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

on a dimensional reduction method i the optimal selection of basis functions abstract this paper is the first in a series of three which analyze an adaptive approximate approach for solving n 1 dimensional boundary value problems by replacing them with systems of equations in n dimensional space in this approach the unknown functions of n 1 variables are projected onto finite linear combinations of functions of just n variables this paper shows how the coefficients of these linear combinations can be chosen optimally author

skeletal reduction of boundary value problems boundary value problems posed over thin solids are amenable to a dimensional reduction in that one or more spatial variables may be eliminated from the governing equation resulting in significant computational gains with minimal loss in accuracy extant dimensional reduction techniques rely on representing the solid as a hypothetical mid surface plus a possibly varying thickness such techniques are however hard to automate since the mid surface is often ill defined and ambiguous we propose here a skeletal representation based dimensional reduction of boundary value problems the proposed technique has the computational advantages of mid surface reduction but can be easily automated a systematic methodology is presented for reducing boundary value problems to lower dimensional problems over the skeleton of a solid the theoretical properties of the proposed method are derived and supported by representative numerical experiments

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

incidence of nonextensive thermodynamics in temporal scaling at feigenbaum points abstract recently in phys rev lett 95 2005 140601 grassberger addresses the interesting issue of the applicability of q statistics to the renowned feigenbaum attractor he concludes there is no genuine connection between the dynamics at the critical attractor and the generalized statistics and argues against its usefulness and correctness yet several points are not in line with our current knowledge nor are his interpretations we refer here only to the dynamics on the attractor to point out that a correct reading of recent developments invalidates his basic claim

applying the hp d version of the fem to locally enhance dimensionally reduced models abstract a modification of the hp d method is presented which allows to locally enhance a dimensionally reduced finite element approximation with an ansatz based on a more accurate i e dimensionally higher model the procedure is based on a hierarchical domain decomposition which is applied to couple discretizations of different types of mechanical models like beams plates and solids thereby the error of the underlying mathematical model can be reduced in critical regions by introducing locally a physically more sophisticated model two examples are given to demonstrate the basic properties of the proposed method in the first example the solution of a one dimensional beam model is improved locally by a two dimensional finite element computation in the second example the local enhancement of a reissner mindlin plate by a fully three dimensional continuum approach is considered yielding a significant reduction of the model error

local error estimation for dimensionally reduced models of elliptic boundary value problems in this paper we present an approach to the local estimation of errors in dimensionally reduced models of elliptic boundary value problems posed on thin domains in this approach errors are measured in quantities of interest whose computation is typically the goal of the analysis explicit bounds and estimates are derived for errors in quantities characterized by continuous linear functionals a representative numerical example showing the accuracy of the bounds and estimates is also presented

a note on the g g expansion method abstract we demonstrate that the g g expansion method which is often used in finding exact solutions of nonlinear differential equation is equivalent to the well known tanh method and application of these methods gives the same exact solutions of nonlinear differential equations

approximation of skewed interfaces with tensor based model reduction procedures in this article we introduce a procedure which allows to recover the potentially very good approximation properties of tensor based model reduction procedures for the solution of partial differential equations in the presence of interfaces or strong gradients in the solution which are skewed with respect to the coordinate axes the two key ideas are the location of the interface either by solving a lower dimensional partial differential equation or by using data functions and the subsequent removal of the interface of the solution by choosing the determined interface as the lifting function of the dirichlet boundary conditions we demonstrate in numerical experiments for linear elliptic equations and the reduced basis hierarchical model reduction approach that the proposed procedure locates the interface well and yields a significantly improved convergence behavior even in the case when we only consider an approximation of the interface

polynomials in logistic function and solitary waves of nonlinear differential equations properties of polynomials in logistic function are studied it is demonstrated that these polynomials can be used for construction of exact solutions to nonlinear differential equations nonlinear differential equations with exact solutions in the form of polynomials in logistic function are found it is shown there are solitary waves of nonlinear differential equations described by polynomial in logistic function with many maximum and minimum

tsallis q index and mori s q phase transitions at the edge of chaos we uncover the basis for the validity of the tsallis statistics at the onset of chaos in logistic maps the dynamics within the critical attractor is found to consist of an infinite family of mori s q phase transitions of rapidly decreasing strength each associated with a discontinuity in feigenbaum s trajectory scaling function sigma the value of q at each transition corresponds to the same special value for the entropic index q such that the resultant sets of q lyapunov coefficients are equal to the tsallis rates of entropy evolution

application of an automatic hp adaptive finite element method for thin walled structures the paper investigates the possibility of solving beam and thin walled structures with an automatic hp adaptive finite element method for three dimensional linear elasticity this technique generates sequences of meshes with nonuniform distribution of element sizes h and their spectral orders p to minimize the interpolation error for the hypothetical exact solution with constant number of degrees of freedom the intent of using hp method for beam and thin walled structures is that it should automatically create fe meshes with adequate distribution of the approximation orders to prevent the numerical locking phenomenon and to provide accurate solutions the method also reduces the human work spent on building the numerical model to generation of the description of the geometry we discuss energy based and goal oriented versions of hp adaptivity the paper presents a broad spectrum of examples of engineering applications

temporal scaling at feigenbaum points and nonextensive thermodynamics we show that recent claims for the nonstationary behavior of the logistic map at the feigenbaum point based on nonextensive thermodynamics are either incorrect or can be easily deduced from well known properties of the feigenbaum attractor in particular there is no generalized pesin identity for this system the existing attempts at proofs being based on misconceptions about basic notions of ergodic theory in deriving several new scaling laws of the feigenbaum attractor thorough use is made of its detailed structure but there is no obvious connection to nonextensive thermodynamics

criticality in nonlinear one dimensional maps rg universal map and nonextensive entropy abstract we consider the period doubling and intermittency transitions in iterated nonlinear one dimensional maps to corroborate unambiguously the validity of tsallis nonextensive statistics at these critical points we study the map x n 1 x n u x n z z 1 as it describes generically the neighborhood of all of these transitions the exact renormalization group rg fixed point map and perturbation static expressions match the corresponding expressions for the dynamics of iterates the time evolution is universal in the rg sense and the nonextensive entropy s q associated to the fixed point map is maximum with respect to that of the other maps in its basin of attraction the degree of nonextensivity the index q in s q and the degree of nonlinearity z are equivalent and the generalized lyapunov exponent q q 2 q 1 is the leading map expansion coefficient u the corresponding deterministic diffusion problem is similarly interpreted we discuss our results

on a dimensional reduction method ii some approximation theoretic results abstract this paper is the second in a series of three that analyze a method of dimensional reduction it contains some results for approximation of functions on the interval 1 1 with elements from the nullspace of p to the n power n greater than or equal to 1 where p is a second order ordinary differential operator a special case of this is approximation by polynomials the one dimensional results are used as a tool to prove similar versions in several dimensions these multi dimensional results are directly related to the approximate method of dimensional reduction and they lead to statements about the convergence properties of this approach the third paper which analyzes the adaptive aspects of the method is forthcoming

exact solutions to the two dimensional korteweg de vries burgers equation two methods are described for obtaining exact travelling monotone shock wave solutions to the two dimensional korteweg de vries burgers equation they are compared with two methods that have been reported recently by other authors

exact solutions of some systems of nonlinear partial differential equations using symbolic computation abstract in this paper we present a solution method that utilizes symbolic computation to obtain exact traveling wave solutions of some systems of nonlinear partial differential equations the solution method is demonstrated by obtaining solutions to the variant shallow water wave equations

exact solutions of one pattern formation model abstract the nonlinear evolution equation for describing the pattern formation processes on the semiconductor surfaces under ion beam bombardment is studied the painleve analysis of equation is considered the new elliptic solution of this equation is obtained

exact travelling wave solutions of nonlinear evolution equation of surface waves in a convecting fluid exact travelling wave solutions in terms of the weierstrass p function of the equation eta t a0 eta x a1 eta eta x a2 eta xxx b0 eta xx b 1 eta eta x x b2 eta xxxx 0 are obtained these solutions may describe particularly both the evolution of a bounded conoidal wave and kink shaped waves

on a dimensional reduction method iii a posteriori error estimation and an adaptive approach abstract this paper analyzes an adaptive approximate approach for solving n 1 dimensional boundary value problems by replacing them with systems of equation in n dimensional space the discussion shows how to find reliable a posteriori estimates for the error and how these can also be used in the design of an adaptive strategy various numerical examples are presented

analytical properties and exact solutions of the lotka volterra competition system the lotka volterra competition system with diffusion is considered the painleve property of this system is investigated exact traveling wave solutions of the lotka volterra competition system are found periodic solutions expressed in terms of the weierstrass elliptic function are also given

pq adaptive solid finite elements for three dimensional plates and shells this article presents a pq adaptive finite element procedure for three dimensional computation of thin walled structures like plates and shells the approach is based on a hexahedral element with high order shape functions the polynomial degree of the shape functions in the three spatial directions of the element is adjusted by means of an adaptive procedure to this end a hierarchical error indicator is applied to automatically control the polynomial degree q in thickness and p in the in plane direction of plate and shell like structures the efficiency and reliability of the proposed adaptive method is demonstrated using two numerical examples

equivalence between the mobility edge of electronic transport on disorderless networks and the onset of chaos via intermittency in deterministic maps we exhibit a remarkable equivalence between the dynamics of an intermittent nonlinear map and the electronic transport properties obtained via the scattering matrix of a crystal defined on a double cayley tree this strict analogy reveals in detail the nature of the mobility edge normally studied near not at the metal insulator transition in electronic systems we provide an analytical expression for the conductance as a function of the system size that at the transition obeys a q exponential form this manifests as power law decay or few and far between large spike oscillations according to different kinds of boundary conditions

exact periodic solutions of the complex ginzburg landau equation three new exact periodic solutions of the complex ginzburg landau equation are obtained in terms of the weierstrass elliptic function furthermore the new periodic solutions and other shock solutions appear as their bounded limits along the real axis for particular relationships between the coefficients in the equation in the general case bounded limits are nothing but the already known pulse hole and shock solutions it is also shown that the shapes of the solutions are quite different from the shape of the usual envelope wave solution in particular the spatial structure of the new bounded periodic solutions varies in time while the pulse solution may exhibit breather like behavior

modified method of simplest equation and its applications to the bogoyavlenskii equation in this paper we construct the exact traveling wave solutions of the bogoyavlenskii equation using modified method of simplest equation the simplest equation herewith is a special riccati equation in the meantime the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields

the renormalization group and optimization of non extensive entropy criticality in non linear one dimensional maps abstract we examine the pitchfork and tangent bifurcations in unimodal maps to illustrate a connection between renormalization group rg fixed points and entropy extremal properties we observe that the exact rg solution for the tangent bifurcation is also applicable to the period doubling cascade and assess its physical meaning since the expression for the fixed point map can be put into the form of the non extensive expressions for the temporal evolution of phase space volume and sensitivity of initial conditions we conclude that the map critical points possess the properties of this formalism the universality of the rg solution makes this interpretation inclusive to all one dimensional maps of non linearity z 1

generalized tanh method with symbolic computation and generalized shallow water wave equation abstract we report that symbolic computation with the generalized tanh method leads to new soliton like solutions for a 2 1 dimensional generalization of the shallow water wave equations

closer look at time averages of the logistic map at the edge of chaos the probability distribution of sums of iterates of the logistic map at the edge of chaos has been recently shown see u tirnakli c beck and c tsallis phys rev e 75 040106 r 2007 to be numerically consistent with a q gaussian the distribution which under appropriate constraints maximizes the nonadditive entropy sq the basis of nonextensive statistical mechanics this analysis was based on a study of the tails of the distribution we now check the entire distribution in particular its central part this is important in view of a recent q generalization of the central limit theorem which states that for certain classes of strongly correlated random variables the rescaled sum approaches a q gaussian limit distribution we numerically investigate for the logistic map with a parameter in a small vicinity of the critical point under which conditions there is convergence to a q gaussian both in the central region and in the tail region and find a scaling law involving the feigenbaum constant our results are consistent with a large number of already available analytical and numerical evidences that the edge of chaos is well described in terms of the entropy sq and its associated concepts

exact solutions for the nonlinear schrodinger equation with variable coefficients using the generalized extended tanh function the sine cosine and the exp function methods abstract in this article we find the exact traveling wave solutions of the generalized nonlinear schrodinger gnls equation with variable coefficients using three methods via the generalized extended tanh function method the sine cosine method and the exp function method the main objective of this article is to compare the efficiency of these methods by delivering the exact traveling wave solutions of the proposed nonlinear equation

hierarchical models of thin elastic structures overview and recent advances in error estimation and adaptivity abstract in the numerical approximation of hierarchical models of thin bodies there are two separate contributions to the total error a modeling error is the error due to dimensional reduction and b discretization error is the error due to the numerical approximation traditionally estimation of these errors has been performed in terms of abstract quantities like the global energy norm which does not give a clear representation of the error in the local features of the solution recently a new class of finite element error estimation techniques has emerged wherein errors in a simulation are measured not in norms but in quantities of interest to the analyst following a brief overview of the literature we focus on extending the theory of a posteriori estimation of the modeling error to local quantities of interest that have physical meaning independent estimates are obtained for the modeling and discretization errors in quantities of interest these local estimates are used to develop goal oriented adaptive modeling strategies in which the model is automatically adapted by controlling the two errors independently and effectively to obtain local quantities of interest to within a preset level of accuracy numerical examples are presented to demonstrate the accuracy of the error estimates and the adaptive strategy

adaptive hpq finite element methods of hierarchical models for plate and shell like structures abstract a four step adaptive strategy for hpq finite element approximations of hierarchical models of the thin plate and shell like structures is derived based on a priori and a posteriori error estimations this strategy controls three parameters the model level q the mesh size h and the approximation order p so as to produce quasi optimal hierarchical models and finite element meshes numerical results are given supporting the theoretical results

a posteriori estimation of dimension reduction errors for elliptic problems on thin domains a new a posteriori error estimator is presented for the verification of the dimensionally reduced models stemming from the elliptic problems on thin domains the original problem is considered in a general setting without any specific assumptions on the domain geometry coefficients and the right hand sides for the energy norm of the error of the zero order dimension reduction method the proposed estimator is shown to always provide a guaranteed upper bound in the case when the original domain has constant thickness but possibly nonplane upper and lower faces the estimator demonstrates the optimal convergence rate as the thickness tends to zero it is also flexible enough to successfully cope with infinitely growing right hand sides in the equation when the domain thickness tends to zero the numerical tests indicate the efficiency of the estimator and its ability to accurately represent the local error distribution needed for an adaptive improvement of the reduced model

a new korn s type inequality for thin domains and its application to iterative methods we present a new korn s type inequality which estimates the ratio of the energy norm to the sobolev norm in a given subspace in terms of the angle it forms with an explicitly extracted finite dimensional subspace this inequality provides crucial information for improving the convergence of various iterative algorithms for elasticity problems in thin domains this is demonstrated in the case of the semi discrete iterative algorithm edra see e e ovtchinnikov and l s xanthis effective dimensional reduction for elliptic problems c r acad sci paris series i 320 879 884 1995 we show both theoretically and numerically that the convergence of the modified edra is independent of the thickness of the domain and of the semi discretisation parameters

application of the kudryashov method for finding exact solutions of the high order nonlinear evolution equations abstract the application of the kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth sixth and seventh order are obtained the efficiency of the kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

extended simplest equation method for nonlinear differential equations abstract the modified simplest equation method to look for the exact solutions of the nonlinear differential equations is presented our approach is applied to search for the exact solutions of the sharma tasso olver and the burgers huxley equations the new exact solutions of these equations are obtained

q deformed statistical mechanical property in the dynamics of trajectories en route to the feigenbaum attractor we demonstrate that the dynamics toward and within the feigenbaum attractor combine to form a q deformed statistical mechanical construction the rate at which ensemble trajectories converge to the attractor and to the repellor is described by a q entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor the values of the q indices involved are given by the unimodal map universal constants while the thermodynamic structure is closely related to that formerly developed for multifractals as an essential component in our demonstration we expose in great detail the features of the dynamics of trajectories that either evolve toward the feigenbaum attractor or are captured by its matching repellor the dynamical properties of the family of periodic superstable cycles in unimodal maps are seen to be key ingredients for the comprehension of the discrete scale invariance features present at the period doubling transition to chaos elements in our analysis are the following i the preimages of the attractor and repellor of each of the supercycles appear entrenched into a fractal hierarchical structure of increasing complexity as period doubling develops ii the limiting form of this rank structure results in an infinite number of families of well defined phase space gaps in the positions of the feigenbaum attractor or of its repellor iii the gaps in each of these families can be ordered with decreasing width in accordance with power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions iv the power law with log periodic modulation associated with the rate of approach of trajectories toward the attractor and to the repellor is explained in terms of the progression of gap formation v the relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated vi a mean field evaluation of the atypical partition function a thermodynamic interpretation of the time evolution process and a crossover to ordinary exponential statistics are given we make clear the dynamical origin of the anomalous thermodynamic framework existing at the feigenbaum attractor

an algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations an algebraic method is devised to uniformly construct a series of exact solutions for general integrable and nonintegrable nonlinear evolution equations compared with most existing tanh methods the jacobi function expansion method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the solutions according to the values of some parameters the solutions obtained in this paper include a polynomial solutions b exponential solutions c rational solutions d triangular periodic wave solutions e hyperbolic and solitary wave solutions and f jacobi and weierstrass doubly periodic wave solutions the efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper new 2 1 dimensional calogero kdv equation 3 1 dimensional jimbo miwa equation symmetric regular long wave equation drinfel d sokolov wilson equation 2 1 dimensional generalized dispersive long wave equation double sine gordon equation calogero degasperis fokas equation and coupled schrodinger boussinesq equation in addition the links among our proposed method the tanh method the extended method and the jacobi function expansion method are also clarified generally

explicit exact solutions of some nonlinear evolution equations with their geometric interpretations in this paper the simplest equation method is applied to obtain multiple explicit exact solutions of the combined dispersion equation the hirota satsuma korteweg de vries system and the generalized burgers huxley equation these solutions are constructed on the basis of solutions of bernoulli equation which is used as simplest equation it is shown that this method is very powerful tool for obtaining exact solutions of a large class of nonlinear partial differential equations the geometric interpretation for some of these solutions are introduced

a dimensional reduction approach based on the application of reduced basis methods in the framework of hierarchical model reduction in this article we introduce a new dimensional reduction approach which is based on the application of reduced basis rb techniques in the hierarchical model reduction hmr frame work considering problems that exhibit a dominant spatial direction the idea of hmr is to perform a galerkin projection onto a reduced space which combines the full solution space in the dominant direction with a reduction space in the transverse direction the latter is spanned by modal or thonormal basis functions while so far the basis functions in the hmr approach have been chosen a priori s perotto a ern and a veneziani multiscale model simul 8 2010 pp 1102 1127 for instance as legendre or trigonometric polynomials in this work a highly nonlinear approximation is employed for the construction of the reduction space to this end we first derive a lower dimen sional parametrized problem in the transverse direction from the full problem where the parameters reflect the influence from the unknown solution in the dominant direction exploiting the good ap proximation properties of rb methods we then construct a reduction space by applying a proper orthogonal decomposition to a set of snapshots of the parametrized partial differential equation for an efficient construction of the snapshot set we apply adaptive refinement in parameter space based on an a posteriori error estimate that is also derived in this article we introduce our method for general elliptic problems such as advection diffusion equations in two space dimensions numerical experiments demonstrate a fast convergence of the proposed dimensionally reduced approximation to the solution of the full dimensional problem and the computational efficiency of our new adaptive approach

modified simple equation method for nonlinear evolution equations abstract this paper reflects the implementation of a reliable technique which is called modified simple equation method msem for solving evolution equations the proposed algorithm has been successfully tested on two very important evolution equations namely fitzhugh nagumo equation and sharma tasso olver equation numerical results are very encouraging

existence of solitary solutions in a class of nonlinear differential equations with polynomial nonlinearity the inverse balancing method for the determination of the necessary conditions of existence of solitary solutions to mth order differential equations with nth order polynomial nonlinearity is presented in this paper it is shown that the order of possible solitary solutions does not increase if orders of the differential equation and the polynomial nonlinearity increase furthermore the relationships between the order of the solitary solution and the order of the equation and the nonlinearity are given in the explicit form

weierstrass traveling wave solutions for dissipative benjamin bona and mahony bbm equation in this paper the effect of a small dissipation on waves is included to find exact solutions to the modified benjamin bona and mahony bbm equation by viscosity using lyapunov functions and dynamical systems theory we prove that when viscosity is added to the bbm equation in certain regions there still exist bounded traveling wave solutions in the form of solitary waves periodic and elliptic functions by using the canonical form of abel equation the polynomial appell invariant makes the equation integrable in terms of weierstrass functions we will use a general formalism based on ince s transformation to write the general solution of dissipative bbm in terms of functions from which all the other known solutions can be obtained via simplifying assumptions using ode ordinary differential equations analysis we show that the traveling wave speed is a bifurcation parameter that makes transition between different classes of waves

new solitary wave solutions to nonlinear evolution equations by the exp function method a new application of the exp function method in combination with the dependent variable transformation from singularity analysis is proposed for constructing new generalized solitary wave solutions and periodic wave solutions for nonlinear evolution equations the korteweg de vries equation is chosen to illustrate the validity and applicability of the suggested approach

fluctuating dynamics at the quasiperiodic onset of chaos tsallis q statistics and mori s q phase thermodynamics abstract we analyze the fluctuating dynamics at the golden mean transition to chaos in the critical circle map and find that trajectories within the critical attractor consist of infinite sets of power laws mixed together we elucidate this structure assisted by known renormalization group rg results next we proceed to weigh the new findings against tsallis entropic and mori s thermodynamic theoretical schemes and observe behavior to a large extent richer than previously reported we find that the sensitivity to initial conditions t has the form of families of intertwined q exponentials of which we determine the q indexes and the generalized lyapunov coefficient spectra q further the dynamics within the critical attractor is found to consist of not one but a collection of mori s q phase transitions with a hierarchical structure the value of mori s thermodynamic field variable q at each transition corresponds to the same special value for the entropic index q we discuss the relationship between the two formalisms and indicate the usefulness of the methods involved to determine the universal trajectory scaling function and or the occurrence and characterization of dynamical phase transitions

analytical and numerical solutions of the generalized dispersive swift hohenberg equation the generalization of the swift hohenberg equation is studied it is shown that the equation does not pass the kovalevskaya test and does not possess the painleve property exact solutions of the generalized swift hohenberg equation which are very useful to test numerical algorithms for various boundary value problems are obtained the numerical algorithm which is based on the crank nicolson adams bashforth scheme is developed this algorithm is tested using the exact solutions the selforganization processes described by the generalization of the swift hohenberg equation are studied

how far one can go with the exp function method a criterion determining if an exact solution of a differential equation can be expressed in a form comprising a finite number of exponential functions is constructed in this paper this criterion is based on the concept of ranks of hankel matrixes constructed from sequences of coefficients produced by symbolic multiplicative operator techniques the employment of this criterion also gives an answer on the structure of the solution several examples are used to illustrate this concept

on elliptic solutions of nonlinear ordinary differential equations the problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied an algorithm for finding elliptic solutions in explicit form is presented

nonlinear differential equations of the second third and fourth order with exact solutions abstract we present the class of nonlinear ordinary differential equations of the second third and fourth order in a polynomial form with exact solutions this class has exact solutions in the form of solitary waves kinks and one periodic solutions

periodic solutions of the non integrable convective fluid equation in the present paper we have obtained and analyzed a family of exact periodic solutions of the nonlinear evolution partial differential convecting fluid equation cfe by applying a modification of the bilinear transformation method this modification is used in view of the circumstance that cfe is a non integrable nonlinear equation a detailed consideration has been given to the quite important case of balance between the dispersion and nonlinear effects establishing that this balance changes the structure of the equation itself the exact periodic solutions of cfe have been also found in the important case when cfe is identical with the kuramoto sivashinsky equation

exact solutions of systems describing interacting populations with low critical densities the generalization of the simplest equation method to look for exact solutions of systems of nonlinear differential equations is presented the exact solutions of nde systems describing the evolution of two interacting populations in two cases both populations have the low critical density or low critical density is typical for only one of populations are obtained

exact solutions of the equation for surface waves in a convecting fluid abstract a method for finding exact solutions and the first integrals is presented the basic idea of the method is to use the value of the fuchs index that appears in the painleve test to construct the auxiliary equation for finding the first integrals and exact solutions of nonlinear differential equations it allows us to obtain the first integrals and new exact solutions of some nonlinear ordinary differential equations the main feature of the method is that we do not assign a solution function at the beginning we find this function during calculations this approach is conceptually equivalent to the third step of the painleve test and sometimes allows us to change this step our approach generalizes a number of other methods for finding exact solutions of nonlinear differential equations we demonstrate a method for finding the traveling wave solutions and the first integrals of the well known nonlinear evolution equation for description of surface waves in a convecting liquid the general solution of this equation at some conditions on parameters and new traveling wave solutions of the fourth order equation are found

convergence to the critical attractor of dissipative maps log periodic oscillations fractality and nonextensivity for a family of logisticlike maps we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space we found that the phase space volume occupied by the ensemble w t depicts a power law decay with log periodic oscillations reflecting the multifractal character of the critical attractor we explore the parametric dependence of the power law exponent and the amplitude of the log periodic oscillations with the attractor s fractal dimension governed by the inflection of the map near its extremal point further we investigate the temporal evolution of w t for the circle map whose critical attractor is dense in this case we found w t to exhibit a rich pattern with a slow logarithmic decay of the lower bounds these results are discussed in the context of nonextensive tsallis entropies

traveling waves and statistical distributions connected to systems of interacting populations we discuss the following two issues from the dynamics of interacting populations i density waves for the case or negligible random fluctuations of the population densities ii probability distributions connected to the model equations for spatially averaged population densities for the case of significant random fluctuations of the independent quantity that can be associated with the population density for the case of issue i we consider model equations containing polynomial nonlinearities such nonlinearities arise as a consequence of interaction among the populations for the case of large population densities or as a result of a taylor series expansion for the case of small density of interacting populations by means of the modified method of the simplest equation we obtain exact traveling wave solutions of the model equations and these solution for the case of issue ii we discuss model equations of the fokker planck kind for the evolution of the statistical distributions of population densities we derive a few stationary distributions for the population density and calculate the expected exit time associated with the extinction of the studied population

elliptic solutions of the quintic complex one dimensional ginzburg landau equation the conte musette method has been modified for the search of only elliptic solutions to systems of differential equations a key idea of this a prior restriction is to simplify calculations by means of the use of a few laurent series solutions instead of one and the use of the residue theorem the application of this approach to the quintic complex one dimensional ginzburg landau equation cgle5 allows us to find elliptic solutions in the wave form restrictions on coefficients which are necessary conditions for the existence of elliptic solutions for the cgle5 have been found as well we demonstrate that to find elliptic solutions the analysis of a system of differential equations is preferable to the analysis of the equivalent single differential equation

chemical turbulence equivalent to nikolavskii turbulence we find evidence that a certain class of reaction diffusion rd systems can exhibit chemical turbulence equivalent to nikolaevskii turbulence we study an extended complex ginzburg landau cgl equation derived from this class of rd systems first we show numerically that the power spectrum of this cgl equation in the neighborhood of a codimension two turing char21 benjamin feir point is qualitatively quite similar to that of the nikolaevskii equation then we demonstrate that the nikolaevskii equation can in fact be obtained from this cgl equation through a phase reduction procedure

exact solitary waves in a convecting fluid the perturbed korteweg de vries equation ut lambda 1uux lambda 2uxxx lambda 3uxxxx lambda 4 uux x lambda 5uxx 0 which describes the evolution of long shallow waves in a convecting fluid when the critical rayleigh number slightly exceeds its critical value admits two types of exact solitary wave solution

an improvement on the exp function method when balancing the highest order linear and nonlinear terms abstract the exp function method has been widely used to solve different kinds of nonlinear partial differential equations these nonlinear partial differential equations are transformed first into nonlinear ordinary differential equations and then the ansatz of the exp function method u n c p a n exp n m d q b m exp m is applied to obtain the solution however a part of the solution using this method is to construct the relations between c p d and q by balancing the highest order linear term with the highest order nonlinear term it is proved in the present paper that c d and p q are the only relations that can be obtained by applying this method to any nonlinear ordinary differential equation therefore the additional calculations of balancing the highest order linear term with the highest order nonlinear term are not longer required in future hence the method becomes more straightforward

polygon wise constant curvature natural element approximation of kirchhoff plate model abstract a polygon wise constant curvature natural element approximation is presented in this paper for the numerical implementation of the abstract kirchhoff plate model the strict c 1 continuity requirement in the displacement field is relaxed by converting the area integral of the curvatures into the boundary integral along the voronoi boundary curvatures and bending moments are assumed to be constant within each voronoi polygon and the voronoi polygon wise constant curvatures are derived in a selective manner for the sake of the imposition of essential boundary conditions the numerical results illustrating the proposed method are also given

travelling wave solutions of nonlinear evolution equations using the simplest equation method in this paper the simplest equation method has been used for finding the exact solutions of three nonlinear evolution equations namely the vakhnenko parkes equation the generalized regularized long wave equation and the symmetric regularized long wave equation all three of these equations arise in fluids science so finding their exact solutions is of great importance

all traveling wave exact solutions of the variant boussinesq equations in this article we employ the complex method to obtain all meromorphic solutions of complex variant boussinesq equations 1 then find out related traveling wave exact solutions of system vb the idea introduced in this paper can be applied to other nonlinear evolution equations our results show that all rational and simply periodic solutions w r 1 k x t w r 2 k x t w s 1 k x t and w s 2 k x t of system vb are solitary wave solutions and there exist some rational solutions wr 2 z and simply periodic solutions ws 2 z which are not only new but also not degenerated successively by the elliptic function solutions we believe that this method should play an important role for finding exact solutions in the mathematical physics we also give some computer simulations to illustrate our main results

painleve analysis and exact solutions for the modified korteweg de vries equation with polynomial source the modified korteweg de vries equation with polynomial source is considered using the painleve test we show that the generalized korteweg de vries equation is not integrable by the inverse scattering transform however there are some expansions of solution in the laurent series and some exact solutions can exist some traveling wave solutions of the modified korteweg de vries equation with polynomial source are found

modified method of simplest equation and its application to nonlinear pdes abstract we search for traveling wave solutions of the class of pdes p 1 n 1 a p q p q t p r 2 n 2 b r q q t r s 1 n 3 c s q s q x s u 2 n 4 d u q q x u f q 0 where a p q b r q c s q d u q and f q are polynomials of q the basis of the investigation is a modification of the method of simplest equation the equations of bernoulli riccati and the extended tanh function equation are used as simplest equations the obtained general results are illustrated by obtaining exact solutions of versions of the generalized kuramoto sivashinsky equation reaction diffusion equation with density dependent diffusion and the reaction telegraph equation

new approach to find exact solutions of time fractional kuramoto sivashinsky equation abstract in the present paper we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics viz time fractional kuramoto sivashinsky equation by a new proposed method via fractional complex transform as a result new types of exact analytical solutions are obtained here the fractional derivative is described in jumarie s modified riemann liouville sense

painleve analysis and backlund transformation in the kuramoto sivashinsky equation the authors apply the painleve test to the kuramoto sivashinsky nonlinear partial differential equation which describes many interesting fluid motions the equation passes the test in a weak sense although the painleve analysis does not yield the general solution they perform the backlund transformation in order to obtain new particular solutions the authors find explicitly all the solutions of the set of equations which define the painleve backlund transformation they reduce to only one type namely the already known kink shaped solitary wave