a second wronskian formulation of the boussinesq equation a wronskian formulation leading to rational solutions is presented for the boussinesq equation it involves third order linear partial differential equations whose representative systems are systematically solved the resulting solutions formulas provide a direct but powerful approach for constructing rational solutions positon solutions and complexiton solutions to the boussinesq equation various examples of exact solutions of those three kinds are computed the newly presented wronskian formulation is different from the one previously presented by li et al which does not yield rational solutions

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 the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

fractal measures of first passage time of a simple random walk abstract we consider random walks starting at the site i 1 on a one dimensional lattice segment with an absorbing boundary at i 0 and a reflecting boundary at i l we find that the typical value of first passage time fpt is independent of system size l while the mean value diverges linearly with l the qth moment of the fpt diverges with system size as l2q 1 for q 1 2 for a finite but large l the fpt distribution has an 1 t tail cut off by an exponential of the form exp t l2 however if l is set equal to infinity the distribution has an algebraic tail given by t 1 2 we find that the generalised dimensions d q have a nontrivial dependence on q this shows that the fpt distribution is a multifractal we also calculate the singularity spectrum f

generalized multiple riccati equations rational expansion method with symbolic computation to construct exact complexiton solutions of nonlinear partial differential equations in this paper based on symbolic computation and the idea of rational expansion method a generalized multiple riccati equations rational expansion method gmrere is proposed to uniformly construct a series of exact complexiton solutions for nonlinear partial differential equations compared with most existing tanh function methods and rational expansion methods the proposed method not only recovers some known solutions but also finds some new and general complexiton solutions the efficiency of the method can be demonstrated on 2 1 dimensional burgers equation

darboux transforms deep reductions and solitons explicit formalisms for deep reductions of matrix differential equations and darboux covariance properties are presented the matrix 2 2 spectral problem of the second order is considered this problem with the reduction constraints being imposed on the potentials is the first equation of the lax representation of the hirota satsuma system 1981 the two reduction cases are treated with the help of the bilinear delta forms the covariance of these forms with respect to the darboux transforms under restrictions gives rise to explicit formulas of n soliton solutions in particular the two parameter soliton solutions of the hirota satsuma system are obtained the specific feature of such solutions evolution is that the singularity appears in some parameter region the yajima oikawa system 1976 is given as an example of the technique application to a 3 3 spectral problem

analytical positon negaton and complexiton solutions for the coupled modified kdv system new analytical positon negaton and complexiton solutions for the coupled modified kdv system are constructed by means of the darboux transformation with a constant seed solution and given out both analytically and graphically

initiation of damage in the kauffman model an alternate method is presented for initiating damage to the kauffman model a numerical study suggests that this modification increases the efficiency of generating damage clusters without changing their fractal properties

multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system compared with most of the existing tanh methods the jacobi elliptic function 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 travelling wave solutions according to the values of some parameters the solutions obtained in this paper include a kink shaped and bell shaped soliton solutions b rational solutions c triangular periodic solutions and d jacobi and weierstrass doubly periodic wave solutions among them the jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition the efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper kdv mkdv ito s fifth mkdv hirota nizhnik novikov veselov broer kaup generalized coupled hirota satsuma coupled schrodinger kdv 2 1 dimensional dispersive long wave 2 1 dimensional davey stewartson equations in addition as an illustrative sample the properties of the soliton solutions and jacobi doubly periodic solutions for the hirota equation are shown by some figures the links among our proposed method the tanh method extended tanh method and the jacobi elliptic function method are clarified generally

new complexiton solutions of the nonlinear evolution equations using a generalized rational expansion method with symbolic computation abstract in this paper we study complexiton solutions of nonlinear evolution equations by using a generalized ansatz with the help of symbolic computation maple we obtain new types of complexiton solutions of the 2 1 dimensional burgers equation the solutions contain the combination of hyperbolic function and elliptic function trigonometric function and elliptic function

toward a hybrid trefftz element with a hole for elasto plasticity the paper deals with the modelling of riveted assemblies for full scale complete aircraft crashworthiness many comparisons between experiments and fe computations of bird impacts onto aluminium riveted panels have shown that macroscopic plastic strains were not sufficiently developed and localised in the riveted shell fe in the impact area consequently fe models never succeed in initialising and propagating the rupture in the sheet metal plates and along rivet rows as shown by experiments without calibrating the input data especially the damage and failure properties of the riveted shell fe to model the assembly correctly it appears necessary to investigate on fe techniques such as hybrid trefftz finite element method h t fem indeed perforated fe plates developed for elastic problems based on a hybrid trefftz formulation have been found in the open literature our purpose is to find a way to extend this formulation so that the super element can be used for crashworthiness to reach this aim the main features of an elastic hybrid trefftz plate are presented and are then followed by a discussion on the possible extensions finally the interpolation functions of the element are evaluated numerically

a high precision element with a central circular hole an eight node rectangular membrane element with a central circular hole has been developed using complex potentials to define stress and strain functions this new element type can be combined with the conventional elements e g isoparametric elements with no modifications required two numerical examples were employed to illustrate the superiority and versatility of the proposed element type the results obtained using the said element type are closer to the theoretical results as compared with those obtained by similar types of elements

n soliton solutions and elastic interaction of the coupled lattice soliton equations for nonlinear waves with the aid of symbolic computation a coupled set of the lattice soliton equations is investigated via darboux transformation dt method the n fold dt and conservation laws are constructed based on its lax representation the n soliton solutions in terms of the vandermonde like determinants are derived structures of the one two three and four soliton solutions are shown graphically elastic interactions among the four solitons are discussed solitonic shapes and amplitudes have not changed after the interaction results in this paper might be helpful for understanding the propagation of nonlinear waves

analytic solutions darboux transformation operators and supersymmetry for a generalized one dimensional time dependent schrodinger equation abstract in this paper analytically investigated is a generalized one dimensional time dependent schrodinger equation using darboux transformation operator technique we construct the first order darboux transformation and the real valued condition of transformed potential for the generalized schrodinger equation to prove the equivalence of the supersymmetry formalism and the darboux transformation we investigate the relationship among first order darboux transformation supersymmetry and factorization of the corresponding effective mass hamiltonian furthermore the nth order darboux transformations are constructed by means of different method finally by using darboux transformation some analytical solutions are generated in a recursive manner for some examples of the schrodinger equation

emergent properties in random complex automata abstract studies of large randomly assembled binary boolean node automata have demonstrated that such systems can spontaneously exhibit enormously ordered dynamical behavior an important approach to characterizing these behaviors has consisted of studying ensembles of automata ensemble specifications have been based on 1 choice of boolean function regulating each node in the automaton 2 random or biased mappings of the 2 n automata states into themselves 3 the numbers of inputs k per node this article briefly reviews these approaches reports new results on the expected behavior of threshold automata and automata with a particular class of biased mappings in addition it discusses emergent highly ordered dynamical behavior in random automata rich in a specific class of canalizing boolean functions due to the crystallization of powerful subautomata called forcing structures the ordered behaviors may be of importance in biological evolution in physics and in the design of adaptive automata

reply to comments by r piltner on development of two dimensional elements with a central circular hole comput methods appl mech engrg 188 2000 431 440

multifractality and the kauffman model the author simulates the spread of an initial damage in the kauffman cellular automata at its critical point on the square lattice he calculates various moments of the probability that a site has been damaged n times and checks for multifractality in the critical exponents for different ensembles specifically he finds no evidence for multifractality when the moments of the probabilities are evaluated with lattice size l but multifractal behaviour occurs when the moments are monitored as a function of time t

multifractal nature of diffusion on hierarchical structures the authors study the moments of the first passage time t of random walks on hierarchical structures using the exact enumeration method they find that the different moments scale with the size of the system with an infinite hierarchy of exponents characterising a multifractal behaviour

modulational instability and exact solutions of the nonlinear schrodinger equation coupled with the nonlinear klein gordon equation the modulational instability mi of the coupled nonlinear schrodinger and nonlinear klein gordon equations is investigated it is found that there are a number of possibilities for the mi regions due to the generalized dispersion relation which relates the frequency and wavenumber of the modulating perturbations some exact travelling wave solutions are constructed via the solutions of a 4 model through a simple mapping relation furthermore we present five different types of solutions representing possible final states of modulationally unstable perturbations the profiles of solitary wave structures are displayed for some fixed parameters

types of solutions of the variable coefficient nonlinear schrodinger equation with symbolic computation by using hirota s bilinear method and symbolic computation solutions for a variable coefficient nonlinear schrodinger equation are obtained theoretically it is found that the type of the solutions changes with the different choices of the group velocity dispersion coefficient beta 2 z according to those solutions the relevant properties and features of physical and optical interest are illustrated in addition an effective technique for controlling the shape of the pulses is presented the results of this paper will be valuable to the study of the future development of ultrahigh rate and long distance optical communication systems

a new coupled sub equations expansion method and novel complexiton solutions of 2 1 dimensional burgers equation in this paper based on symbolic computation and the idea of rational expansion method a new coupled sub equations method is devised to uniformly construct a series of exact complexiton solutions for nonlinear evolution equations compared with most existing tanh function methods and other sophisticated methods the proposed method not only recover some known solutions but also find some new and general solutions which include many new types of complexiton solutions the efficiency of the method can be demonstrated on 2 1 dimensional burgers equation

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

line soliton interactions of a nonisospectral and variable coefficient kadomtsev petviashvili equation an extension of the dressing method is developed to solve a nonisospectral and variable coefficient generalization of the kadomtsev petviashvili equation nonstandard dynamics of the one soliton solutions are presented decomposition of the two solitons and their interactions are studied in detail

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

dromion interactions of 2 1 dimensional nonlinear evolution equations starting from two line solitons the solution of integrable 2 1 dimensional mkdv system and kdv system in bilinear form yields a dromion solution or a solitoff solution such a dromion solution is localized in all directions and the solitoff solution decays exponentially in all directions except a preferred one for the physical field or a suitable potential the interactions between two dromions and between the dromion and solitoff are studied by the method of figure analysis for a 2 1 dimensional modified kdv equation and a 2 1 dimensional kdv type equation our analysis shows that the interactions between two dromions may be elastic or inelastic for different forms of solutions

extended multilinear variable separation approach and multivalued localized excitations for some 2 1 dimensional integrable systems the multilinear variable separation approach and the related universal formula have been applied to many 2 1 dimensional nonlinear systems starting from the universal formula abundant 2 1 dimensional localized excitations have been found in this paper the universal formula is extended in two different ways one is obtained for the modified nizhnik novikov veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the 2 1 dimensional symmetric sine gordon system the other is for the dispersive long wave equation the broer kaup kupershmidt system the higher order broer kaup kupershmidt system and the burgers system where arbitrary number of variable separated functions can be involved because of the existence of the arbitrary functions in both the original universal formula and its extended forms the multivalued functions can be used to construct a new type of localized excitations folded solitary waves fsws and foldons the fsws

vortices circumfluence symmetry groups and darboux transformations of the 2 1 dimensional euler equation the euler equation ee is one of the basic equations in many physical fields such as fluids plasmas condensed matter astrophysics and oceanic and atmospheric dynamics a symmetry group theorem of the 2 1 dimensional ee is obtained via a simple direct method which is thus utilized to find exact analytical vortex and circumfluence solutions a weak darboux transformation theorem of the 2 1 dimensional ee can be obtained for an arbitrary spectral parameter from the general symmetry group theorem possible applications of the vortex and circumfluence solutions to tropical cyclones especially hurricane katrina 2005 are demonstrated

a generalized hirota satsuma coupled kdv system darboux transformations and reductions a darboux transformation is constructed for the generalized hirota satsuma coupled kdv system and the result is compared with the recent work of geng and his collaborators x g geng et al phys rev e 79 056602 2009 and x g geng and g l he j math phys 51 033514 2010 it is shown that our darboux transformation may be applied to three interesting reductions of the general system in addition the iteration of this darboux transformation is worked out and some solutions to the associated systems are obtained

some novel nonlinear coherent excitations of the davey stewartson system exact solutions of many integrable 2 1 2 spatial and 1 temporal dimensional systems of nonlinear evolution equations e g the davey stewartson model can be obtained by a special separation of variables procedure by choosing the jacobi elliptic functions as the building blocks exact doubly periodic solutions are obtained analytically here two sets of elliptic functions with two different independent moduli are employed and the resulting wave packets are expressed as rational functions of elliptic functions by taking the long wave limit in one spatial variable peculiar wave patterns localized in one direction but periodic in the other direction will arise by taking the long wave limit in both spatial variables exponentially localized wave patterns which differ from the known dromions will result the boundary conditions relating to these localized structures are studied

the modified kdv equation with variable coefficients exact uni bi variable travelling wave like solutions in this paper the modified korteweg de vries mkdv equation with variable coefficients vc mkdv equation is investigated via two kinds of approaches and symbolic computation on the one hand we firstly reduce the vc mkdv equation to a second order nonlinear nonhomogeneous ode using travelling wave like similarity transformation and then we obtain its many types of exact fractional solutions with one travelling wave like variable by applying some fractional transformations to the obtained nonlinear ode on the other hand we reduce the vc mkdv equation to two nonlinear pdes with variable coefficients using the anti tangent and anti hypertangent function transformations respectively and then we given its many types of exact solutions with two different travelling wave like variables by studying the obtained nonlinear pde with variable coefficients c 2008 elsevier inc all rights reserved

coupled kdv equations derived from two layer fluids some types of coupled korteweg de vries kdv equations are derived from a two layer fluid system in the derivation procedure an unreasonable y average trick usually adopted in the literature is removed the derived models are classified by means of the painleve test three types of function and multiple soliton solutions of the models are explicitly given via the exact solutions of the usual kdv equation it is also discovered that a non painleve integrable coupled kdv system can have multiple soliton solutions

darboux transformation for a generalized hirota satsuma coupled korteweg de vries equation a darboux transformation for the generalized hirota satsuma coupled korteweg de vries kdv equation is derived with the aid of the gauge transformation between the corresponding 4x4 matrix spectral problems with three potentials by which some explicit solutions of the generalized hirota satsuma coupled kdv equation are constructed as a reduction a darboux transformation of the complex coupled kdv equation and its explicit solutions are obtained

localized excitations of the 2 1 dimensional sine gordon system the 2 1 dimensional sine gordon system is investigated via a generalized bilinear operator representation plateau basin bowl and saddle type ring solitons are thereby constructed it is indicated how iterated moutard transformations may be employed to extend the range of the method

an application of the modified adomian decomposition method for n 1 dimensional sine gordon field abstract based upon the modified adomian decomposition method a scheme is developed to obtain approximate jacobi elliptic function solutions of the n 1 dimensional sine gordon field with initial conditions as a result we derive the approximate solution and exact jacobi elliptic function solution of the n 1 dimensional sine gordon field with initial conditions moreover the approximate solution is compared with the exact solution with n 2 and with different modulus k 0 25 0 5 and 1

length of state cycles of random boolean networks an analytic study in this paper we consider the mean length of transients and the length of state cycles in random boolean networks we present an approximate calculation of these quantities as a function of the size and connectivity of the network where using an annealed approximation we derive a recursive formula for the length of steps of the system using the mean step length and an effective momentary state space we calculate an approximate formula for the probability distribution function pdf of the length of state cycles and transients we compare this pdf with analytical results in special cases and with simulations by the monte carlo procedure

high order compact alternating direction implicit method for the generalized sine gordon equation high order compact alternating direction implicit scheme is given for solving the generalized sine gordon equation in a two dimensional rectangular domain we apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives and we give a linearized three time level algorithm for solving the original nonlinear equation error estimate is given by the energy method numerical results are provided to verify the accuracy and efficiency of this algorithm

darboux transformation and explicit solutions for the satsuma hirota coupled equation abstract a darboux transformation for the satsuma hirota coupled equation is obtained with the help of the gauge transformation between the lax pairs as an application of the darboux transformation we give some new explicit solutions including rational solutions soliton solutions and periodic solutions and others of the satsuma hirota coupled equation

scaling properties of diffusion on comb like structures the authors study the scaling properties of the probability density of random walks on comb like structures they find that in contrast to the first passage time in which several exponents are needed to characterise their distribution the probability density can be described by a single exponent

negaton and complexiton solutions of sine gordon equation abstract the negaton and complexiton solutions are introduced for the sine gordon equation by means of the darboux transformation dt the mutual interactions of multinegaton and interactions of negaton soliton and negaton positon are discussed in detail the superposition formulae for negations and complexitons are constructed noticing the relation of the sine gordon equation and the sinh gordon equation we also obtain the real valued singular negaton and complexiton solution for the sinh gordon equation

development of two dimensional elements with a central circular hole abstract two new two dimensional 2d elements one is a eight node rectangular and the other is a eight node circular element each with a central circular hole have been developed using complex potentials to define stress and strain functions these new elements can be combined with the conventional elements e g isoparametric elements with no modifications required two numerical examples were employed to illustrate the potential accuracy and versatility of the proposed elements the results obtained using these two elements are in good agreement with the theoretical results moreover the proposed rectangular element performs better than the proposed circular element

nonlinear resistor fractal networks topological distances singly connected bonds and fluctuations the authors consider a fractal network of nonlinear resistors with the voltage v behaving as a power of the current i mod v mod r mod i mod alpha the resistance between two points at a distance l is r l varies as lzeta alpha they prove that zeta 0 describes the scaling of the topological chemical distance while zeta infinity describes that of the number of singly connected red bonds for random resistors they also consider the width of the resistance distribution delta r varies as l zeta 2 alpha values for zeta and zeta 2 are explicitly derived for two model fractals and delta r r is found to grow with l for the sierpinski gasket and alpha 1 612 the relevance of the results to percolation clusters is discussed

complexiton solutions to integrable equations abstract complexiton solutions or complexitons for short are exact solutions newly introduced to integrable equations are newly introduced starting with the solution classification for a linear differential equation the korteweg de vries equation and the toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations the crucial step in the solution process is to apply the wronskian and casoratian techniques for hirota s bilinear equations correspondence between complexitons of the korteweg de vries equation and complexitons of the toda lattice equation is provided

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

complexiton solutions of the toda lattice equation a set of coupled conditions consisting of differential difference equations is presented for casorati determinants to solve the toda lattice equation one class of the resulting conditions leads to an approach for constructing complexiton solutions to the toda lattice equation through the casoratian formulation an analysis is made for solving the resulting system of differential difference equations thereby providing the general solution yielding eigenfunctions required for forming complexitons moreover a feasible way is presented to compute the required eigenfunctions along with examples of real complexitons of lower order

interactions among periodic waves and solitary waves of the n 1 dimensional sine gordon field exact solutions of the n 1 dimensional sine gordon field equation are studied with help of those of the cubic nonlinear klein gordon fields the mapping relations among the sine gordon field equation and the cubic nonlinear klein gordon fields are pure algebraic by solving the cubic nonlinear klein gordon equations many new types of exact explicit solutions such as the periodic periodic interaction waves periodic kink interaction waves periodic perturbed snake shape solitary waves etc are displayed both analytically and graphically

remarks on a system of coupled nonlinear wave equations it is shown that a generalized system of coupled kdv mkdv equation exhibits solitary wave solutions it has been shown that an increase in nonlinearity in one variable in a particular fashion does not affect the existence of solitary wave solutions the periodic traveling wave solutions of the system have also been investigated

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

the full positive flows of manakov hierarchy hamiltonian structures and conservation laws based on four sets of lenard recursion sequences and zero curvature equation we derive the full positive flows of the manakov hierarchy associated with a 3x3 matrix spectral problem from which some new nonlinear evolution equations are proposed with the help of the darboux transformation soliton solutions of two new nonlinear evolution equations in the manakov hierarchy are constructed as two special reductions the full positive flows of the coupled modified korteweg de vries hierarchy and the sasa satsuma hierarchy are deduced in which some new nonlinear evolution equations are included and then we construct the hamiltonian structures of the manakov hierarchy and infinite conservation laws of several nonlinear evolution equations

numerical complexiton solutions for the complex kdv equation by the homotopy perturbation method in this paper the homotopy perturbation method is extended to investigate the numerical complexiton solutions of the complex kdv equation by constructing special forms of initial conditions three new types of realistic numerical solutions are obtained numerical positon solution expressed by the trigonometric functions numerical negaton solution expressed by the hyperbolic functions and particularly the numerical analytical complexiton solutions expressed by combinations of the two kinds of functions all these numerical solutions obtained can rapidly converge to the exact solutions obtained by lou et al illustrative numerical figures are exhibited the efficiency of the proposed method

soliton solutions for two nonlinear partial differential equations using a darboux transformation of the lax pairs two darboux transformations of the 1 1 dimensional wu zhang wz equation and the two component camassa holm 2ch system with the reciprocal transformation are obtained one loop and two loop soliton solutions and multisoliton like solutions of the 2ch system are obtained by using the darboux transformations and selecting different seed solutions of the corresponding equations the bidirectional soliton solutions of the 1 1 dimensional wz equation are also obtained the interactions of two soliton head on and overtaking collisions for the wz equation and the evolution of the two soliton like solutions for the 2ch system are studied

multiple kink solutions and multiple singular kink solutions for the 2 1 dimensional burgers equations abstract in this work we study the 2 1 dimensional painleve integrable burgers equation multiple kink solutions and multiple singular kink solutions are formally derived for this equation the analysis depends mainly on the hirota s bilinear method

the relations among a special type of solutions in some d 1 dimensional nonlinear equations among a special type of solutions of some kinds of nonlinear equations in d 1 dimensions such as the cubic nonlinear klein gordon sine gordon double sine gordon ginzburg landau equations etc some mapping relations exist that allow one to find a solution of one system from a solution of another furthermore one can find some new solutions from a known one within the same system some new n kinklike solutions are also presented

search for multifractality in damage spreading for kauffman cellular automata the authors check how an initial disturbance called the damage spreads through a square lattice of kauffman cellular automata at their critical point they determine the moments of the probability that a site has been damaged n times and check for multifractality in the fractal dimensions or critical exponents of these moments of the damage probability

backlund transformation lax pairs and explicit exact solutions for the shallow water waves equation in this paper we consider the hirota satsuma equation for shallow water waves we first obtain the backlund transformation and lax pairs by using the extended homogeneous balance method then we find some explicit exact solutions by means of backlund transformation and the extended hyperbolic function method these solutions include the solitary wave solution of rational function soliton solutions double soliton solutions n soliton solutions the multiple solitary wave solutions singular solutions and the periodic wave solutions of triangle function type

scaling of the first passage time of biased diffusion on hierarchical comb structures biased diffusion on hierarchical comb structures is studied within an exact renormalisation group scheme the scaling exponents of the moments of the first passage time for random walks are obtained it is found that the scaling properties of the diffusion depend only on the direction of bias in a particular case the presence of bias may give rise to a new multifractality

damage spreading and multifractality in the travelling salesman problem the authors study numerically the spreading of damage in the euclidean travelling salesman problem there is a critical temperature tc approximately 0 20 0 25 below which damage does not spread they search for multifractal behaviour in the moments of the damage probability distribution multifractality is found in the frozen phase when the moments are monitored as a function of time if they are evaluated with the number of cities multifractal behaviour occurs in the chaotic phase

new analytical positon negaton and complexiton solutions of a coupled kdv mkdv system new positon negaton and complexiton solutions of a coupled kdv mkdv system are constructed directly from zero seed solution by means of darboux transformation these new exact solutions are all analytical with different spectral parameters and constants selection

complexitons of the modified kdv equation by darboux transformation abstract darboux transformation dt a comprehensive approach to construct the explicit solutions of the nonlinear evolutionary equation is applied to construct the new complexiton solution of the negative mkdv equation we find that the complexiton solution is related to the single complex spectral parameter and that it is completely different from the breather solution for the positive mkdv equation consequently we generalize the concept of complexiton of the kdv equation to the mkdv equation in addition the relationship of the spectral parameter and the known solutions of the mkdv equation is clarified and the multi complexiton solution multi complexiton positon multi complexiton negaton and multi complexiton soliton solutions are obtained in the uniform manner by dt the interaction of complexiton and soliton is also discussed in detail it is shown that the new complexiton and soliton remain unchanged except for phase shifts after their interaction at the same time the superreflectionless property of one positon potential for the mkdv equation is shown in detail

dromion like structures in a 3 1 dimensional kdv type equation dromion structures have been found in many 2 1 dimensional models the similar structure in 3 1 dimensions is studied in this paper for a kdv type equation starting from the bilinear form of the model we found that there are five types of multi dromion solutions for its potentials say the first type of multi dromion solution is driven by multi camber solitons with one of them being non parallel to the x axis the second third and fourth types of multi dromion solutions are driven by two camber solitons and one two and three sets of parallel plane solitons respectively the fifth type of multi dromion solution is driven by one set of parallel plane solitons one set of camber solitons which are parallel to the x axis and one camber soliton which is non parallel to the x axis phase shifts may be involved in the interactions among the multi dromions for the first and fifth types of solutions a single dromion solution of the model may possess quite a free shape for instance the point like dromions ring type dromions extended and sharp dromions and oscillatory dromions can be obtained by selecting some arbitrary functions appropriately

painleve property soliton like solutions and complexitons for a coupled variable coefficient modified korteweg de vries system in a two layer fluid model abstract as a model derived from a two layer fluid system which describes the atmospheric and oceanic phenomena a coupled variable coefficient modified korteweg de vries system is concerned in this paper with the help of symbolic computation its integrability in the painleve sense is investigated furthermore hirota s bilinear method is employed to construct the bilinear forms through the dependent variable transformations and soliton like solutions and complexitons are derived finally effects of variable coefficients are discussed graphically and it is concluded that the variable coefficients control the propagation trajectories of solitons and complexitons