symmetries and algebras of the integrable dispersive long wave equations in 2 1 dimensional spaces similarly to the kadomtsev petviashvili kp equation a set of generalized symmetries with arbitrary functions of t is given by a simple constructable formula for the integrable dispersive long wave equations in 2 1 space dimensions these symmetries constitute an infinite dimensional lie algebra which is a generalization to the known winfinity algebra

complexiton and resonant multiple wave solutions to the 2 1 dimensional konopelchenko dubrovsky equation abstract in this paper the 2 1 dimensional konopelchenko dubrovsky kd equation is investigated using the hirota direct method and linear superposition principle the complexiton and resonant multiple wave solutions are successfully constructed for the hirota direct method it is essential that finding the bilinear form to the kd equation and applying its 2 n soliton formulation in real field to build the n complexiton solutions under the action of pairs of conjugate wave variables the linear superposition in real field of exponential traveling waves can generate the resonant multiple wave solutions and then generalize linear superposition principle to complex field we obtain some new resonant multiple wave type solutions the phenomena of complexiton resonant multiple wave and new resonant multiple wave type solutions are presented by figures

a method for constructing nontraveling wave solutions for 1 1 dimensional evolution equations a method is proposed to construct nontraveling wave solution for 1 1 dimensional evolution equations by extending the linear traveling wave transformation of the g g expansion method into a nonlinear transformation owing to a built in arbitrary function included in the solution abundant solutions can be excited taking the vakhnenko equation as an example a series of nontraveling wave solutions with variable separation is obtained some new solutions are excited and the known solitary wave solutions are special cases of the nontraveling wave solutions

application of the exp function method for nonlinear differential difference equations in this paper we established abundant travelling wave solutions for some nonlinear differential difference equations it is shown that the exp function method with the help of symbolic computation provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics

differential stackel matrices we show that additive separation of variables for linear homogeneous equations of all orders is characterized by differential stackel matrices generalizations of the classical stackel matrices used for multiplicative separation of second order schrodinger equations and additive separation of hamilton jacobi equations we work out the principal properties of these matrices and demonstrate that even for second order laplace equations additive separation may occur when multiplicative separation does not

interactions among periodic waves and solitary waves for a higher dimensional system the general variable separated approach is successfully extended to the 2 1 dimensional physical model and we obtain a universal formula involving arbitrary number of variable separated functions because of the existence of the arbitrary functions in the universal formula many new types of structures of periodic waves for example the periodic periodic interaction waves periodic kink interaction waves periodic peakon interaction waves periodic compacton interaction waves periodic foldon interaction waves etc are investigated both analytically and graphically some novel features or interesting behaviours are revealed

interaction solutions of a 2 1 dimensional dispersive long wave system abstract with the help of the consistent tanh expansion this paper obtains the interaction solutions between solitons and potential burgers waves of a 2 1 dimensional dispersive long wave system based on some known solutions of the potential burgers equation the multiple resonant soliton wave solutions soliton error function wave solutions soliton rational function wave solutions and soliton periodic wave solutions are obtained directly

functional separation of variables for laplace equations in two dimensions the authors say that a solution psi of a partial differential equation in two real variables x1 x2 is functionally separable in these variables if psi x1 x2 phi a x1 b x2 for single variable functions phi a b such that phi a b not 0 they classify all possibilities for regular functional separation in local coordinates for equations of the form delta 2 psi f psi x1 x2 where delta 2 is the laplace beltrami operator on a two dimensional riemannian or pseudo riemannian space if the dependence of f on x1 x2 is non trivial then separation can occur for conformal cartesian coordinates on any space if f g psi then for orthogonal coordinates they find that true functional separation i e separation other than additive or multiplicative occurs precisely for cartesian coordinates in the euclidean and pseudo euclidean planes the sine gordon equation provides an example of this separation for many of these cases the separated solutions a b can be expressed in terms of elliptic functions for non orthogonal coordinates and f g psi true functional separation occurs precisely for cartesian coordinates in the pseudo euclidean plane and for a coordinate system on the hyperboloid of one sheet a pseudo riemannian space of constant curvature

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

dispersionless scalar integrable hierarchies whitham hierarchy and the quasiclassical dressing method the quasiclassical limit of the scalar nonlocal problem is derived and a quasiclassical version of the dressing method is presented dispersionless kadomtsev petviashvili kp modified kp and dispersionless two dimensional toda lattice 2dtl hierarchies are discussed as illustrative examples it is shown that the universal whitham hierarchy is nothing but the ring of symmetries for the quasiclassical problem the reduction problem is discussed and in particular the d2dtl equation of b type is derived

variable separation approach for a differential difference system special toda equation a bilinear variable separation approach is used to construct some special solutions for a differential difference toda equation the semi discrete form of the continuous formula which describes some types of special solutions for many 2 1 dimensional continuous systems is found for a suitable quantity of the differential difference toda equation thus abundant semi discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions

symmetry analysis and exact solutions of the 2 1 dimensional sine gordon system according to the space variable exchange symmetry we change the 2 1 dimensional sine gordon system obtained by konopelchenko and rogers kr to a variant form some types of similarity reductions are obtained by using some lie symmetry analysis from these similarity reductions we find that the soliton structure of the system possesses quite a rich structure the line solitons parallel to the lines x y 0 and x y 0 may have an arbitrary shape in addition to the well known dromion solution which is constructed by two line solitons one may find many other kinds of soliton solutions localized in all directions for instance some kinds of ring type basin like plateau like and bowl like and instanton type soliton solutions can be found directly by selecting the arbitrary functions included in the single soliton solution the dromion solutions may also be constructed by straight line and curved line solitons

on the oscillation of differential transforms of eigenfunction expansions this paper continues the study of p6lya and wiener wille and szegb into the connections between the oscillation of derivatives of a real function and its analytic character in the present paper a sturm liouville operator l is applied successively to an infinitely differentiable function which admits a certain eigenfunction expansion the eigenfunction expansion is assumed to be conservative in the sense of hille several theorems are given which link the frequency of oscillation of latf x to the size of the coefficients of f x and thus to its analytic character

from the special 2 1 toda lattice to the kadomtsev petviashvili equation the nonlinearization of the eigenvalue problems associated with the toda hierarchy and the coupled korteweg de vries kdv hierarchy leads to an integrable symplectic map s and an integrable hamiltonian system h0 respectively it is proved that s and h0 have the same integrals hk the quasi periodic solution of the 2 1 dimensional kadomtsev petviashvili equation is split into three hamiltonian systems h0 h1 h2 while that of the special 2 1 dimensional toda equation is separated into h0 h1 plus the discrete flow generated by the symplectic map s a clear evolution picture of various flows is given through the window of abel jacobi coordinates the explicit theta function solutions are obtained by resorting to this separation technique

the applications of the gauge transformation for the bkp hierarchy abstract in this paper we investigated four applications of the gauge transformation for the bkp hierarchy firstly it is found that the orbit of the gauge transformation for the constrained bkp hierarchy defines a special 2 1 dimensional toda lattice equation structure then the tau function of the bkp hierarchy generated by the gauge transformation is shown to be the pfaffian and the higher fay like identities for the bkp hierarchy is also obtained through the gauge transformation at last the compatibility between the additional symmetry and the gauge transformation of the bkp hierarchy is proven

dromion can be remote controlled the one dromion solution of the davey stewartson 1 equations with driving boundaries is analysed numerically it is shown that the dromion can follow the motion of the crosspoint of the mean flows even if it is not of exact solutions moreover it is found that the localized structure keeps its shape when it is exposed to the forced motion of the mean flows in the case where dromion runs on a circle path the decay ratio of the localized structure versus the radius is given several types of movement of mean flows are also studied it is expected that dromion can be controlled arbitrarily by driving mean flows at the boundaries

novel variable separation solutions and exotic localized excitations via the etm in nonlinear soliton systems in this paper first the etm is applied to obtain variable separation solutions of 2 1 dimensional systems a common formula with some arbitrary functions is derived to describe suitable physical quantities for some 2 1 dimensional models such as the generalized nizhnik novikov veselov davey stewartson broer kaup kupershmidt boiti leon pempinelli integrable kortweg de vries kdv breaking soliton and burgers models the universal formula in tang lou and zhang phys rev e 66 046601 2002 can be simplified to the common formula in the present paper which indicates that redundant process is included there since the easier variable separation form can be employed without loss of generality second this method is successfully generalized to 1 1 dimensional systems such as coupled integrable dispersionless long wave short wave resonance interaction and negative kdv models and obtain another common formula to describe suitable physical fields or potentials of these 1 1 dimensional model

some new fractals behaviors in 2 1 dimensional integrable system by introducing the extended homogeneous balance approach into the 2 1 dimensional integrable system a linearized form of this physical model is established in this paper subsequently after applied the backlund transformation in the system a variable separation solution with the entrance of different arbitrary functions is obtained furthermore by using the weierstrass bessel and jacobian elliptic functions some interesting fractal structures are produced

generalized stackel matrices stackel and differential stackel matrices are generalized so that the matrix elements may be functions of the derivatives of the dependent variable as well as the independent variable the inverses of these matrices are characterized and it is shown that for significant classes of linear and nonlinear partial differential equations variable separation is accomplished via this generalized stackel mechanism

2 1 dimensional compacton solutions with and without completely elastic interaction properties taking the 2 1 dimensional broer kaup kupershmidt system as a simple example some special types of 2 1 dimensional compacton solutions are constructed it is shown that there is quite rich interaction behaviour between two travelling compactons for some types of compactons the interactions among them may not be completely elastic for some others the interactions are completely elastic there is no phase shift for the interactions of the 2 1 dimensional compactons discussed in this paper

the multicomponent eckhaus equation the n component eckhaus equation is c integrable here the dependent variable is an n vector and and are two constant matrices restricted by the condition that both and be hermitian some special cases and some explicit solutions are displayed

formal variable separation approach for nonintegrable models using a formal variable separation approach a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations taking the kdv burgers and modified kdv burgers equations with background interaction as simple examples some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained

localized solutions with chaotic and fractal behaviours in a 2 1 dimensional dispersive long wave system it is known that in high dimensions there exist abundant localized excitations such as dromions lumps ring soliton solutions and so on in this paper the possible chaotic and fractal localized structures are revealed for the 2 1 dimensional dispersive long wave equation the chaotic and fractal dromion and lump patterns of the model are constructed by some types of lower dimensional chaotic and fractal patterns

a new 2 1 dimensional generalization of the modified korteweg devries equation and its infinitely many symmetries starting from the nonlinear version of bilinear negative kadomtsev petviashvili system a new 2 1 dimensional generalization of the modified korteweg devries equation is obtained for this new system a set of infinitely many generalized symmetries is found by applying a formal series symmetry approach and these symmetries constitute a closed infinite dimensional lie algebra which is a generalization of w type algebra

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

extension of the discrete kp hierarchy in this paper we discuss possible generalizations of the discrete kp toda lattice hierarchy as a result we introduce the integrable hierarchy which can be considered as the proper extension of the discrete kp hierarchy such extension forces us to introduce an infinite number of additional multi times t n t n 1 x n t n 2 n 2 whereas ordinary discrete kp is relevant to t 1 it is shown that any subsystem of extended discrete kp attached to multi time t n is in fact equivalent to a bi infinite sequence of continuous kp hierarchies whose lax operators are glued together by compatible gauge transformations this paper can be thought of as a natural continuation and generalization of our previous paper svinin a k 2001 j phys a math gen 34 10559 68

generalized symmetries and w infinity algebras in 3 dimensional toda field theory after establishing a formal theory for getting solutions of one type of high dimensional partial differential equation two sets of generalized symmetries of the 3d toda theory which arises from a particular reduction of the 4d self dual gravity equation are obtained concretely by a simple formula each set of symmetries constitutes a generalized w algebra which contains three types of the usual w algebras as special cases some open questions are discussed

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

multiple rogue wave and breather solutions for the 3 1 dimensional kpi equation abstract by a symbolic computation approach rouge wave solutions with controllable center of the 3 1 dimensional kadomtsev petviashvili i equation kpi can be presented the first and the second order rouge wave will be obtained applying to hirota bilinear method we obtain multiple breather solutions and interaction solutions for the 3 1 dimensional kpi equation

2 1 dimensional m n component akns system painleve integrability infinitely many symmetries similarity reductions and exact solutions the 2 1 dimensional m n component akns system that is derived from the inner parameter dependent symmetry constraint of the kp equation is studied in detail first the painleve integrability of the model is proved by using the standard wtc and kruskal approach using the formal series symmetry approach the generalized kmv symmetry algebra and the related symmetry group are found the two dimensional similarity partial differential equation reductions and the ordinary differential equation reductions are obtained from the generalized kmv symmetry algebra and the direct method abundant localized coherent structures are revealed by the variable separation approach some special types of the localized excitations like the multiple solitoffs dromions lumps ring solitons breathers and instantons are plotted also

separation of variables in the nonlinear wave equation we develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations using it we obtain a number of new two dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions

separation of variables and exact solutions to quasilinear diffusion equations with nonlinear source abstract a solution of a partial differential equation with two real variables t and x is functionally separable in these variables if q u x t for some single variable functions q and in this paper the generalized conditional symmetry approach is used to study the separation of variables of quasilinear diffusion equations with nonlinear source we obtain a complete list of canonical forms for such equations which admit the functionally separable solutions as a result we get broad families of exact solutions to some quasilinear diffusion equations with nonlinear source the behavior and blow up properties of some solutions are described

functionally separable solutions to nonlinear wave equations by group foliation method abstract we apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force u t t a x d u u x x b x q u a x 0 a classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases

a new general type of solitary wave of a 2 1 dimensional system using the extended homogeneous balance method ehbm and variable separation approach vsa an exact variable separation excitation of a 2 1 dimensional system is derived based on the derived excitation a new general type of solitary wave i e semifolded solitary waves and semifoldons sfsws is defined and studied we investigate the behaviours of the interactions for the semifolded localized structures and find that the interactions possess some novel and interesting features

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

the conserved quantities and symmetries of the two dimensional toda lattice hierarchy the conserved quantities and symmetries of the soliton equations included in the 2dtl two dimensional toda lattice hierarchy are considered by means of the theory of the function a typical example is presented for the 2dtl equation that is the simplest nontrivial equation in the hierarchy then by applying suitable reductions the cases of the toda lattice equation and the sine gordon equation are discussed

separation of variables for scalar evolution equations in one space dimension the flow of an autonomous kth order evolution equation in one space dimension is generated by a kth order ordinary differential operator on the space of fields the characteristic fields are the solutions of the characteristic equation of the operator a solution of the evolution equation is separable if and only if it is a curve in the k c 1 parameter space of characteristic fields the non stationary characteristic fields with separable evolution are those which remain characteristic under small dilation every characteristic field has local separable evolution if and only if the characteristic equation is infinitesimally dilation invariant this is the case when the evolutionary generator is an infinitesimal symmetry of the characteristic equation so that its flow stabilizes the space of characteristic fields the evolution equation is then locally homogeneous if and only if the characteristic value is an invariant of the restricted flow which is a non generic property the separable kth order evolution equations are parametrized by the kth order invariants of the locally homogeneous k c 1 th order ordinary differential equations

group theoretical analysis of dispersive long wave equations in two space dimensions abstract the symmetry algebra of an integrable dispersive long wave equation in two space dimensions is shown to be infinite dimensional and to have a kac moody virasoro structure the corresponding symmetry group is used to generate a large number of new solutions in particular solitons kinks and periodic waves these waves have wave crests of quite general shapes the results are compared to those for a nonintegrable two space dimensional dispersive long wave equation for which the symmetry group is finite dimensional and wave crests are always straight lines

applications of linear superposition principle to resonant solitons and complexitons we apply the linear superposition principle to hirota bilinear equations and generalized bilinear equations by extending the linear superposition principle to complex field we construct complex exponential wave function solutions first and then get complexions by taking pairs of conjugate parameters a few examples of mixed resonant solitons and complexitons to hirota and generalized bilinear differential equations are presented

separation of variables of a generalized porous medium equation with nonlinear source this paper considers a general form of the porous medium equation with nonlinear source term ut d u u n x f u n 1 the functional separation of variables of this equation is studied by using the generalized conditional symmetry approach we obtain a complete list of canonical forms for such equations which admit the functional separable solutions as a consequence some exact solutions to the resulting equations are constructed and their behavior are also investigated 2002 elsevier science usa all rights reserved

symbolic computation of jacobi elliptic function solutions to nonlinear differential difference equations in this paper an algorithm is presented to find exact polynomial solutions of nonlinear differential difference equations ddes in terms of the jacobi elliptic functions the key steps of the algorithm are illustrated by the discretized mkdv lattice a maple package jacobi is developed based on the algorithm to automatically compute special solutions of nonlinear ddes the effectiveness of the package is demonstrated by applying it to a variety of equations

nonlinear evolution equations admitting multilinear variable separable solutions based on the hirota s bilinear transformation method it is shown that a family of nonlinear partial differential equations admit multilinear variable separation solutions this extends the class of nonlinear partial differential equations which admit the multilinear variable separation solutions

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

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

remarks on several 2 1 dimensional lattices in this paper five 2 1 dimensional lattices considered by several authors are revisited again first of all we will show that two lattices proposed by blaszak and szum j math phys 42 225 2001 become the so called differential difference kp equation due to date jimbo and miwa j phys soc jpn 51 4116 1982 51 4125 1982 52 388 1983 52 761 1983 52 766 1983 by simple variable transformations while another lattice found by blaszak and szum can be viewed as a higher dimensional generalization of a lattice given by wu and hu j phys a 32 1515 1999 some integrable properties on these three lattices are derived second it is shown that a 2 1 dimensional toda like lattice studied by cao geng and wu j phys a 32 8059 1999 can be transformed into the bilinear equation given by hu clarkson and bullough j phys a 30 l669 1997 for this bilinear version we also present some rational solutions and lie symmetries finally a lattice due to levi ragnisco and shabat can j phys 72 439 1994 is transformed into coupled bilinear equations it is shown that these coupled bilinear equations do not have two soliton solutions this further confirms that the lattice under consideration is not completely integrable

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

infinitely many symmetries of the davey stewartson equation using a formal series symmetry method for the davey stewartson equation dse we find that there exist two sets of infinitely many formal series symmetries of the dse however different from the kadomtsev petviashvili equation kpe we failed to get infinitely many truncated symmetries only six truncated symmetries can be obtained simply from the formal series symmetries

fractal structure of ferromagnets the singularity structure analysis following the weiss tabor carnevale approach j weiss m tabor and g carnevale j math phys 24 522 1983 10 1063 1 525721 j weiss m tabor and g carnevale j math phys 25 13 1984 10 1063 1 526009 designed for studying the integrability properties of nonlinear partial differential equations we investigate the singularity structure of a 2 1 dimensional wave equation describing the propagation of polariton solitary waves in a ferromagnetic slab as a result we show that out of any damping instability the system above is integrable looking forward to unveiling its complete integrability we derive its backlund transformation and hirota s bilinearization useful in generating a set of soliton solutions in the wake of such results using the arbitrary functions to enter into the laurent series of solutions to the above system we discuss some typical class of excitations generated from the previous solutions in account of a classification based upon the different expressions of a gene

a new algebraic procedure to construct exact solutions of nonlinear differential difference equations this paper presents a new algebraic procedure to construct exact solutions of selected nonlinear differential difference equations the discrete sine gordon equation and differential difference asymmetric nizhnik novikov veselov equations are chosen as examples to illustrate the efficiency and effectiveness of the new procedure where various types of exact travelling wave solutions for these nonlinear differential difference equations have been constructed it is anticipated that the new procedure can also be used to produce solutions for other nonlinear differential difference equations

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

derivative dependent functional separable solutions for the kdv type equations abstract a complete list of canonical forms for a special kdv type of equations which admit derivative dependent functional separable solutions is obtained by using the generalized conditional symmetry approach some exact solutions to the resulting equations are explicitly obtained

on the new approach to variable separation in the time dependent schrodinger equation with two space dimensions we suggest an effective approach to separation of variables in the schrodinger equation with two space variables using it we classify inequivalent potentials v x1 x2 such that the corresponding schrodinger equations admit separation of variables besides that we carry out separation of variables in the schrodinger equation with the anisotropic harmonic oscillator potential v k1x21 k2x22 and obtain a complete list of coordinate systems providing its separability most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free schrodinger equation v 0

separation of variables in 1 2 dimensional schrodinger equations using our classification of separable schrodinger equations with two space dimensions published in j math phys 36 5506 1995 we give an exhaustive description of the coordinate systems providing their separability furthermore we apply these results to separate variables in the heat hamilton jacobi and fokker planck equations

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

equations of arbitrary order invariant under the kadomtsev petviashvili symmetry group by means of a simple new approach a general kadomtsev petviashvili kp family with an arbitrary function of group invariants of arbitrary order is proposed it is proved that the general kp family possesses a common infinite dimensional kac moody virasoro lie point symmetry algebra the known fourth order one can be re obtained as a special example the finite transformation group is presented in a clearer form the kac moody virasoro group invariant solutions and the kac moody group invariant solutions of the kp family are determined by the boussinesq and kdv families respectively

the oscillation of derivatives the bernstein problem for fourier integrals in 1914 s bernstein discovered that if a real valued function f x and all its derivatives are of a constant sign in a closed interval z then f is the restriction to z of an analytic function the problem of determining the analytic character of a function f x from the oscillatory properties of its derivatives has thus become known as the bernstein problem for functions periodic on the real axis the important results are due to polya and wiener and ego

full symmetry groups painleve integrability and exact solutions of the nonisospectral bkp equation abstract based on the generalized symmetry group method presented by lou and ma lou and ma non lie symmetry groups of 2 1 dimensional nonlinear systems obtained from a simple direct method j phys a math gen 38 2005 l129 firstly both the lie point groups and the full symmetry group of the nonisospectral bkp equation are obtained at the same time a relationship is constructed between the new solutions and the old ones of equation secondly the nonisospectral bkp can be proved to be painleve integrability by combining the standard wtc approach with the kruskal s simplification some solutions are obtained by using the standard truncated painleve expansion finally based on the relationship by the generalized symmetry group method and some solutions by using the standard truncated painleve expansion some interesting solution are constructed

complexiton solutions to soliton equations by the hirota method we apply the hirota direct method to construct complexiton solutions complexitons the key is to use hirota bilinear forms we prove that taking pairs of conjugate wave variables in the 2n soliton solutions generates n complexion solutions the general theory is used to construct multi complexion solutions to the korteweg de vries equation