nonlocal symmetries related to backlund transformation and their applications starting from nonlocal symmetries related to backlund transformation bt many interesting results can be obtained taking the well known potential kdv pkdv equation as an example a new type of nonlocal symmetry in an elegant and compact form which comes from bt is presented and used to perform research works in two main subjects the nonlocal symmetry is localized by introducing suitable and simple auxiliary dependent variables to generate new solutions from old ones and to consider some novel group invariant solutions some other models both in finite and infinite dimensions are generated under new nonlocal symmetry the finite dimensional models are completely integrable in liouville sense which are shown equivalent to the results given through the nonlinearization method for lax pair

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

explicit solutions from eigenfunction symmetry of the korteweg de vries equation in nonlinear science it is very difficult to find exact interaction solutions among solitons and other kinds of complicated waves such as cnoidal waves and painlevwaves actually even if for the most well known prototypical models such as the kortewet de vries kdv equation and the kadomtsev petviashvili kp equation this kind of problem has not yet been solved in this paper the explicit analytic interaction solutions betweensolitarywavesandcnoidalwavesareobtainedthroughthelocalizationprocedureofnonlocalsymmetries which are related to darboux transformation for the well known kdv equation the same approach also yields some other types of interaction solutions among different types of solutions such as solitary waves rational solutions bessel function solutions and or general painlev e ii solutions

from time series to linear system part iii approximate modelling abstract in the third part of this paper the problem of finding a linear time invariant complete system which models an observed time series will be continued however in this part it will be assumed that the model is required to explain the observations only approximately this raises the question of what is meant by an approximate model and how one should judge a model against data which in a strict sense falsify the model the initial sections of this paper introduce the methodology either the maximal admissible complexity is fixed and the best fitting model in the model class is sought or the maximally tolerated misfit is fixed and the least complex model in the model class is sought this is illustrated by means of finite dimensional models where natural complexity and misfit functions suggest themselves subsequently what is meant by the complexity of a linear time invariant complete system is discussed a concrete complexity function is proposed which is defined in terms of the richness of the behaviour on finite time intervals and is in one to one correspondence with the number of inputs the number of states and the observability indices of the underlying minimal i s o system the next definition is that of the misfit between an observed time series and a system a concrete misfit function is proposed which takes into consideration the lag structure of the system and in how far the observations fail to corroborate the ar equations which define the system s behaviour how the misfit may be computed by means of the correlation function of the data is then shown these concepts are subsequently used in order to set up two algorithms for approximate modelling of an observed q dimensional time series the first algorithm fixes the maximal complexity in a sense this comes down to fixing the number and the lags of the required ar equations the second algorithm fixes the tolerated misfit it is adaptive in nature in the sense that the number and the structure of the resulting ar equations will depend on the observations these ideas and algorithms are illustrated by means of two simulated examples

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

state maps for linear systems modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential algebraic equations as such state space models are not a natural starting point for modeling while they have utmost importance in the simulation and control phase the paper addresses the problem of computing state variables for systems of linear differential algebraic equations of various forms the point of view from which the problem is considered is the behavioral one as put forward in j c willems automatica j ifac 22 1986 pp 561 580 dynamics reported 2 1989 pp 171 269 ieee trans automat control 36 1991 pp 259 294

every storage function is a state function it is shown that for linear dynamical systems with quadratic supply rates a storage function can always be written as a quadratic function of the state of an associated linear dynamical system this dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate

non local symmetries via darboux transformations darboux transformations are used to find non local symmetries of integrable nonlinear models which include the kdv kp cdgks and 2 1 dimensional cdgks equations

on the approximate solutions of fragmentation equations a numerical model based on the finite volume scheme is proposed to approximate the binary breakage problems initially it is considered that the particle fragments are characterized by a single property i e particle s volume we then investigate the extension of the proposed model for solving breakage problems considering two properties of particles the efficiency to estimate the different moments with good accuracy and simple extension for multi variable problems are the key features of the proposed method moreover the mathematical convergence analysis is performed for one dimensional problems all mathematical findings and numerical results are validated over several test problems for numerical validation we propose the extension of bourgade filbet 2008 math comput 77 851 882 doi 10 1090 s0025 5718 07 02054 6 model for solving two dimensional pure breakage problems in this aspect numerical treatment of the two dimensional binary breakage models using finite volume methods can be treated to be the first instance in the literature

behavioral realizations using companion matrices and the smith form classical procedures for the realization of transfer functions are unable to represent uncontrollable behaviors in this paper we use companion matrices and the smith form to derive explicit observable realizations for a general not necessarily controllable linear time invariant behavior we then exploit the properties of companion matrices to efficiently compute trajectories and the solutions to lyapunov equations for the realizations obtained the results are motivated by the important role played by uncontrollable behaviors in the context of physical systems such as passive electrical and mechanical networks

kinetics of fragmentation a general discussion of the kinetics of continuous irreversible fragmentation processes is presented for a linear process where particle breakup is driven by an external force the authors develop a scaling theory to describe the evolution of the cluster size distribution they treat the general case where the breakup rate of a cluster of mass x varies as xlambda when lambda 0 corresponding to larger clusters more likely to break up the scaled cluster size distribution phi x decays with the scaled mass x as x 2 exp axlambda as x for small mass phi x has the log normal form exp a ln2 x if the breakup kernel has a small size cutoff while phi x has a power law tail in the absence of a cutoff they also show that a conventional scaling picture applies only for the case lambda 0 for lambda 0 they develop an alternative formulation for the cluster size distribution in which the typical mass scale is determined by the initial condition in this regime they also investigate the nature of a shattering transition where mass is lost to a dust phase of zero mass particles they also study the kinetics of a nonlinear collision induced fragmentation process they analyse the asymptotic behaviour of a simple minded class of models in which a two particle collision results in either 1 both particles splitting into two equal pieces 2 only the larger particle splitting in two or 3 only the smaller particle splitting they map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations scaling is found to hold for different ranges of homogeneity index for the three models

controllability of linear passive network behaviors abstract classical rlc realization procedures e g bott duffin result in networks with uncontrollable driving point behaviors with this motivation we use the behavioral framework of jan willems to provide a rigorous analysis of rlc networks and passive behaviors we show that the driving point behavior of a general rlc network is stabilizable and controllable if the network contains only two types of elements in contrast we show that the full behavior of the rlc network need not be stabilizable but is marginally stabilizable these results allow us to formalize the phasor approach to rlc networks using the notion of sinusoidal trajectories and to address an assumption of conventional phasor analysis finally we show that any passive behavior with a hybrid representation is stabilizable this paper relies substantially on the fundamental work of our late friend and colleague jan willems to whom the paper is dedicated

integrable systems associated with the schrodinger spectral problem in the plane associated with the schrodinger spectral problem in the plane the nonlinear integrable systems are investigated the commutative properties of equations of the flow and their symmetries and mastersymmetries are derived in spite of the noncanonicality of the related extended recursion operator by applying the inverse scattering transform the systems can be solved and then the correspondences between commuting resp noncommuting flows and isospectral resp nonisospectral deformations of the spectral problem are found

soliton solutions for a negative order non isospectral akns equation abstract in this paper the bilinear form of a negative order non isospectral akns equation is given the soliton solutions are obtained through hirota s direct method and the wronskian technique respectively

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

a theory of passive linear systems with no assumptions we present two linked theorems on passivity the passive behavior theorem parts 1 and 2 part 1 provides necessary and sufficient conditions for a general linear system described by a set of high order differential equations to be passive part 2 extends the positive real lemma to include uncontrollable and unobservable state space systems

proper elimination of latent variables we consider behaviors in which we distinguish two types of variables manifest variables the variables that are of interest to the user and latent variables the variables that are introduced to obtain a first representation the problem is to find a representation of the manifest behavior that is we want to eliminate the latent variables if the original behavior can be represented by linear differential equations with constant coefficients then under certain conditions the same is true for the manifest behavior in this note we formulate and study these conditions the results are illustrated by means of some examples as an application we study behaviors in image representation

negative powers of olver recursion operators olver s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator some negative order kdv equations are derived in this way in particular the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated schrodinger equation used in the inverse scattering solution of the kdv equation and this formulation is used to show that this new equation is the miura transform of a sinh gordon equation

positive real lemma necessary and sufficient conditions for the existence of solutions under virtually no assumptions in this note the celebrated positive real lemma equations are considered with the purpose of obtaining necessary and sufficient conditions for the existence of solutions under the mildest system theoretic assumptions more precisely necessary and sufficient conditions are established under the very weak assumption of sign controllability it is then shown that the order of the equations may be suitably reduced by restricting attention to a subspace related to a certain observable subsystem this reduction beside being interesting per se permits us to weaken the assumption of sign controllability finally the assumptions are further weakened as to include the case when the state matrix has uncontrollable eigenvalues on the imaginary axis

a theoretical explanation of grain size distributions in explosive rock fragmentation we have measured grain size distributions of the results of laboratory decompression explosions of volcanic rock the resulting distributions can be approximately represented by gamma distributions of weight per cent as a function of formula see text where d is the grain size in millimetres measured by sieving with a superimposed long tail associated with the production of fines we provide a description of the observations based on sequential fragmentation theory which we develop for the particular case of self similar fragmentation kernels and we show that the corresponding evolution equation for the distribution can be explicitly solved yielding the long time lognormal distribution associated with kolmogorov s fragmentation theory particular features of the experimental data notably time evolution advection truncation and fines production are described and predicted within the constraints of a generalized reductive fragmentation model and it is shown that the gamma distribution of coarse particles is a natural consequence of an assumed uniform fragmentation kernel we further show that an explicit model for fines production during fracturing can lead to a second gamma distribution and that the sum of the two provides a good fit to the observed data

using symbolic computation to exactly solve for the bogoyavlenskii s generalized breaking soliton equation abstract in this paper the symbolic computation systems help us find some exact solutions for the bogoyavlenskii s generalized breaking soliton equation which are the sums of the arbitrary functions of time and parameterized solitary waves

cellular automaton for migration in ecosystem application of traffic model to a predator prey system in most cases physicists have studied the migration of biospecies by the use of random walk in the present article we apply cellular automaton of traffic model for simplicity we deal with an ecosystem contains a prey and predator and use one dimensional lattice with two layers preys stay on the first layer but predators uni directionally move on the second layer the spatial and temporal evolution is numerically explored it is shown that the migration has the important effect on populations of both prey and predator without migration the phase transition between a prey phase and coexisting phase occurs in contrast the phase transition disappears by migration this is because predator can survive due to migration we find another phase transition for spatial distribution in one phase prey and predator form a stripe pattern of condensation and rarefaction while in the other phase they uniformly distribute the self organized stripe may be similar to the migration patterns in real ecosystems

an observation on the positive real lemma abstract we give a necessary and a sufficient condition that the transfer function of an exponentially stable linear finite dimensional system be a real positive matrix the condition does not assume controllability observability properties

on kalman yakubovich popov lemma for stabilizable systems the kalman yakubovich popov kyp lemma has been a cornerstone in system theory and network analysis and synthesis it relates an analytic property of a square transfer matrix in the frequency domain to a set of algebraic equations involving parameters of a minimal realization in time domain this note proves that the kyp lemma is also valid for realizations which are stabilizable and observable

kac moody virasoro symmetry algebra and symmetry reductions of the bilinear sinh gordon equation in 2 1 dimensions by means of the formal series symmetry approach proposed in 1 infinite many symmetries and the corresponding kac moody virasoro lie symmetry algebra of a new bilinear 2 1 dimensional sinh gordon equation are given then the obtained symmetries are used to get the symmetry reductions of the model from one of the special reduction we know that the bilinear form of the first member of the negative kadomtsev petviashvili hierarchy is not only a 2 1 dimensional sinh gordon extension but also a novel 2 1 dimensional classical boussinesq extension

new family of overturning soliton solutions for a typical breaking soliton equation the breaking soliton equations are of current interest while the application of computer algebra to sciences has a bright future in this paper a new family of overturning soliton solutions for a typical breaking soliton equation is obtained via a computer algebra based method an example of explicit solutions from the family is given solitary waves are also shown to be merely a simple case belonging to the family

dissipativity of uncontrollable systems storage functions and lyapunov functions dissipative systems have played an important role in the analysis and synthesis of dynamical systems the commonly used definition of dissipativity often requires an assumption on the controllability of the system in this paper we use a definition of dissipativity that is slightly different and less often used in the literature to study a linear time invariant possibly uncontrollable dynamical system we provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not mixed i e no pair of uncontrollable poles is symmetric about the imaginary axis this condition is known to be related to the solvability of a lyapunov equation we link lyapunov functions for autonomous systems to storage functions of an uncontrollable system the set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature we show that for an uncontrollable system the set of storage functions is unbounded and that the unboundedness arises precisely due to the set of lyapunov functions for an autonomous linear system being unbounded further we show that stabilizability of a system results in this unbounded set becoming bounded from below positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive in this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions in most of the results in this paper we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity

a 2 1 dimensional extension for the sine gordon equation starting from the breaking soliton equation the author obtains a new integrable equation in 2 1 dimensions though the equation has no exchange symmetry of the space variables x and y the model reduces back to the known 1 1 dimensional sine gordon or liouville equation

on the solvability of the positive real lemma equations abstract in this paper we consider the classical equations of the positive real lemma under the sole assumption that the state matrix a has unmixed spectrum a a without any other system theoretic assumption observability reachability stability etc we derive a necessary and sufficient condition for the solvability of the positive real lemma equations