hodograph transformations for a camassa holm hierarchy in 2 1 dimensions a generalization of the negative camassa holm hierarchy to 2 1 dimensions is presented under the name chh 2 1 several hodograph transformations are applied in order to transform the hierarchy into a system of coupled cbs calogero bogoyavlenskii schiff equations in 2 1 dimensions that pass the painleve test a non isospectral lax pair for chh 2 1 is obtained through the above mentioned relationship with the cbs spectral problem

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

lax pair and first integrals of the traveling wave reduction for the kdv hierarchy abstract the traveling wave reduction of the korteweg de vries hierarchy is considered the linear system of equations associated with this hierarchy is found the lax pair is used to obtain the first integrals of the traveling wave reduction for the korteweg de vries hierarchy exact formulas for the first integrals of the hierarchy are given the first three members of the hierarchy are considered in more detail these first integrals are the examples of the integrable nonlinear differential equations with the painleve property exact solutions in the form of the soliton for the traveling wave reduction of the korteweg de vries hierarchy and its first integrals are presented

almost invariant sets and invariant manifolds connecting probabilistic and geometric descriptions of coherent structures in flows abstract we study the transport and mixing properties of flows in a variety of settings connecting the classical geometrical approach via invariant manifolds with a probabilistic approach via transfer operators for non divergent fluid like flows we demonstrate that eigenvectors of numerical transfer operators efficiently decompose the domain into invariant regions for dissipative chaotic flows such a decomposition into invariant regions does not exist instead the transfer operator approach detects almost invariant sets we demonstrate numerically that the boundaries of these almost invariant regions are predominantly comprised of segments of co dimension 1 invariant manifolds for a mixing periodically driven fluid like flow we show that while sets bounded by stable and unstable manifolds are almost invariant the transfer operator approach can identify almost invariant sets with smaller mass leakage thus the transport mechanism of lobe dynamics need not correspond to minimal transport the transfer operator approach is purely probabilistic it directly determines those regions that minimally mix with their surroundings the almost invariant regions are identified via eigenvectors of a transfer operator and are ranked by the corresponding eigenvalues in the order of the sets invariance or leakiness while we demonstrate that the almost invariant sets are often bounded by segments of invariant manifolds without such a ranking it is not at all clear which intersections of invariant manifolds form the major barriers to mixing furthermore in some cases invariant manifolds do not bound sets of minimal leakage our transfer operator constructions are very simple and fast to implement they require a sample of short trajectories followed by eigenvector calculations of a sparse matrix

finite time entropy a probabilistic approach for measuring nonlinear stretching abstract transport and mixing processes in dynamical systems are often difficult to study analytically and therefore a variety of numerical methods have been developed finite time lyapunov exponents ftles or related stretching indicators are frequently used as a means to estimate transport barriers alternatively eigenvectors singular vectors or oseledets vectors of numerical transfer operators find almost invariant sets finite time coherent sets or time asymptotic coherent sets respectively which are minimally dispersed under the dynamics while these families of approaches geometric ftles and the probabilistic transfer operator often give compatible results a formal link is still missing here we present a small step towards providing a mathematical link we propose a new entropy based methodology for estimating finite time expansive behaviour along trajectories in autonomous and nonautonomous dynamical systems we introduce the finite time entropy fte field as a simple and flexible way to capture nonlinear stretching directly from the entropy growth experienced by a small localised density evolved by the transfer operator the fte construction elucidates in a straightforward way the connection between the evolution of probability densities and the local stretching experienced we develop an extremely simple and numerically efficient method of constructing an estimate of the fte field the fte field is instantaneously calculable from a numerical transfer operator a transition matrix of conditional probabilities that describes a discretised version of the dynamical system once one has such a transition matrix the fte field may be computed for free we also show i how to avoid long time integrations in autonomous and time periodic systems ii how to perform backward time computations by a fast matrix manipulation rather than backward time integration and iii how to easily employ adaptive methods to focus on high value fte regions

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

computer assisted bounds for the rate of decay of correlations the rate of decay of correlations quantitatively describes the rate at which a chaotic system mixes the state space we present a new rigorous method to estimate a bound for this rate of mixing the technique may be implemented on a computer and is applicable to both multidimensional expanding and hyperbolic systems the bounds produced are significantly less conservative than current rigorous bounds in some situations it is possible to approximate resonant eigenfunctions and to strengthen our bound to an estimate of the decay rate order of convergence results are stated

new classes of solutions for the schwarzian korteweg de vries equation in 2 1 dimensions we have obtained new classes of solutions for the 2 1 dimensional schwarzian korteweg de vries equation by considering several types of reductions of a system equivalent to this equation the first analysis is done by studying the nonclassical reductions of the system further reductions are attained by means of other types of symmetry reductions or by ansatz based reductions most of the new classes of solutions depend on jacobian elliptic functions and solutions of a riemann wave equation including the cnoidal waves solutions the new classes of solutions can display several types of coherent structures and can exhibit the overturning or intertwining phenomena according to the suitable selection of the functions these solutions depend on

the painleve property and backlund transformations for the sequence of boussinesq equations we investigate the sequence of boussinesq equations by the method of singular manifolds for the boussinesq equation which is known to possess the painleve property a backlund transformation is defined this backlund transformation which is formulated in terms of the schwarzian derivative obtains the system of modified boussinesq equations and the resulting miura type transformation the modified boussinesq equations are found to be invariant under a discrete group of symmetries acting on the dependent variables by linearizing the miura transformation and modified equations the lax pair is readily obtained furthermore by a result of fokas and anderson the recursion operators defining the sequence of higher order boussinesq equations may be constructed from the miura transformation this allows the recursive definition of backlund transformations for this sequence of equations the recursion operator is found to preserve the discrete symmetries of the modified boussinesq equations this leads to the conclusion that the sequences of boussinesq and modified boussinesq equations identically possess the painleve property are meromorphic we also find that by a simple reduction the sequences of caudrey dodd gibbon and kuperschmidt equations are contained within the boussinesq sequence rational solutions are iteratively constructed for the boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems

exact solutions of the bogoyavlenskii equation using the multiple g g expansion method in this work exact traveling wave solutions of the bogoyavlenskii equation are studied by the g g expansion method and its variants the solutions obtained are general solutions which are in the form of hyperbolic trigonometric and rational functions and a variety of special solutions like kink shaped antikink shaped bell type soliton solutions etc can easily be derived from the general results under certain domain these methods are more effective and simple than other methods and a number of solutions can be obtained at the same time in order to visualize the underlying dynamics of the obtained solutions two and three dimensional plots are drawn

the singular manifold method revisited for many completely integrable partial differential equations pdes the singular manifold method of weiss allows the recovery of the lax pair and darboux transformation dt and so also the backlund transformation from a truncated painleve expansion recently the so called two singular manifold method has been proposed in order to handle pdes such as the modified korteweg de vries mkdv equation here we present a more natural extension of the weiss singular manifold method which makes use of only one singular manifold but is capable of dealing with such pdes in this approach we allow the possibility that the dt might in fact correspond to an infinite painleve expansion for a certain choice of the arbitrary coefficients this then leads us to a new and more consistent definition of singular manifold equation sme this can give smes different from those usually presented the summation of infinite painleve expansions is effected by seeking a truncation in a new riccati variable z the use o

on matrix painleve hierarchies abstract we define a matrix first painleve hierarchy and a matrix second painleve p ii hierarchy for our matrix p ii hierarchy we also give auto backlund transformations and consider the iteration of solutions this is the first paper to define matrix painleve hierarchies and to give auto backlund transformations for a matrix painleve hierarchy we also consider amongst other results the derivation of sequences of special integrals and autonomous limits until now it has been unknown how to connect the known matrix p ii equation to the obvious candidates for related completely integrable matrix partial differential equations our matrix p ii hierarchy is placed firmly within the context of a matrix modified korteweg de vries mkdv hierarchy in deriving our matrix p ii hierarchy we make use of the hamiltonian structure of this matrix mkdv hierarchy we thus see once again the importance for painleve hierarchies of the integrability structures of related completely integrable equations

integrable 1 1 dimensional hierarchies arising from reduction of a non isospectral problem in 2 1 dimensions this work presents a classical lie point symmetry analysis of a two component non isospectral lax pair of a hierarchy of partial differential equations in 2 1 dimensions which can be considered as a modified version of the camassa holm hierarchy in 2 1 dimensions a classification of reductions for this spectral problem is performed non isospectral reductions in 1 1 dimensions are considered of remarkable interest

prediction intervals for neural networks via nonlinear regression standard methods for computing prediction intervals in nonlinear regression can be effectively applied to neural networks when the number of training points is large simulations show however that these methods can generate unreliable prediction intervals on smaller datasets when the network is trained to convergence stopping the training algorithm prior to convergence to avoid overfitting reduces the effective number of parameters but can lead to prediction intervals that are too wide we present an alternative approach to estimating prediction intervals using weight decay to fit the network and show via a simulation study that this method may be effective in overcoming some of the shortcomings of the other approaches

the two singular manifold method i modified korteweg de vries and sine gordon equations we address the problem of finding by painleve analysis only the backlund transformation of partial differential equations pdes having two families of movable singularities with opposite principal parts such as the modified korteweg de vries mkdv sine gordon or nonlinear schrodinger equations this first paper gives an almost algorithmic method which extends the singular manifold method of weiss 1986 that is unable to handle these equations first with only one singular manifold at a time we obtain the darboux transformation second we assume that the ratio of two functions defining the singular manifolds satisfies the most general projective riccati system with undetermined coefficients the darboux transformation then generates a very small number of determining equations admitting a unique solution equivalent to the lax pair of the zakharov shabat ablowitz kaup newell segur scheme by the canonical linearization of the riccati system the method is here applied to the mkdv and sine gordon equations

absolute flux optimising curves of flows on a surface abstract given a flow on a surface we consider the problem of connecting two distinct trajectories by a curve of extremal absolute instantaneous flux we develop a complete classification of flux optimal curves accounting for the possibility of the flux having spatially and temporally varying weight this weight enables modelling the flux of non equilibrium distributions of tracer particles pollution concentrations or active scalar fields such as vorticity our results are applicable to all smooth autonomous flows area preserving or not

identification of almost invariant aggregates in reversible nearly uncoupled markov chains the topic of the present paper has been motivated by a recent computational approach to identify metastable chemical conformations and patterns of conformational changes within molecular systems after proper discretization such conformations show up as almost invariant aggregates in reversible nearly uncoupled markov chains numcs most of the former work on this subject treated the direct problem given the aggregates analyze the loose coupling in connection with the computation of the stationary distribution aggregation disaggregation techniques in contrast to that the present paper focuses on the inverse problem given the system as a whole identify the almost invariant aggregates together with the small probabilities of transitions between them a robust algorithm is worked out on the basis of some detailed perturbation analysis and illustrated at a simple molecular system

on an extended second painleve hierarchy abstract we present a new extension of the second painleve hierarchy and study its properties in addition to lax pairs backlund transformations auto backlund transformations and basic special integrals we also consider a new phenomenon whereby we obtain relations between systems of different orders but of the same form the extension made here of the second painleve hierarchy is based on the use of non isospectral scattering problems and so is quite general we thus expect to be able to obtain similar extensions of other painleve hierarchies including not only for continuous examples but also for discrete and differential delay examples we believe that our work is also of relevance for painleve classification since it gives information about classes of equation that may be of interest and in addition provides a key to the possible identification of equations isolated in such a process

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

rational solutions for schwarzian integrable hierarchies we give an approach to finding rational solutions of completely intagrable hierarchies which makes use of the relationship between modifications and the schwarzian equations obtained via the singular manifold method this extends the recent work of kudryashov which allowed a simple derivation of the iteration used to construct sequences of such solutions we also give a closed form for the index polynomial of the schwarzian korteweg de vries hierarchy in addition we consider the representation of rational solutions using lower families of the hierarchy we give a simple representation under which the rational solutions remain solutions of every flow of the hierarchy this representation also allows the inclusion of arbitrary data corresponding to negative indices we use our method to derive an alternative form of the backlund transformation for the hierarchy of the second painleve equation as well as new solutions of a hierarchy of breaking soliton equations we also present here for the first time a schwarzian version of this breaking soliton hierarchy

method for deriving rational solutions of some nonlinear evolution equations maps for nonlinear evolution equations are discussed an algorithmic method for deriving rational solutions is presented this approach is illustrated by examples which have solutions in the form of two truncated expansions

discretization of the frobenius perron operator using a sparse haar tensor basis the sparse ulam method the global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated frobenius perron operator for systems with low dimensional long term dynamics efficient techniques exist for a numerical approximation of the most important eigenfunctions cf m dellnitz and o junge siam j numer anal 36 1999 pp 491 515 they are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor ulam s method in this paper we develop a numerical technique which makes ulam s approach applicable to systems with higher dimensional long term dynamics it is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids c zenger sparse grids in parallel algorithms for partial differential equations kiel 1990 vieweg braunschweig 1991 pp 241 251 h j bungartz and m griebel acta numer 13 2004 pp 147 269 here we use a sparse haar tensor basis as the underlying approximation space we develop the technique establish statements about its complexity and convergence and present two numerical examples

the painleve property for partial differential equations ii backlund transformation lax pairs and the schwarzian derivative in this paper we investigate the painleve property for partial differential equations by application to several well known partial differential equations burgers kdv mkdv bousinesq higher order kdv and kp equations it is shown that consideration of the singular manifold leads to a formulation of these equations in terms of the schwarzian derivative this formulation is invariant under the moebius group acting on dependent variables and is shown to obtain the appropriate lax pair linearization for the underlying nonlinear pde

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

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

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

robust perron cluster analysis in conformation dynamics the key to molecular conformation dynamics is the direct identification of metastable conformations which are almost invariant sets of molecular dynamical systems once some reversible markov operator has been discretized a generalized symmetric stochastic matrix arises this matrix can be treated by perron cluster analysis a rather recent method involving a perron cluster eigenproblem the paper presents an improved perron cluster analysis algorithm which is more robust than earlier suggestions numerical examples are included

lyapunov measure for almost everywhere stability this paper is concerned with the analysis and computational methods for verifying global stability of an attractor set of a nonlinear dynamical system based upon a stochastic representation of deterministic dynamics a lyapunov measure is proposed for these purposes this measure is shown to be a stochastic counterpart of stability transience just as an invariant measure is a counterpart of the attractor recurrence it is a dual of the lyapunov function and is useful for the study of more general weaker and set wise notions of stability in addition to the theoretical framework constructive methods for computing approximations to the lyapunov measures are presented these methods are based upon set oriented numerical approaches several equivalent descriptions including a series formula and a system of linear inequalities are provided for computational purposes these descriptions allow one to carry over the intuition from the linear case with stable equilibrium to nonlinear systems with globally stable attractor sets finally in certain cases the exact relationship between lyapunov functions and lyapunov measures is also given

a coherent structure approach for parameter estimation in lagrangian data assimilation abstract we introduce a data assimilation method to estimate model parameters with observations of passive tracers by directly assimilating lagrangian coherent structures our approach differs from the usual lagrangian data assimilation approach where parameters are estimated based on tracer trajectories we employ the approximate bayesian computation abc framework to avoid computing the likelihood function of the coherent structure which is usually unavailable we solve the abc by a sequential monte carlo smc method and use principal component analysis pca to identify the coherent patterns from tracer trajectory data our new method shows remarkably improved results compared to the bootstrap particle filter when the physical model exhibits chaotic advection

detecting and locating near optimal almost invariant sets and cycles the behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict rather than characterizing dynamical behavior at the level of trajectories we consider following the evolution of sets there are often collections of sets that behave in a very predictable way in spite of the fact that individual trajectories are entirely unpredictable such special collections of sets are invisible to studies of long trajectories we describe a global set oriented method to detect and locate these large dynamical structures our approach is a marriage of new ideas in modern dynamical systems theory and the novel application of graph dissection algorithms

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

estimating long term behavior of flows without trajectory integration the infinitesimal generator approach the long term distributions of trajectories of a flow are described by invariant densities i e fixed points of an associated transfer operator in addition global slowly mixing structures such as almost invariant sets which partition phase space into regions that are almost dynamically disconnected can also be identified by certain eigenfunctions of this operator indeed these structures are often hard to obtain by brute force trajectory based analyses in a wide variety of applications transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system the computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories in this paper we propose to directly work with the infinitesimal generator instead of the operator completely avoiding trajectory integration we propose two different discretization schemes a cell based discretization and a spectral coll