the 2 1 dimensional sine gordon equation integrability and localized solutions in this paper the 2 1 dimensional sine gordon equation 2dsg introduced by konopelchenko and rogers is investigated and is shown to satisfy the painleve property a variable coefficient hirota bilinear form is constructed by judiciously using the painleve analysis with a non conventional choice of the vacuum solutions first the line kinks are constructed then exact localized coherent structures in the 2dsgi equation are generated by the collision of two non parallel ghost solitons which drive the two non trivial boundaries present in the system also the reason for the difficulty in identifying localized solutions in the 2dsgii equation is indicated we also highlight the significance of the asymptotic values of the boundaries of the system

comment on application of exp function method for 3 1 dimensional nonlinear evolution equations comput math appl 56 2008 1451 1456 we show that boz and bekir a boz a bekir application of exp function method for 3 1 dimensional nonlinear evolution equations comput math appl 56 2008 1451 1456 obtained some incorrect solutions for the equations studied by means of the exp function method we verify our assertion by direct substitution and pole order analysis in addition we provide the correct results using the same approach

a note on the exp function method for traveling waves of nonlinear evolution equations abstract in this note we analyze the paper of noor et al 2010 1 using the exp function method noor et al found the generalized solitary and periodic solutions of zakharov kuznetsov and zakharov kuznetsov modified equal width equations we have checked noor s solutions and proved that seven from ten of them does not satisfy to equations considered the general solution of the zakharov kuznetsov equation in the traveling waves was obtained along ago we give this solutions in this letter for the reference source

a new note on exact complex travelling wave solutions for 2 1 dimensional b type kadomtsev petviashvili equation abstract exact solutions of the 2 1 dimensional kadomtsev petviashvili by zhang huiqun zhang a note on exact complex travelling wave solutions for 2 1 dimensional b type kadomtsev petviashvili equation appl math comput 216 2010 2771 2777 are considered to look for new types of exact solutions travelling wave solutions of equation zhang has used the g g expansion method we demonstrate that there is the general solution for the reduction by zhang from the 2 1 dimensional kadomtsev petviashvili equation and all solutions by zhang are found as partial cases from the general solution

the non local delta problem and 2 1 dimensional soliton equations a method of constructing 2 1 dimensional non linear integrable equations and their solutions by means of the non local delta problem is developed a basic set of equations is obtained by using different normalisations of the non local delta problem and the lagrangian of the set is found other integrable equations which are degenerate cases of the basic set are also lagrangian

modified extended tanh function method and its applications to nonlinear equations new exact travelling wave solutions for the generalized shallow water wave equation the improved boussinesq equation and the coupled system for the approximate equations for water waves are found using a modified extended tanh function method the obtained results include rational periodic singular and solitary wave solutions

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

stickiness in generic low dimensional hamiltonian systems a recurrence time statistics approach we analyze the structure and stickiness in the chaotic components of generic hamiltonian systems with divided phase space following the method proposed recently in lozej and robnik phys rev e 98 022220 2018 2470 004510 1103 physreve 98 022220 the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid we perform extensive numerical studies of three example systems the chirikov standard map the family of robnik billiards and the family of lemon billiards the filling of the cells is compared to the random model of chaotic diffusion introduced in robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 for the description of transport in the phase spaces of ergodic systems the model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential in generic systems the stickiness induces correlations in the cell visits the distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping we model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times we introduce the variable s which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness we use s to globally assess the distributions of recurrence times we find that in the bulk of the chaotic sea s 1 while s 1 in areas of stickiness we present the results in the form of animated grayscale plots of the variable s in the largest chaotic component for the three example systems included as supplemental material to this paper

a note on travelling wave solutions to lax s seventh order kdv equation ganji and abdollahzadeh d d ganji m abdollahzadeh appl math comput 206 2008 438 444 derived three supposedly new travelling wave solutions to lax s seventh order kdv equation each solution was obtained by a different method it is shown that any two of the solutions may be obtained trivially from the remaining solution furthermore it is noted that one of the solutions has been known for many years

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

exact soliton solutions of the one dimensional complex swift hohenberg equation abstract using painleve analysis the hirota multi linear method and a direct ansatz technique we study analytic solutions of the 1 1 dimensional complex cubic and quintic swift hohenberg equations we consider both standard and generalized versions of these equations we have found that a number of exact solutions exist to each of these equations provided that the coefficients are constrained by certain relations the set of solutions include particular types of solitary wave solutions hole dark soliton solutions and periodic solutions in terms of elliptic jacobi functions and the weierstrass function although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic swift hohenberg equations those presented here are the first examples of exact analytic solutions found thus far

new kink shaped solutions and periodic wave solutions for the 2 1 dimensional sine gordon equation new exact solutions including the kink shaped solutions bell shaped solutions periodic solutions singular soliton solutions and rational solution for the 2 1 dimensional sine gordon equation are obtained using the tanh method the g g expansion method and the auxiliary function method respectively

symmetry group and non propagating solitons in the 2 1 dimensional sine gordon equation starting from the compact version of the 2 1 dimensional sine gordon equation the symmetry group and then the lie symmetries and the related algebra can be obtained via a simple combination of a gauge transformation of the spectral function and the transformations of the space time variables thereout applying the group transformation theorem on a two straight line soliton solution one can derive two types of meaningful non propagating dromion lattice excitations

a unified approach to infinitesimal loewner and geroch transformations and the ernst and einstein maxwell equations it is demonstrated that the dual ernst equation of general relativity constitutes a stationary loewner system in an analogous manner it is shown that the einstein maxwell equations for stationary axisymmetric space times and their extension to einstein s equations coupled with an arbitrary number of u 1 gauge fields may be interpreted as generalized loewner systems moreover it is recorded that the base geroch transformation for the dual ernst equation may be located in a class of infinitesimal backlund transformations introduced by loewner in 1952 a geroch type transformation for a generic class of 2 1 dimensional loewner systems is set down and it is shown how the base geroch and hoenselaers kinnersley xanthopoulos transformations are naturally retrieved

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

letter to the editor comment on new types of exact solutions for nonlinear schrodinger equation with cubic nonlinearity in this comment we analyze the paper abdelhalim ebaid s m khaled new types of exact solutions for nonlinear schrodinger equation with cubic nonlinearity j comput appl math 235 2011 1984 1992 using the traveling wave ebaid and khaled have found new types of exact solutions for nonlinear schrodinger equation with cubic nonlinearity we demonstrate that the authors studied the well known nonlinear ordinary differential equation with the well known general solution we illustrate that ebaid and khaled have looked for some exact solution for the reduction of the nonlinear schrodinger equation taking the general solution of the same equation into account

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

on generalized loewner systems novel integrable equations in 2 1 dimensions a reinterpretation and generalization of a class of infinitesimal backlund transformations originally introduced in a gas dynamics context by loewner in 1952 leads to a linear representation for a novel class of 2 1 dimensional nonlinear equations the latter may be parametrized in terms of a triad of eigenfunctions moreover like the well known integrable davey stewartson and nizhnik novikov veselov equations the nonlinear systems typically contain the two spatial variables on an equal footing the dressing method is outlined for these generalized loewner systems it is noted that basic reductions lead to 2 1 dimensional versions of the principal chiral fields model toda lattice system and notably the classical sine gordon equation loewner systems on grassmannian the projective cpn and rpn manifolds are considered in particular the 2 1 dimensional integrable sine gordon system in which x and y occur in a symmetric manner arises naturally in this context a gauge equivalent system likewise eme

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

ermakov systems of arbitrary order and dimension structure and linearization ermakov systems of arbitrary order and dimension are constructed these inherit an underlying linear structure based on that recently established for the classical ermakov system as an application alignment of a 2 1 dimensional ermakov and integrable ernst system is shown to produce a novel integrable hybrid of a 2 1 dimensional sinh gordon system and of a conventional ermakov system

on the geometry of an integrable 2 1 dimensional sine gordon system it is recorded that darboux9s method of linking the classical lame system governing triply orthogonal systems of surfaces with an integrable 2 1 dimensional sine gordon equation may be extended and applied to the integrable two component generalization of the latter introduced by konopelchenko and rogers thus in a reinterpretation this 2 1 dimensional sine gordon system is shown to define particular integrable motions of surfaces

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

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

the double f expansion approach and novel nonlinear wave solutions of soliton equation in 2 1 dimension abstract the aim of this paper is to construct the double nonlinear wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov annv equation by means of double f expansion approach as a result families of double jacobi elliptic function like solutions and double solitary wave like solutions of the 2 1 dimensional annv equation are successfully constructed this method is very valid to some other 2 1 dimensional soliton equations

stickiness in hamiltonian systems from sharply divided to hierarchical phase space we investigate the dynamics of chaotic trajectories in simple yet physically important hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures we show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one parameter families of marginally unstable periodic orbits and is characterized by an exponent ensuremath gamma 2 for the asymptotic power law decay of the distribution of recurrence times generic perturbations lead to systems with hierarchical phase space where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and cantori in this case we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws depending on the relative contribution of the primary and secondary structures of the hierarchy numerical verification of our main results are provided for area preserving maps mushroom billiards and the newly defined magnetic mushroom billiards

a note on a capillarity model and the nonlinear schrodinger equation a system of two equations governing the irrotational flow of a capillary fluid is shown for a particular class of free energy functions to reduce to a nonlinear schrodinger equation

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

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

virasoro structure and localized excitations of the lkr system a symmetry analysis is conducted for a master 2 1 dimensional soliton system the classical symmetries are shown to constitute an infinite dimensional kac moody virasoro algebra finite symmetry group transformations are then used to construct localized excitations of the system

a travelling wave solution to the ostrovsky equation in this paper we consider a nonlinear evolution equation like ostrovsky equation by using the hyperbolic tangent method and an exponential function approach a travelling wave solution for the ostrovsky equation is presented it is observed that both methods lead to the same type of solution

linear series with cusps and n fold points a linear series v s on a curve x has an n fold point along a divisor d of degree n if dim vr h x d dim v 1 the linear series has a cuspofordere atapoint p if dim frch0 xs e l p dim z l linear series with cusps and n fold points are shown to exist if certain inequalities are satisfied the dimensions of the families of linear series with cusps are determined for general curves introduction throughout this paper we work over the complex numbers c let x be a smooth projective curve of genus g a gd on x is a linear series of degree d and dimension r on x i e a pair v s consisting of a line bundle s of degree d on x and an r l dimensional subspace vof h x s when p g r d g r 1 g r d o then x has gd s and the family of gd s on x forms a projective scheme gd x of dimension p g r d furthermore if x is general in moduli then gd x is smooth of dimension p g r d acgh definitic n if d is an effective divisor of degree n 2 on x we say that a gd v s has an n fold point along d if dim v n h x 2 d r we say that a gd has a cusp of order e at a point p if it has an e l fold point along the divisor e 1 p the aim of this paper is to prove the following i if g n p g r d n 1 r n o and p g r 1 d n o then there exists a gd on x with an n fold point along some divisor d of degree n ii if p g r d n 1 r o p g r 1 d n 0 and d is any divisor of degree n on x then there exists a gd on x with an n fold point along d iii if p g r d er 1 0 and p g r 1 d e 1 o then there exists a gd on x with a cusp of order e at some point p iv if x is general in moduli then the family of gd s on x with a cusp of order e has dimension p g r d er 1 if it is nonempty assertion iv has been proved independently by marc coppens c it is easy to see that the hypotheses p g r 1 d n 0 in assertions i and ii and p g r 1 d e 1 0 in assertion iii are necessary if x is general in moduli received by the editors january 20 1987 1980 mclthemcltlcs subject clclsslficcltion 1985 revision primary 14h10 1987 american mathematical society 0002 9947 87 1 00 25 per page

novel integrable reductions in nonlinear continuum mechanics via geometric constraints the nonlinear equations that describe solitonic behavior in physical systems have to date typically been derived by approximation or expansion methods here by contrast hidden integrable structure is revealed in diverse areas of nonlinear continuum mechanics through natural geometric constraints

a note on new travelling wave solutions to the ostrovsky equation in a recent paper by yasar e yasar new travelling wave solutions to the ostro vsky equation appl math comput 216 2010 3191 3194 new travelling wave solutions to the transformed reduced ostrovsky equation are presented in this note it is shown that some of these solutions are disguised versions of known solutions

solitary solutions of some nonlinear evolution equations in this paper the traveling wave reduction combined with the homogeneous balance method hbm is used to find new exact solutions by using less studied nonlinear partial differential equations npdg of higher order the usual starting point is a special transformation converting the npdg in its two variables x and t into an nonlinear ordinary differential equation node in the single variable x using the hyperbolic tangent method new exact solutions for the three npdgs are studied the new feature of this paper is of course the fact that we are dealing with npdgs which cannot be found by studying the relevant literature therefore we believe it is time for publishing some special solutions of this interesting kind of equations clearly we do not use a new formalism but with the aid of the tanh method we are able to calculate at least exact traveling wave solutions solitary waves which can physically interpreted as an action of wave propagation simultaneous we point out the necessity of such sophisticated methods since a general theory of npdgs does not exist the existence of nontrivial noncomplex valued solutions of the ramani equation enables us to generate families of real valued solutions of a nonlinear sixth order system special differential transformations are leading to connections between traveling wave solutions of associated nonlinear evolution equations

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

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

a note on new kink shaped solutions and periodic wave solutions for the 2 1 dimensional sine gordon equation exact solutions of the nizhnik novikov veselov equation by li new kink shaped solutions and periodic wave solutions for the 2 1 dimensional sine gordon equation appl math comput 215 2009 3777 3781 are analyzed we have observed that fourteen solutions by li from 30 do not satisfy the equation the other 16 exact solutions by li can be found from the general solutions of the well known solution of the equation for the weierstrass elliptic function

semiempirical theory of level spacing distribution beyond the berry robnik regime modeling the localization and the tunneling effects in this work we study the level spacing distribution in the classically mixed type quantum systems which are generic exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions in the asymptotic regime of the sufficiently deep semiclassical limit sufficiently small effective planck constant the berry and robnik 1984 j phys a math gen 17 2413 picture applies which is very well established we present a new quasi universal semiempirical theory of the level spacing distribution in a regime away from the berry robnik regime the near semiclassical limit by describing both the dynamical localization effects of chaotic eigenstates and the tunneling effects which couple regular and chaotic eigenstates the theory works extremely well in the 2d mixed type billiard system introduced by robnik 1983 j phys a math gen 16 3971 and is also tested in other systems mushroom billiard and prosen billiard

structure size and statistical properties of chaotic components in a mixed type hamiltonian system we perform a detailed study of the chaotic component in mixed type hamiltonian systems on the example of a family of billiards introduced by robnik in j phys a math gen 16 3971 1983 jphac50305 447010 1088 0305 4470 16 17 014 the phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times the structure of the chaotic component is discerned from the cells visited by the chaotic orbit the fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box counting method the cell filling dynamics is compared to a model of uncorrelated motion the so called random model robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 and deviations attributed to sticky objects in the phase space are found the statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run the stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times the recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed the standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter lastly the decay of correlations and the diffusion of momenta is analyzed

dynamical localization of chaotic eigenstates in the mixed type systems spectral statistics in a billiard system after separation of regular and chaotic eigenstates we study the quantum mechanics of a billiard robnik 1983 j phys a math gen 16 3971 in the regime of mixed type classical phase space the shape parameter 0 15 at very high lying eigenstates starting at about 1 000 000th eigenstate and including the consecutive 587654 eigenstates by calculating the normalized poincare husimi functions of the eigenstates and comparing them with the classical phase space structure we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates and the corresponding energies the chaotic eigenstates appear all to be dynamically localized meaning that they do not uniformly occupy the entire available chaotic classical phase space component but are localized on a proper subset we find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the poisson statistics and the chaotic ones obey the brody statistics as anticipated in a recent paper by batistic and robnik 2010 j phys a math theor 43 215101 where the entire spectrum was found to obey the berry robnik brody statistics there are no effects of dynamical tunneling in this regime due to the high energies as they decay exponentially with the inverse effective planck constant which is proportional to the square root of the energy

two dimensional schro dinger operator inverse scattering transform and evolutional equations abstract the inverse problem for the two dimensional schrodinger operator on the data from one energy level is solved in a special class of finite zone or algebraic operators this class seems to be dense among all smooth periodic schrodinger operators evolutional equations associated with this problem are constructed

on laplace darboux type sequences of generalized weingarten surfaces two novel classes of integrable surfaces generalized weingarten surfaces of class 1 and 2 which may be defined in a coordinate independent manner or equivalently via an extension of the classical lelieuvre formulas are introduced the classes 1 and 2 generalize bianchi surfaces and surfaces of harmonic inverse mean curvature respectively and turn out to be governed by analogs of the ernst weyl equation descriptive of gravitational and neutrino fields in general relativity the latter connection is exploited to derive laplace darboux type transformations for class 2 via the associated loewner konopelchenko rogers representation the laplace darboux type transformations are shown to have remarkable geometric properties in terms of sphere congruences connections with the pohlmeyer lund regge vortex model and an inhomogeneous heisenberg spin equation are recorded a backlund transformation for a particular painleve iii equation is derived in a purely geometric manner and a geometric interpretation of a

doubly periodic and multiple pole solutions of the sinh poisson equation application of reciprocal transformations in subsonic gas dynamics vortex patterns associated with the sinh poisson equation arise in a remarkable manner as relaxation states of the navier stokes equations here doubly periodic and multiple pole solutions of the sinh poisson equation are generated via the hirota bilinear operator formalism and exploitation of the phenomenon of coalescence of wave numbers it is then shown how the multi parameter reciprocal transformations of gas dynamics may be applied to a seed doubly periodic solution of the sinh poisson equation to generate associated periodic vortex structures valid in the subsonic flow of a generalized karman tsien gas

linear series with an n fold point on a general curve a linear series v s on a curve x has an n fold point along a divisor d of degree n if dim v n h x s d dimv 1 the dimensions of the families of linear series with an n fold point are determined for general curves we work over the field of complex numbers c at x be a smooth projective curve a gd on x is a linear series of dimension r and degree d on x i e a pair v s consisting of a line bundle y of degree d and an r 1 dimensional subspace v c h x s the gd s on x are parameterized by a projective scheme gd x if x is general in moduli then dimgd x p g r d g r l g r d acgh defixlition we say that a gd v s has an n fold point along a divisor d of degree n 2 in x if dim vnh x s d r if z x v is a flat proper irreducible family of smooth curves there exists a scheme gd xlv which is projective over v whose fiber over each v e v is gd xu where xv ot 1 v eh 2 it is easy to see that by considering the construction of gd x xv xn xn as the degeneracy locus of a vector bundle map there is a closed subscheme n x v c gd x xv xn xn such that the fiber over each point p1 pn e xvn consists of the gd s on xv with an n fold point along e pj we will write n x if v is a point let xg denote the moduli space of smooth curves of genus g 3 and let u be the open subset of xg corresponding to curves without nontrivial automorphisms let z u be the universal curve over u we will prove the following theorem theorem with g n 2 g 3 r 2 and the notation as above dimn zu p g r d r n 1 n when u is a sufficiently general point of u

letter to the editor a note on new exact solutions for the kawahara equation using exp function method exact solutions of the kawahara equation by assas l m b assas new exact solutions for the kawahara equation using exp function method j comput appl math 233 2009 97 102 are analyzed it is shown that all solutions do not satisfy the kawahara equation and consequently all nontrivial solutions by assas are wrong

the classical korteweg capillarity system geometry and invariant transformations a class of invariant transformations is presented for the classical korteweg capillarity system the invariance is an extension of a kind originally introduced in an anisentropic gasdynamics context in a particular instance application of the invariant transformation leads to a deformed one parameter class of karman tsien type capillarity laws associated with a deformation of an integrable nonlinear schrodinger type equation which incorporates a de broglie bohm potential the latter and another integrable case associated with the classical boussinesq equation may be linked to the motion of curves in euclidean and projective space so that both the invariant transformation and the galilean invariance of the capillarity system may be interpreted in a geometric and soliton theoretic manner the work is set in the broader context of other connections of invariant transformations in gasdynamics with soliton theory

new exact homoclinic wave and periodic wave solutions for the ginzburg landau equation abstract new exact solutions including homoclinic wave and periodic wave solutions for the 2d ginzburg landau equation are obtained using the auxiliary function method and the g g expansion method respectively the solutions are expressed by the hyperbolic functions and the trigonometric functions there result shows that there exists a kink wave solution which tends to one and the same periodic wave solution as time tends to infinite

backlund transformations for the nizhnik novikov veselov equation the backlund transformations for the nizhnik novikov veselov equation are presented it is shown that these transformations can be iterated and that the resulting sequence can be described by the volterra equations the relationships between the nizhnik novikov veselov equation and the volterra hierarchy are discussed

the bilinear negative kadomtsev petviashvili system and its kac moody virasoro symmetry group the symmetry transformation group of the bilinear negative kadomtsev petviashvili system is studied by means of a direct method the kac moody virasoro type lie point symmetry algebra is found to be a special infinitesimal form of the symmetry group

multifractality stickiness and recurrence time statistics we identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics rts which is based on the concept of poincare recurrences the projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near integrable system the annular billiard these microstructures interfere in the effective transport of a particle in the phase space which can be observed through rts this technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea

on the application of the exp function method to the kp equation for n soliton solutions abstract we observe that the form of the kadomstev petviashvili equation studied by yu 2011 s yu n soliton solutions of the kp equation by exp function method appl math comput 2011 doi 10 1016 j amc 2010 12 095 is incorrect we claim that the n soliton solutions obtained by means of the basic exp function method and some of its known generalizations do not satisfy the equation considered we emphasize that yu s results except only one cannot be solutions of the correct form of the kadomstev petviashvili equation in addition we provide some correct results using the same approach

quantum localization of chaotic eigenstates and the level spacing distribution the phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos or wave chaos and thus plays an important role in general quantum mechanics or even in general wave mechanics in this work we propose two different localization measures characterizing the degree of quantum localization and study their relation to another fundamental aspect of quantum chaos namely the energy spectral statistics our approach and method is quite general and we apply it to billiard systems one of the signatures of the localization of chaotic eigenstates is a fractional power law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance s goes like proportionality s for small s where 0 1 and 1 corresponds to completely extended states we show that there is a clear functional relation between the exponent and the two different localization measures one is based on the information entropy and the other one on the correlation properties of the husimi functions we show that the two definitions are surprisingly linearly equivalent the approach is applied in the case of a mixed type billiard system m robnik j phys a math gen 16 3971 1983 in which the separation of regular and chaotic eigenstates is performed

menelaus theorem clifford configurations and inversive geometry of the schwarzian kp hierarchy it is shown that the integrable discrete schwarzian kp dskp equation which constitutes an algebraic superposition formula associated with for instance the schwarzian kp hierarchy the classical darboux transformation and quasi conformal mappings encapsulates nothing but a fundamental theorem of ancient greek geometry thus it is demonstrated that the connection with menelaus theorem and more generally clifford configurations renders the dskp equation a natural object of inversive geometry on the plane the geometric and algebraic integrability of dskp lattices and their reductions to lattices of menelaus darboux schwarzian kdv schwarzian boussinesq and schramm type are discussed the dskp and discrete schwarzian boussinesq equations are shown to represent discretizations of families of quasi conformal mappings

new universal aspects of diffusion in strongly chaotic systems we study some new universal aspects of diffusion in chaotic systems especially those having very large lyapunov coefficients on the chaotic indecomposable topologically transitive component we do this by discretizing the chaotic component on the surface of section sos in a large number n of simplectically equally big cells in the sense of equal relative invariant ergodic measure normalized so that the total measure of the chaotic component is unity by iterating the transition of the chaotic orbit through the sos where j counts the number of iterations discrete time and assuming complete lack of correlation even between consecutive crossings which can be justified due to the very large lyapunov exponents we show the universal approach of the relative measure of the occupied cells denoted by to the asymptotic value of unity in the following way so that in the limit of big n we have for j n fixed the exponential law this analytic result is verified numerically in a variety of specific systems for a plane billiard robnik 1983 for a 3d billiard prosen 1997 for an ergodic logistic map tent map for a standard map k 400 and for the hydrogen atom in a strong magnetic field the agreement is almost perfect except in the latter two systems for some long time deviations on very small scales however for henon heiles system and for the standard map k 3 the deviations are noticeable though not very big only about 1 we have tested the random number generators press et al 1986 and confirmed that some are almost perfect ran0 and ran3 whilst two of them ran1 and ran2 exhibit big deviations

statistics of return times a general framework and new applications in this paper we provide general estimates for the errors between the distribution of the first and more generally the k th return time suitably rescaled and the poisson law for measurable dynamical systems in the case that the system exhibits strong mixing properties these bounds are explicitly expressed in terms of the speed of mixing using these approximations the poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval

exact travelling solutions for the lax s seventh order kdv equation by sech method and rational exp function method in this paper we establish exact solutions for a nonlinear evolution equation the sech method and the exp function method are used to construct the solitary travelling wave solutions of lax s seventh order kdv equation these solutions may be important of significance for the explanation of some practical physical problem