The converse problem for the multipotentialisation of evolution equations and systems

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the {\it converse problem}. Although we mainly study a method for $(1+1)$-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-B\"acklund transformations for the equations/systems. In this respect we define the {\it triangular-auto-B\"acklund transformation} and derive its connections to the converse problem. Several explicit examples are given. In particular we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.Comment: 31 Pages, 7 diagrams, submitted for consideratio

Linearisable third order ordinary differential equations and generalised Sundman transformations

We calculate in detail the conditions which allow the most general third order ordinary differential equation to be linearised in X'''(T)=0 under the transformation X(T)=F(x,t), dT=G(x,t)dt. Further generalisations are considered.Comment: 33 page

On differential equations invariant under two-variable Moebius transformations

We compute invariants for the two-variable M ̈obius transformation. In particular we are interested in partial differential equations in two dependent and two independent variables that are kept invariant under this transformation

A Nonliearly Dispersive Fifth Order Integrable Equation and its Hierarchy

In this paper, we study the properties of a nonlinearly dispersive integrable system of fifth order and its associated hierarchy. We describe a Lax representation for such a system which leads to two infinite series of conserved charges and two hierarchies of equations that share the same conserved charges. We construct two compatible Hamiltonian structures as well as their Casimir functionals. One of the structures has a single Casimir functional while the other has two. This allows us to extend the flows into negative order and clarifies the meaning of two different hierarchies of positive flows. We study the behavior of these systems under a hodograph transformation and show that they are related to the Kaup-Kupershmidt and the Sawada-Kotera equations under appropriate Miura transformations. We also discuss briefly some properties associated with the generalization of second, third and fourth order Lax operators.Comment: 11 pages, LaTex, version to be published in Journal of Nonlinear Mathematical Physics, has expanded discussio

A tree of linearisable second-order evolution equations by generalised hodograph transformations

We present a list of (1+1)-dimensional second-order evolution equations all connected via a proposed generalised hodograph transformation, resulting in a tree of equations transformable to the linear second-order autonomous evolution equation. The list includes autonomous and nonautonomous equations.Comment: arXiv version is already officia

Novel potential interacting partners of fibronectin in spontaneous animal model of interstitial cystitis

Feline idiopathic cystitis (FIC) is the only spontaneous animal model for human interstitial cystitis (IC), as both possess a distinctive chronical and relapsing character. Underlying pathomechanisms of both diseases are not clearly established yet. We recently detected increased urine fibronectin levels in FIC cases. The purpose of this study was to gain further insight into the pathogenesis by assessing interacting partners of fibronectin in urine of FIC affected cats. Several candidate proteins were identified via immunoprecipitation and mass spectrometry. Considerable changes in FIC conditions compared to physiological expression of co-purified proteins were detected by Western blot and immunohistochemistry. Compared to controls, complement C4a and thioredoxin were present in higher levels in urine of FIC patients whereas loss of signal intensity was detected in FIC affected tissue. Galectin-7 was exclusively detected in urine of FIC cats, pointing to an important role of this molecule in FIC pathogenesis. Moderate physiological signal intensity of galectin-7 in transitional epithelium shifted to distinct expression in transitional epithelium under pathophysiological conditions. I-FABP expression was reduced in urine and urinary bladder tissue of FIC cats. Additionally, transduction molecules of thioredoxin, NF-κB p65 and p38 MAPK, were examined. In FIC affected tissue, colocalization of thioredoxin and NF-κB p65 could be demonstrated compared to absent coexpression of thioredoxin and p38 MAPK. These considerable changes in expression level and pattern point to an important role for co-purified proteins of fibronectin and thioredoxin-regulated signal transduction pathways in FIC pathogenesis. These results could provide a promising starting point for novel therapeutic approaches in the future

Robotic control of the seven-degree-of-freedom NASA laboratory telerobotic manipulator

A computationally efficient robotic control scheme for the NASA Laboratory Telerobotic Manipulator (LTM) is presented. This scheme utilizes the redundancy of the seven-degree-of-freedom LTM to avoid joint limits and singularities. An analysis to determine singular configurations is presented. Performance criteria are determined based on the joint limits and singularity analysis. The control scheme is developed in the framework of resolved rate control using the gradient projection method, and it does not require the generalized inverse of the Jacobian. An efficient formulation for determining the joint velocities of the LTM is obtained. This control scheme is well suited for real-time implementation, which is essential if the end-effector trajectory is continuously modified based on sensory feedback. Implementation of this scheme on a Motorola 68020 VME bus-based controller of the LTM is in progress. Simulation results demonstrating the redundancy utilization in the robotic mode are presented

On nonlocal symmetries, nonlocal conservation laws and nonlocal transformations of evolution equations

We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We define higher-degree potential symmetries which then lead to nonlocal conservation laws and nonlocal transformations for the equations. We demonstrate our approach by the Burgers' hierarchy and the Calogero-Degasperis-Ibragimov-Shabat hierarchy

A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also devise a method to derive explicit general solutions of the nonlinear ODEs. Interestingly, many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models.Comment: 12 pages. J. Phys. A: Math. Gen. 39 (2006) in pres