P. G. Estévez

Darboux transformation and solutions for an equation in 2+1 dimensions

Painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2+1 dimensions. This procedure has allowed us to obtain the Lax pair, Darboux transformation and τ functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively.

Calcified root cells in Miocene pedogenic carbonates of the Madrid Basin: evidence for the origin of Microcodium b

Abstract Calcified root cells, forming microspar and pseudospar mosaics of calcite and/or dolomite crystals, constitute a major component of calcretes and dolocretes from the Miocene of the Madrid Basin. The calcified cells occur in massive nodules or fill root tubes in the calcrete-dolocrete profiles. The arrangement of the cells within the mosaics and their internal features, together with the isotopic data, clearly indicate that the crystals formed through the calcification of root cells and not through recrystallization or dolomitization. Calcified root cells formed in a favourable microenvironment caused by biochemical phenomena associated with plant growth. In these examples, the calcification is incomplete as only in the innermost part of the root the cells were totally calcified, whereas in the root cortex only the cell walls were calcified. The distribution of the calcified cells within the roots was controlled by the different ionic environments which prevail within an active root system. In the inner part the ionic conditions were mostly controlled by the cellular activity of the root creating a suitable microenvironment for the biomineralization of the cells. This differs notably from most published examples of calcified root cells in which it is usual for the cortical cells of roots to be completely calcified. The calcified root cells of the Madrid Basin resemble unequivocally the problematic Microcodium (b), which suggests that this type of Microcodium formed through calcification of root cells.

DARBOUX TRANSFORMATIONS VIA PAINLEVE ANALYSIS

The interesting result obtained in this paper involves using the generalized singular manifold method to determine the Darboux transformations for the equations. It allows us to establish an iterative procedure to obtain multisolitonic solutions. This procedure is closely related to the Hirota -function method. In this paper, we report how to improve the singular manifold method when the equation has more than one Painlev? branch. The singular manifold method generalized in such a way is applied to a pair of equations in 2 + 1 dimensions

A wave equation in 2+1: Painleve analysis and solutions

In this paper the nonlinear equation mty=(myxx+mxmy)x is thoroughly analysed. The Painleve test is performed yielding a positive result. The Backlund transformations are found and the Darboux-Moutard-Matveev formalism arises in the context of this analysis. The singular manifold method, based upon the Painleve analysis, is proved to be a useful tool for generating solutions. Some interesting explicit expressions for one and two solitons are obtained and analysed in such a way.

Non-classical symmetries and the singular manifold method : the Burgers and the Burgers-Huxley equations

In this paper a generalization of the direct method of Clarkson and Kruskal (1989) for finding similarity reductions of partial differential equations is found and discussed for the Burgers and Burgers-Huxley equations. The generalization incorporates the singular manifold method largely based upon the Painleve property. This singular manifold can be used as a reduced variable. Furthermore, a sort of inverse procedure is hereby developed through which we find the equations that yield the vector field components associated to the symmetries of the PDE. This procedure also displays the profound relationship among the symmetries and the singular manifold as a reduced variable. The symmetries found in this way are shown to be those corresponding to the so-called non-classical symmetries by Bluman and Cole (1974), and Olver and Rosenau (1986).

Darboux transformations for a Bogoyavlenskii equation in 2+1 dimensions

Painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2+1 dimensions. This procedure has allowed us to obtain the Lax pair, Darboux transformation and τ functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively.

Painleve analysis of the generalized Burgers-Huxley equation

A complete Painleve test (1900) is applied to the generalized Burgers-Huxley equation using the version of the Painleve analysis recently developed by Weiss, Tabor and Carnevale (1983) for partial nonlinear differential equations. In so doing, the authors are able to find a complete set of new solutions as well as recovering some previous particular solutions already found by using ad hoc methods which have been recently published.

Modified Singular Manifold Expansion: Application to the Boussinesq and Mikhailov-Shabat Systems

We present a modified treatment of the singular manifold method as an improved variant of the Painleve analysis for partial differential equations with two branches in the Painleve expansion. We illustrate the method by fully applying it to the Classical Boussinesq system and to the Mikhailov-Shabat system.

Separation of variables of a generalized porous medium equation with nonlinear source

This paper considers a general form of the porous medium equation with nonlinear source term: ut = (D(u)u n)x + F( u), n � 1. The functional separation of variables of this equation is studied by using the generalized conditional symmetry approach. We obtain a complete list of canonical forms for such equations which admit the functional separable solutions. As a consequence, some exact solutions to the resulting equations are constructed, and their behavior are also investigated.  2002 Elsevier Science (USA). All rights reserved.

A Generalization of the Sine-Gordon Equation to 2+1 Dimensions

Abstract The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions [13] that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painlevé branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a B¨acklund-gauge transformation. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.