (3D) there are very few publications, usually concentrate in some specific result without the details of the used numerical scheme. We present here a stable and fast conservative finite difference scheme to solve the Cahn-Hilliard with two improvements: a splitting potential into a implicit and explicit in time part and a the use of free boundary conditions. We show that gradient stability is achieved in one, two and three dimensions with large time marching steps than normal methods.

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The dual nature of the electronic structure of stripes in La2−xSrxCuO4 was characterized by many experiments. We present here an attempt to characterize this dual behavior based on the Cahn-Hilliard theory of a phase separation transition which is assumed to occur at the high pseudogap temperature. The resulting inhomogeneous low doping system is formed of hole-rich (metallic like) regions embedded in a hole-poor (insulator like). This inhomogeneous configuration is analyzed by a new method within the Bogoliubov-deGennes superconducting theory. This approach describes well the electronic nodal-antinodal dichotomy and parts of the phase diagram.The high critical temperature superconductors (HTSC) represents today one of the greatest challenge of condensed matter physics. It is likely that the main difficulty to understand the physical properties of HTSC is due to the fact that some families seem to have a high inhomogeneous electronic structure while others families appear to be more homogeneous or, at least, without some gross inhomogeneity.An important technique to study the HTSC electronic structure is provided by angle resolved photoemission (ARPES) experiments. With the improvement of the energy and momentum resolution in recent works [1,2,3,4,5], it was possible to distinguish a two component electronic structure in k-space of the La 2−x Sr x CuO 4 (LSCO) family: a metallic quasi particle peak crossing the Fermi level along the zone diagonal following the (0, 0)-(π, π) nodal direction which increases with the doping level [1,5]. On the other hand, the spectral weight at the straight segments in the (π, 0) and (0,π) antinodal regions, is compatible with a quasi one dimensional structure or with static stripes [6]. Due to the d-wave symmetry of the superconducting order parameter, the zero temperature superconducting gap ∆ 0 vanishes at the nodal and is measured at the antinodal directions by the leading edge shift on the Fermi surface by the ARPES spectra. Moreover, the values of ∆ 0 decreases with doping but the quasiparticle spectral weight near the nodal directions, at the Fermi level, increases, showing the distinct behavior of these two aspects of the electronic structure [1,2,3,4].A complementary technique to ARPES is provided by scanning tunneling microscopy (STM) since it probes the differential conductance or ∆ 0 directly on the surface of the compound. Recent STM data have revealed a patchwork of (nanoscale) local spatial variations in the density of states which is used to measure the local superconducting gap [7,8]. With this technique, it was also pos- * evandro@if.uff.br sible to distinguish two distinct behavior: well defined coherent and ill-defined incoherent peaks depending on the spectra location on a Bi 2 Sr2CaCu 2 O 8+δ (Bi2212) surface [9,10,11]. Also, tunneling experiments using superconductor insulator superconductor (SIS) with different insulator layers have shown distinct sets of energy scales and have also led to the idea that the richness of the phase diagram as function of doping ...

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