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

finite difference preconditioning for compact scheme discretizations of the poisson equation with variable coefficients abstract the finite difference preconditioning for higher order compact scheme discretizations of non separable poisson s equation is investigated an eigenvalue analysis of a one dimensional problem is detailed for compact schemes up to the tenth order the analysis concludes that the spectrum is bounded irrespective of the mesh size and the continuous variable coefficient hence combined to a multigrid method the preconditioned richardson method shows a convergence rate which is independent from the mesh size and the variable coefficient several numerical experiments including the simulation of a flow with large density variations confirm that the spectrum of the preconditioned operator remains bounded

similarity and conditional similarity reductions of a 2 1 dimensional kdv equation via a direct method to get the similarity solutions of a nonlinear physical equation one may use the classical lie group approach nonclassical lie group approach and the clarkson and kruskal ck direct method in this paper the direct method is modified to get the similarity and conditional similarity reductions of a 2 1 dimensional kdv type equation ten types of usual similarity reductions including the 1 1 dimensional shallow water wave equation and the variable kdv equation and six types of conditional similarity reductions of the 2 1 dimensional kdv equation are obtained some special solutions of the conditional similarity reduction equations are found to show the nontriviality of the conditional similarity reduction approach the conditional similarity solutions cannot be obtained by using the nonclassical lie group approach in its present form how to modify the nonclassical lie group approach to obtain the conditional similarity solutions is still open

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

infinitely many lax pairs and symmetry constraints of the kp equation starting from a known lax pair one can get some infinitely many coupled lax pairs infinitely many nonlocal symmetries and infinitely many new integrable models in some different ways in this paper taking the well known kadomtsev petviashvili kp equation as a special example we show that infinitely many nonhomogeneous linear lax pairs can be obtained by using infinitely many symmetries differentiating the spectral functions with respect to the inner parameters using a known lax pair and the darboux transformations dt infinitely many nonhomogeneous nonlinear lax pairs can also be obtained by means of the infinitely many lax pairs dt and the conformal invariance of the schwartz form of the kp equation infinitely many new nonlocal symmetries can be obtained naturally infinitely many integrable models in 1 1 dimensions 2 1 dimensions 3 1 dimensions and even in higher dimensions can be obtained by virtue of symmetry constraints of the kp equation related to the infinitely many lax pairs

potential symmetries and direct reduction methods of order two for partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened

semi foldon fission and fusion in the 2 1 dimensional higher order broer kaup system we obtain the general variable separation solution of the 2 1 dimensional higher order broer kaup system via the separation variable method and study semi foldon fission and fusion based on this variable separation solution

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

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

nonlinear schrodinger type equations from multiscale reduction of pdes ii necessary conditions of integrability for real pdes the multiscale method has been applied in j math phys 41 6399 2000 to evolution pdes to obtain various nonlinear schrodinger type equations here these results are exploited to derive necessary conditions of integrability for nonlinear real evolution pdes in 1 1 dimensions these conditions are given by explicit formulas and several examples are detailed

note on same result of different ansatz based on extended tanh function method for nonlinear models the same result of different ansatz based on extended tanh function method for nonlinear models is firstly reported as an example we obtain eleven kinds of variable separation solutions of variable coefficient boiti leon pempinelli system based on the extended tanh function method by considering three different ansatz namely positive power form ansatz positive and negative power symmetric ansatz and radical sign combined ansatz by careful analysis we find that these seemly independent variable separation solutions are essentially same from this perspective different ansatz are not really effective to construct so called new solutions therefore we should check carefully solutions obtained by the same method with different ansatz and avoid casually asserting these essentially same new solutions

multiple soliton solutions and fusion interaction phenomena for the 2 1 dimensional modified dispersive water wave system by means of backlund transformation and the hirota bilinear method the 2 1 dimensional modified dispersive water wave system is investigated with the aid of symbolic computation new multiple soliton solutions with arbitrary functions are derived based on the derived solutions rich coherent structures like dromions peakons and compactons can be derived and moreover the fusion interactions among different types of localized structures are graphically studied which might be helpful to understand the propagation processes for nonlinear and dispersive long gravity waves on shallow waters

application of rational expansion method for stochastic differential equations abstract by means of the hermite transformation a new general ansatz and the symbolic computation system maple we apply the riccati equation rational expansion method 24 to uniformly construct a series of stochastic non traveling wave solutions for stochastic differential equations to illustrate the effectiveness of our method we take the stochastic mkdv equation as an example and successfully construct some new and more general solutions the method can also be applied to solve other nonlinear stochastic differential equation or equations

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

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

generalized dromion solutions of the 2 1 dimensional kdv equation the rich dromion structures for a 2 1 dimensional kdv equation are revealed the dromions in a high dimensional integrable model may have a free shape in one or more directions multi dromion solutions can be driven by perpendicular line non perpendicular line and curved line ghost solitons

fft fmm or multigrid a comparative study of state of the art poisson solvers for uniform and nonuniform grids in the unit cube from molecular dynamics and quantum chemistry to plasma physics and computational astrophysics poisson solvers in the unit cube are used in many applications in computational science and engineering in this work we benchmark and discuss the performance of the scalable methods for the poisson problem which are used widely in practice the fast fourier transform fft the fast multipole method fmm the geometric multigrid gmg and algebraic multigrid amg our focus is on solvers supporting high order highly nonuniform discretizations to allow comparisons with standard libraries we also compare adaptive solvers with solvers specialized for problems on regular grids that is fft and regular stencil multigrid since both are very popular algorithms for several practical applications for the multigrid we use the finite element variant of a high performance geometric multigrid hpgmg benchmark in total we compare five different codes three of which are developed in our group our fft gmg and

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

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

two types of generalized integrable decompositions and new solitary wave solutions for the modified kadomtsev petviashvili equation with symbolic computation the modified kadomtsev petviashvili mkp equation is shown in this paper to be decomposable into the first two soliton equations of the 2n coupled chen lee liu and kaup newell hierarchies by respectively nonlinearizing two sets of symmetry lax pairs in these two cases the decomposed 1 1 dimensional nonlinear systems both have a couple of different lax representations which means that there are two linear systems associated with the mkp equation under the same constraint between the potential and eigenfunctions for each lax representation of the decomposed 1 1 dimensional nonlinear systems the corresponding darboux transformation is further constructed such that a series of explicit solutions of the mkp equation can be recursively generated with the assistance of symbolic computation in illustration four new families of solitary wave solutions are presented and the relevant stability is analyzed

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

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

universal and integrable nonlinear evolution systems of equations in 2 1 dimensions integrable systems of nonlinear partial differential equations pdes are obtained from integrable equations in 2 1 dimensions by means of a reduction method of broad applicability based on fourier expansion and spatio temporal rescalings which is asymptotically exact in the limit of weak nonlinearity the integrability by the spectral transform is explicitly demonstrated because the corresponding lax pairs have been derived applying the same reduction method to the lax pair of the initial equation these systems of nonlinear pdes are likely to be of applicative relevance and have a universal character inasmuch as they may be derived from a very large class of nonlinear evolution equations with a linear dispersive part

explicit exact solutions for the generalized zakharov equations with nonlinear terms of any order in this paper an auxiliary ordinary differential equation with nonlinear terms of any order is introduced and its exact solutions are obtained by means of the auxiliary equation and its solutions abundant explicit exact solutions to the generalized zakharov equations with nonlinear terms of any order are obtained in a concise manner that include new solitary wave solutions and triangular periodic wave solutions

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

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

periodic and localized solutions of the long wave short wave resonance interaction equation in this paper we investigate the 2 1 dimensional long wave short wave resonance interaction lsri equation and show that it possess the painleve property we then solve the lsri equation using painleve truncation approach through which we are able to construct solution in terms of three arbitrary functions utilizing the arbitrary functions present in the solution we have generated a wide class of elliptic function periodic wave solutions and exponentially localized solutions such as dromions multidromions instantons multi instantons and bounded solitary wave solutions

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

scaling to the stars a linearly scaling elliptic solver for p multigrid abstract high order methods gain increased attention in computational fluid dynamics however due to the time step restrictions arising from the semi implicit time stepping for the incompressible case the potential advantage of these methods depends critically on efficient elliptic solvers due to the operation counts of operators scaling with the polynomial degree p times the number of degrees of freedom n dof the runtime of the best available multigrid solvers scales with o p n dof this scaling with p significantly lowers the applicability of high order methods to high orders while the operators for residual evaluation can be linearized when using static condensation schwarz type smoothers require their inverses on fixed subdomains no explicit inverse is known in the condensed case and matrix matrix multiplications scale with p n dof this paper derives a matrix free explicit inverse for the static condensed operator in a cuboidal cartesian subdomain it scales with p 3 per element i e n dof globally and allows for a linearly scaling additive schwarz smoother yielding a p multigrid cycle with an operation count of o n dof the resulting solver uses fewer than four iterations for all polynomial degrees to reduce the residual by ten orders and has a runtime scaling linearly with n dof for polynomial degrees at least up to 48 furthermore the runtime is less than one microsecond per unknown over wide parameter ranges when using one core of a cpu leading to time stepping for the incompressible navier stokes equations using as much time for explicitly treated convection terms as for the elliptic solvers

new similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations the author discusses similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations 1972 both of which arise in several physical applications including shallow water waves both equations are thought to non integrable i e not solvable by inverse scattering since numerical studies show that the interaction of solitary waves in inelastic in particular we determine some new similarity reductions of the symmetric regularised long wave equation these new similarity reductions are not obtainable using the classical lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques it is shown that every similarity reduction of both the symmetric regularised long wave and modified benjamin bona mahoney equations obtained using the classical lie group method reduces the associated partial differential equation to an ordinary differential equation of painleve type whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of painleve type it is also shown that neither the symmetric regularised long wave equation nor the modified benjamin bona mahoney equation possesses the painleve property for partial differential equations as defined by weiss et al 1983

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

on the painleve property of nonlinear field equations in 2 1 dimensions the davey stewartson system with the purpose of clarifying some aspects of the complete integrability of nonlinear field equations a singular point analysis is performed of the davey stewartson system which can be considered as an extension in 2 1 dimensions of the nonlinear schrodinger equation it is found that the system under consideration possesses the painleve property and allows a set of backlund transformations obtained by truncating the series expansions of the solutions about the singularity manifold

nonlinear schrodinger type equations from multiscale reduction of pdes i systematic derivation in this article we begin a systematic investigation via multiscale expansions of nonlinear evolution pdes partial differential equations in this first article we restrict consideration to a single autonomous but otherwise generic pde in 1 1 variables space time of first order in time whose linear part is dispersive and to solutions dominated by a single plane wave satisfying the linear part of the pde the expansion parameter is an assumedly small coefficient multiplying this plane wave the main indeed asymptotically exact effect of the weak nonlinearity is then to cause a modulation of the amplitude of the plane wave and of its harmonics which is generally described in appropriately defined coarse grained time and space variables by evolution equations of nonlinear schrodinger type a systematic analysis of such equations is presented corresponding to various assumptions on the resonances occurring for the first few harmonics

fast immersed interface poisson solver for 3d unbounded problems around arbitrary geometries abstract we present a fast and efficient fourier based solver for the poisson problem around an arbitrary geometry in an unbounded 3d domain this solver merges two rewarding approaches the lattice green s function method and the immersed interface method using the sherman morrison woodbury decomposition formula the method is intended to be second order up to the boundary this is verified on two potential flow benchmarks we also further analyse the iterative process and the convergence behavior of the proposed algorithm the method is applicable to a wide range of problems involving a poisson equation around inner bodies which goes well beyond the present validation on potential flows

a reduced order representation of the poincare steklov operator an application to coupled multi physics problems this work investigates a model reduction method applied to coupled multi physics systems the case in which a system of interest interacts with an external system is considered an approximation of the poinste operator is computed by simulating in an offline phase the external problem when the inputs are the laplace beltrami eigenfunctions defined at the interface in the online phase only the reduced representation of the operator is needed to account for the influence of the external problem on the main system an online basis enrichment is proposed in order to guarantee a precise reduced order computation several test cases are proposed on different fluid structure couplings

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

a fast multi resolution lattice green s function method for elliptic difference equations abstract we propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice green s function flgf method the flgf method exploits the regularity of the cartesian mesh and uses the fast multipole method in conjunction with fast fourier transforms to yield linear complexity and decrease time to solution we extend this method to a multi resolution scheme and allow for locally refined cartesian blocks embedded in the computational domain appropriately chosen interpolation and regularization operators retain consistency between the discrete laplace operator and its inverse on the unbounded domain second order accuracy and linear complexity are maintained while significantly reducing the number of degrees of freedom and hence the computational cost

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

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

new similarity solutions for the modified boussinesq equation in this paper the author presents some new similarity solutions of the modified boussinesq equation which is a completely integrable soliton equation these new similarity solutions include reductions to the second and fourth painleve equations which are not obtainable using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques

letter to the editor the davey stewartson equation with sources derived from nonlinear variable separation method in this paper by taking the davey stewartson equation as an example a new construction procedure based on nonlinear variable separation method is presented to obtain nonlinear evolution equations with sources for the first time

a new integrable davey stewartson type equation a new integrable nonlinear partial differential equation pde in 2 1 dimensions is derived starting from the konopelchenko dubrovsky equation we use an asymptotically exact reduction method based on fourier expansion and spatio temporal rescaling and obtain a new integrable davey stewartson type equation in order to demonstrate the integrability of the new equation by the inverse scattering method we apply the reduction technique to the lax pair of the konopelchenko dubrovsky equation and find the corresponding lax pair of the new equation the new equation reduces to the davey stewartson or the nonlinear schrodinger equation by appropriate limits

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

an integral equation method for the cahn hilliard equation in the wetting problem abstract we present an integral equation approach to solving the cahn hilliard equation equipped with boundary conditions that model solid surfaces with prescribed young s angles the discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step to solve it we split the solution into a volume potential computed with free space kernels plus the solution to a second kind integral equation skie the volume potential is evaluated with the help of a box based volume fmm method for non box domains the source density is extended by solving a biharmonic dirichlet problem the near singular boundary integrals are computed using quadrature by expansion qbx with fmm acceleration our method has linear complexity in the number of surface volume degrees of freedom and can achieve high order convergence in space with adaptive refinement to manage error from function extension

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

analysis of the hydrostatic stokes problem and finite element approximation in unstructured meshes the stability of velocity and pressure mixed finite element approximations in general meshes of the hydrostatic stokes problem is studied where two inf sup conditions appear associated to the two constraints of the problem namely incompressibility and hydrostatic pressure since these two constraints have different properties it is not easy to choose finite element spaces satisfying both from the analytical point of view two main results are established the stability of an anisotropic approximation of the velocity using different spaces for horizontal and vertical velocities with piecewise constant pressures and the unstability of standard isotropic approximations which are stable for the stokes problem like the mini element or the taylor hood element moreover we give some numerical simulations which agree with the previous analytical results and allow us to conjecture the stability of some anisotropic approximations of the velocity with continuous piecewise linear pressure in unstructured meshes

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

similarity solutions of the konopelchenko dubrovsky system using lie group theory this research deals with the similarity solutions of 2 1 dimensional konopelchenko dubrovsky kd system solutions so obtained are derived by using similarity transformations method based on lie group theory the method reduces the number of independent variables by one exploiting lie symmetries and using invariance property thus the kd system can further be reduced to a new system of ordinary differential equations under a suitable choice of functions and the arbitrary constants these new equations yield the explicit solutions of the kd system which are discussed in the similarity solutions section of the article moreover the physical analysis of the solutions is illustrated graphically in the analysis and discussions section based on numerical simulations in order to highlight the importance of the study

some more similarity solutions of the 2 1 dimensional blp system some more similarity solutions of the coupled boiti leon pempinelli system have been obtained by using similarity transformations method via lie group theory the solutions are attained under the restrictions imposed on the functions involved in infinitesimals then solutions are investigated through the graphical representation in different cases the cases depend upon some suitable choice of arbitrary functions involved there i e in the solutions to get physically meaningful structures analytical solutions are supplemented by numerical simulations with appropriate choice of arbitrary functions and constants eventually profiles show single soliton elastic multisolitons quasi periodic parabolic and asymptotic nature of solutions

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

lax pairs galore a formula that yields an apparently but only apparently nontrivial lax pair for any nonlinear evolution pde in 1 1 dimensions possessing a local conservation law is presented several examples are exhibited

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

coherent solutions for relativistic vectorial fields approximate interacting localized solutions of a vectorial massive nonlinear equation are obtained by using the asymptotic perturbation ap method based on fourier expansion and spatio temporal rescaling the amplitude slow modulation of fourier modes is described by a system of nonlinear evolution equations solvable via an appropriate change of variables various types of localized solutions dromions lumps ring solitons and breathers as well as multiple soliton and instanton solutions can be explicitly constructed and their interaction is completely elastic because they pass through each other and preserve their shape the only change being a phase shift

the novel solitary wave structures and interactions in the 2 1 dimensional kortweg de vries system abstract using symbolic and algebra computation the extended tanh function method etm based on mapping method is further extended the new variable separation solutions of the 2 1 dimensional korteweg de vries kdv system are derived from the periodic wave solution and by selecting appropriate functions the evolutional behaviors of dromions and compactons on the background of jacobian elliptic wave and their interaction behaviors are investigated

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

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

drone like dynamics of dromion pairs in the 2 1 akns equation abstract we employ truncated painleve approach tpa to the 2 1 dimensional akns equation and construct the solutions in closed form in terms of lower dimensional arbitrary functions of space and time the highlight of our investigation is that we are able to generate dromions undergoing inelastic and elastic collisions we observe that the conventional dromions undergo inelastic collision not only exchanging their energy but also their phase while the dromion pair undergoes elastic collision in particular we observe that we are able to turn on or off the dynamic property of dromion pair by selectively choosing the lower dimensional arbitrary functions with a suitable initial condition similar to drones unmanned aerial vehicles uavs dromion pairs can be driven anywhere in the two dimensional plane by selectively giving the initial conditions in addition to dromions we have also generated a wide class of localized solutions such as rogue waves and lumps we observe that while the rogue waves are found to be unstable and stationary lumps do not interact with other when they travel in the two dimensional plane

the kadomtsev petviashvili equation as a source of integrable model equations a new integrable and nonlinear partial differential equation pde in 2 1 dimensions is obtained by an asymptotically exact reduction method based on fourier expansion and spatiotemporal rescaling from the kadomtsev petviashvili equation the integrability property is explicitly demonstrated by exhibiting the corresponding lax pair that is obtained by applying the reduction technique to the lax pair of the kadomtsev petviashvili equation this model equation is likely to be of applicative relevance because it may be considered a consistent approximation of a large class of nonlinear evolution pdes

a class of integrable davey stewartson type systems a new integrable davey stewartson type class of systems of partial differential equations in 2 1 dimensions is derived from a previously known integrable equation by means of an asymptotically exact nonlinear reduction method based on fourier expansion and spatio temporal rescaling the integrability by the inverse scattering method is explicitly demonstrated because the reduction technique is applied to the lax pair of the starting equation and the corresponding lax pair of the class of systems of equations is found the particular characteristics of the reduction method imply that the new systems are likely to be of applicative relevance

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