A discrete linearizability test based on multiscale analysis

In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Möbius transformation

Classification of discrete systems on a square lattice

We consider the classification up to a Möbius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A1, A2, and A3 linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice

Invariant differential equations and the Adler-Gel'fand-Dikii bracket

In this paper we find an explicit formula for the most general vector evolution of curves on RP^{n-1} invariant under the projective action of SL(n,R). When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that this evolution is identical to the second KdV Hamiltonian evolution under appropriate conditions. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of SL(n,R), namely, the SL(n,R) invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudo-differential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary n

Multiscale expansion and integrability properties of the lattice potential KdV equation

We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schroedinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007 Conferenc

Point Symmetries of Generalized Toda Field Theories II Applications of the Symmetries

The Lie symmetries of a large class of generalized Toda field theories are studied and used to perform symmetry reduction. Reductions lead to generalized Toda lattices on one hand, to periodic systems on the other. Boundary conditions are introduced to reduce theories on an infinite lattice to those on semi-infinite, or finite ones.Comment: 26 pages, no figure

On the Integrability of the Discrete Nonlinear Schroedinger Equation

In this letter we present an analytic evidence of the non-integrability of the discrete nonlinear Schroedinger equation, a well-known discrete evolution equation which has been obtained in various contexts of physics and biology. We use a reductive perturbation technique to show an obstruction to its integrability.Comment: 4 pages, accepted in EP

Lie point symmetries of difference equations and lattices

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations

Usefulness of Discriminant Analysis in the Morphometric Differentiation of Six Native Freshwater Species from Ecuador

The aim of this research was to find out the morphometric differentiation of six native freshwater species in the Guayas Hydrographic Basin (Ecuador) by means of discriminant analysis. A total of 1355 mature fishes (Cichlasoma festae, Andinoacara rivulatus, Dormitator latifrons, Bryncon dentex, Hoplias microlepis and Leporinus ecuadorensis) were captured and 27 morphometric measurements and 20 landmarks were used. Two-way analysis of variance with species and sex as fixed factors and discriminant analysis were applied. The selection of the most discriminant variables was made applying the F of Snedecor, Wilks’-Lambda and the 1-Tolerance. While sex within species had no significant effect on the morphology, differences among species were significant. Twenty-seven morphological variables showed highly significant differences among six native freshwater species. Nine biometric variables with high discriminant power were selected. The six species analyzed were discriminated by the morphometric models generated, thus showing that discriminant analysis was useful for differentiating species. The morphometric differentiation by discriminant analysis is a direct, simple and economic methodology to be applied in situ in rural communities. It favors the implementation of a livestock development program and it could be used with other native freshwater species in the Guayas Hydrographic Basin

Morphological Variations of Wild Populations of Brycon dentex (Characidae, Teleostei) in the Guayas Hydrographic Basin (Ecuador). The Impact of Fishing Policies and Environmental Conditions

The Guayas, located in Ecuador, is the largest basin in the Pacific Ocean and has an inventory of 123 native freshwater species. Most of these are endemic species that are threatened or at-risk due to anthropogenic activity and the modification, fragmentation, and destruction of habitats. The aim of this study was to determine the morphometric variation in three wild populations of Brycon dentex in the Guayas basin rivers and their connections to fishing management and environmental conditions. A total of 200 mature fish were captured, and 26 morphometric parameters were measured. The fishing policies (Hypothesis 1) and environmental conditions (Hypothesis 2) were considered fixed factors and were validated by t-tests. The morphological variation among the three populations (Hypothesis 3) was validated through a discriminant analysis. Fishing policies and resource management were found to generate morphological differences associated with body development. In addition, the environmental conditions were found to influence the size and structure of Brycon dentex populations. The analyzed populations were discriminated by the generated morphometric models, which differentiated Cluster 1 (Quevedo and Mocache rivers) with high fishing pressure from Cluster 2 (Pintado river) with medium–low fishing pressure. Morphometric differentiation by discriminant analysis is a direct and economic methodology that can be applied as an indicator of diversity maintenance