explicit series solutions to nonlinear evolution equations the sine cosine method in this paper a sine cosine method is used to construct many periodic and solitary wave solutions to two nonlinear evolution systems the coupled quadratic nonlinear equations and the coupled klein gordon schrodinger equations under different parameter conditions explicit formulas for some new periodic and solitary wave solutions are successfully obtained the proposed solutions are found to be important for the explanation of some practical physical problems

fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

application of the extended g g expansion method to solve the pochhammer chree equations abstract in the present paper the extended g g expansion method is used for a reliable treatment of the pochhammer chree equations which include variety of models travelling wave solutions of distinct structures namely bell shaped solitons kink shaped solitons and periodic solutions are formally derived the method demonstrates power reliability and efficiency

the existence and uniqueness of the local solution for a camassa holm type equation a shallow water equation of camassa holm type containing nonlinear dissipative effect is investigated using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself we establish the existence and uniqueness of its local solution in sobolev space h s r with s 32 meanwhile a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order sobolev space h s with 1

the extended tanh method for new solitons solutions for many forms of the fifth order kdv equations abstract the extended tanh method is used to derive new solitons solutions for several forms of the fifth order nonlinear kdv equation the forms include the lax sawada kotera sk sawada kotera parker dye skpd kaup kupershmidt kk kaup kupershmidt parker dye kkpd and the ito equations the criteria established in a m wazwaz abundant solitons solutions for several forms of the fifth order kdv equation by using the tanh method appl math comput in press doi 10 1016 j amc 2006 02 047 to build up reliable relations between the parameters of the equation are confirmed by using this new approach entirely new bell shaped solitons are determined

a regular model for bond percolation the approach towards the threshold a two dimensional regular model is constructed for bond percolation via a rule of bond deletions on the square lattice the model describes the approach towards the threshold the critical bond concentration pc and the correlation length exponent nu are found the fractal geometry of the infinite percolating network and the scaling property of the cluster size distribution are studied above the threshold the infinite percolating network is shown to be reconstructed by the mandelbrot koch curve proposed as the non random model for the infinite cluster at criticality

self dual fractal models for the low field hall effect near percolation threshold fractal lattices with the self duality property are proposed to imitate the geometric texture of two dimensional percolating networks at percolation threshold the exponents describing the power law dependence on the scale length of the hall and ohmic conductivities are found the exponent for the hall conductivity agrees with twice that for the ohmic conductivity the model is extended to describe the approach towards the threshold it is found that the model shows the typical percolation behaviour as a function of a parameter p the bond concentration

n soliton solutions for shallow water waves equations in 1 1 and 2 1 dimensions abstract a variety of shallow water waves equations in 1 1 and 2 1 dimensions are investigated we first show that these models are completely integrable we next determine multiple soliton solutions for each equation the simplified hirota s bilinear method developed by hereman will be employed to achieve this goal a comparison between dispersion relations and the phase shifts will be conducted but possess the same coefficients for the polynomials of exponentials

analysis of 3 1 dimensional boitileonmannapempinelli equation via lax pair investigation and group transformation method singular manifolds method smm is utilized to detect the lax pair and bcklund transformation of 3 1 dimensional boitileonmannapempinelli blmp equation an extension of group transformation method with three parameters is exploited in transforming the lax pair directly to a system of ordinary differential equations odes two cases of similarity variables are considered resulting in solvable systems of odes new analytical solutions are attained and graphed containing some soliton waves some of these solutions are applicable of plasma explosions and fluid dynamics both of investigation of lax pair of 3 1 dimensional evolution equations and three parameters group method are seldom utilized in this kind of evolution equations

a variety of exact travelling wave solutions for the 2 1 dimensional boiti leon pempinelli equation abstract in this paper the 2 1 dimensional boiti leon pempinelli blp equation will be studied the tanh coth method will be used to obtain exact travelling wave solutions for this equation the exp function method will also be applied to the blp equation to derive a new variety of travelling wave solutions with distinct physical structures

a repository of equations with cosine sine compactons nonlinear evolution equations with cosine sine compacton solutions are reviewed including the rosenau hyman equation and generalizations of korteweg de vries camassa holm boussinesq benjamin bona mahony klein gordon and other equations each equation is generalized to three dimensions and the conditions for its cosine solitary waves to be either a compacton or a soliton are determined several equations claimed in the literature to be different among them are found to be equivalent

on existence and uniqueness of the global weak solution for a shallow water equation abstract a shallow water equation including the camassa holm and degasperis procesi equations as special cases is investigated provided that initial value u 0 h 1 r u 0 l 1 r and 1 x 2 u 0 does not change sign the existence and uniqueness of global weak solution for the equation are established

embedding problems over large fields in this paper we study galois theoretic properties of a large class of fields a class which includes all fields satisfying a universal local global principle for the existence of rational points on varieties as applications we give a positive answer to a long standing conjecture which originates in an unpublished note of roquette and asserts that the absolute galois group of a countable hilbertian pac field is profinite free see f j problem 24 41 secondly we give new evidence for the conjecture of shafarevich which asserts that the absolute galois group of qab is profinite free finally one of the most interesting applications of the theory we develop here is the insight in the galois structure of the totally eg adic numbers

observations on the tanh coth expansion method for finding solutions to nonlinear evolution equations abstract the tanh coth expansion method for finding solitary travelling wave solutions to nonlinear evolution equations has been used extensively in the literature it is a natural extension to the basic tanh function expansion method which was developed in the 1990s it usually delivers three types of solution namely a tanh function expansion a coth function expansion and a tanh coth expansion it is known that for every tanh function expansion solution there is a corresponding coth function expansion solution it is shown that there is a tanh coth expansion solution that is merely a disguised version of the coth solution in many papers such tanh coth solutions are erroneously claimed to be new however other tanh coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh function method similar remarks apply to tan cot and tan cot expansion solutions

solutions of the generalized kdv equation with time dependent damping and dispersion solitary wave solutions are obtained for the generalized korteweg de vries gkdv equation with time dependent damping and dispersion by using the tanh coth method the exp function method and the modified sine cosine method these methods are useful and efficient and a variety of solitary wave solutions are obtained that possess variable coefficients

phase transitions on fractals ii sierpinski gaskets for pt i see ibid vol 16 p 1267 1983 the authors construct and investigate a family of fractals which are generalisations of the sierpinski gaskets sgs to all euclidean dimensionalities these fractal lattices have a finite order of ramification and can be considered marginal between one dimensional and higher dimensional geometries physical models defined on them are exactly solvable the authors argue that short range spin models on the sg show no finite temperature phase transitions as examples they solve a few spin models and study the resistor network and percolation problems on these lattices

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

general solutions with solitary patterns for the defocusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this work the defocusing branch of the genuinely nonlinear dispersive k n n equation that exhibits solutions with solitary patterns is investigated the equation with constant coefficients to the derivatives is examined in one two and three dimensional spaces general formulas for solutions with solitary patterns are developed for all positive integers n n 1 two sets of entirely new formulas one set produces solutions for even integers n whereas the second set produces solutions in opposite directions for odd integers n n 1 are established

on exact solutions of the regularized long wave equation a direct approach to partially integrable equations i solitary wave and solitons hitherto the bilinear transformation method has been used to obtain analytic solutions of completely integrable or nonlinear evolution equations in this first of two studies designated i and ii it is shown that the method may apply equally well to those nonlinear wave equations dubbed partially integrable here the regularized long wave rlw and rlw boussinesq rlwb equations are considered as examples in each case the bilinear form is found to have an extra equation which would appear to signal their nonintegrability their solitary wave solutions are reconstructed and the nonexistence of multisoliton solutions is demonstrated the present work provides the basis for the construction of periodic solutions in ii

the hirota s direct method for multiple soliton solutions for three model equations of shallow water waves abstract multiple soliton solutions for three model equations for shallow water waves are determined the three models are completely integrable the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these equations the tanh coth method is used to obtain single soliton solutions and other solutions for these three models the three models have different linear dispersion relations but possess the same coefficients for the polynomials of exponentials

series solution to the pochhammer chreeequation and comparison with exact solutions in this study a decomposition method for approximating the solution of the pochhammer chreeequation is implemented by using this scheme explicit exact solution is calculated in the form of a convergent power series with easily computable components to illustrate the application of this method numerical results are derived by using the calculated components of the decomposition series the obtained results are found to be in good agreement with the exact solutions known for some special cases

fractal models for diffusion controlled aggregation a two dimensional fractal model is constructed for diffusion controlled deposition on a surface the fractal geometry of the deposit and the power law behaviour of the quantities characterising the non equilibrium cluster size distribution are shown to be consequences of the competition generally present in a nonlinear growth process a qualitative agreement with previous numerical results is found and the scaling laws for the critical exponents of the problem are shown to be satisfied exactly

definable sets in algebraically closed valued fields elimination of imaginaries it is shown that if k is an algebraically closed valued field with valuation ring r then th k has elimination of imaginaries if sorts are added whose elements are certain cosets in k n of certain definable r submodules of k n for all n 1 the proof involves the development of a theory of independence for unary types which play the role of 1 types followed by an analysis of germs of definable functions from unary sets to the sorts contents

general compactons solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this paper we will study the focusing branch of the genuinely nonlinear dispersive k n n equation that exhibits compactons solitons with finite wavelengths the equation will be studied in one two and three dimensional spaces general formulas for compactons solutions are developed for all positive integers n n 1 two sets of entirely new formulas one set produces compactons for even integers n whereas the second set produces compactons and anticompactons for odd integers n n 1 are established

multiple soliton solutions of two extended model equations for shallow water waves abstract multiple soliton solutions for two extensions of model equations for shallow water waves presented in m j ablowitz d j kaup a c newell h segur the inverse scattering transform fourier analysis for nonlinear problems stud appl math 53 1974 249 315 r hirota j satsuma n soliton solutions of model equations for shallow water waves j phys soc jpn 40 2 1976 611 612 are obtained the two extensions possess identical dispersion relations but with different structures for the n soliton solutions for n 1 the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these extended equations the tanh coth method is used to obtain single soliton solutions

a fractal model for band structures just above percolation threshold a two dimensional regular fractal model is proposed to imitate the geometric texture of percolating networks just above percolation threshold the system is self similar fractal on length scales smaller than the connectedness length but becomes homogeneous non fractal square lattice on larger length scales the scaling of vibrational density of states and band structures are studied by using exact renormalisation group methods it is shown that an infinite number of narrow bands appear to have self similar structures of typical cantor sets at low frequency

exact fractals with adjustable fractal and fracton dimensionalities a family of exact fractals is presented by adjusting two external parameters a wide range of fractal and fracton dimensionalities can be achieved this includes the case of fracton dimensionality of 2 which is critical for diffusion

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

difference galois theory of linear differential equations abstract we develop a galois theory for linear differential equations equipped with the action of an endomorphism this theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation the galois groups here are linear difference algebraic groups i e matrix groups defined by algebraic difference equations

dynamical survey of a generalized zakharov equation and its exact travelling wave solutions we investigate the dynamical behavior of a generalized zakharov equation for the complex envelope of the high frequency wave and the real low frequency field by analyzing its phase portraits following a dynamical system approach in different parameter regions we depict phase portraits of a travelling wave system through discussing the bifurcation of phase portraits we unwrap explicit miscellaneous travelling waves including localized and periodic ones

single and multiple soliton solutions for the 2 1 dimensional kdv equation abstract the 2 1 dimensional kdv equation is studied by four distinct methods the hirota s bilinear method is used to derive multiple soliton solutions for this completely integrable equation the tanh coth method and the cosh ansatz are used to obtain single soliton solutions and travelling wave solutions the exp function method gives travelling wave solutions with some free parameters the work highlights the reliability of the four methods

solitary wave solutions of the generalized shallow water wave gsww equation by hirota s method tanh coth method and exp function method abstract the generalized short wave equation gsww is studied by three distinct methods the hirota s bilinear method is used to derive multiple soliton solutions for the completely integrable forms of the gsww equation the tanh coth method is used to obtain single soliton solutions for this equation the exp function method is also applied to derive a variety of travelling wave solutions with distinct physical structures for this non linear evolution equation

the tanh coth and the sine cosine methods for kinks solitons and periodic solutions for the pochhammer chree equations abstract in this paper the tanh coth method and the sine cosine method are used for a reliable treatment of the pochhammer chree equations which include a variety of models travelling wave solutions of distinct structures namely bell shaped solitons kink shaped solitons and periodic solutions are formally derived the two methods demonstrate power reliability and efficiency

application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media we consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media it is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions in addition to the fractal dimensiond f 1 d the fractal dimension of the backbone arises in connection with electric current flow 2 d red the fractal dimension of the singly connected bonds in the backbone arises in connection with its equivalence to the thermal scaling power 3 d e the fractal dimension of the of the elastic backbone 4 d u the fractal dimension of the unscreened perimeter arises in connection with the viscosity singularity at the gelation threshold 5 d min the fractal dimension of the minimum path or chemical distance between two sites arises in co nnection with the aharony stauffer conjecture 6 dw the fractal dimension of a random walk 7 d g the fractal dimension of growth sites that arise as a random walk creates a cluster relations among these fractal dimensions are discussed some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

a study on nonlinear dispersive partial differential equations of compact and noncompact solutions in this paper we study two genuinely nonlinear dispersive partial differential equations in one and higher dimensional spaces we show that the focusing branch and the defocusing branch of these equations each leads to a different physical structure in addition we develop general solutions of compact and noncompact support which are not obtainable through classic techniques

imaginaries and invariant types in existentially closed valued differential fields we answer two open questions about the model theory of valued differential fields introduced by scanlon we show that they eliminate imaginaries in the geometric language introduced by haskell hrushovski and macpherson and that they have the invariant extension property these two result follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields finally we show that this theory is metastable

diffusion on fractal lattices and the fractal einstein relation finitely ramified fractal lattices show anomalous diffusion with r2 varies as t2h there is a hierarchy of transit times which are shown by monte carlo simulation to satisfy tau n varies as alpha n for large n where alpha b1h and b is the base of the lattice the lattice resistivity scales with size as rho varies as rn where r is characteristic of the lattice exact renormalisation group methods yield alpha and r analytically and verify the einstein relation for fractals alpha rn