supersymmetric moyal lax representation the super moyal lax representation and the super moyal momentum algebra are introduced and the properties of simple and extended supersymmetric integrable models are systematically investigated it is shown that much like in the bosonic cases the super moyal lax equation can be interpreted as a hamiltonian equation and can be derived from an action similarly we show that the parameter of non commutativity in this case is related to the central charge of the second hamiltonian structure of the system the super moyal lax description allows us to go to the dispersionless limit of these models in a singular limit and we discuss some of the properties of such systems

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

superconformally covariant operators and super w algebras superdifferential operators of order 2n 1 which are covariant with respect to superconformal changes of coordinates on a compact super riemann surface are studied herein it is shown that all such operators arise from super mobius covariant ones a canonical matrix representation is presented and applications to classical super w algebras are discussed

the supersymmetric camassa holm equation and geodesic flow on the superconformal group we study a family of fermionic extensions of the camassa holm equation within this family we identify three interesting classes a equations which are inherently hamiltonian describing geodesic flow with respect to an h1 metric on the group of superconformal transformations in two dimensions b equations which are hamiltonian with respect to a different hamiltonian structure and c supersymmetric equations classes a and b have no intersection but the intersection of classes a and c gives a system with interesting integrability properties we demonstrate the painleve property for some simple but nontrivial reductions of this system and also discuss peakon type solutions

on darboux transformation of the supersymmetric sine gordon equation darboux transformation is constructed for superfields of the super sine gordon equation and the superfields of the associated linear problem the darboux transformation is shown to be related to the super backlund transformation and is further used to obtain n super soliton solutions

moveout analysis of wave equation extended images conventionalvelocityanalysisappliedtoimagesproducedby wave equationmigrationwithacrosscorrelationimagingconditionusesmoveoutinformationfromspacelagsorfocusinginformationfromtimelag however morerobustvelocity estimation methodscanbedesignedtosimultaneouslytakeadvantageofthe semblanceandfocusinginformationprovidedbymigratedimages such a velocity estimation requires characterization of the moveout surfaces defined jointly for space and time lags extended images the analytic solutions to the moveout surfaces can be derived by solving the system of equations representing the shifted source and receiver wavefields the superposition of the surfaces from many experiments shots is equivalent to the envelope for the family of the individual surface the envelope forms a shape that can be characterized as a cone in the extended space of depth space lag and time lag when imaged with the correct velocity the apex of the cone is located at the correct reflectiondepthandatzerospaceandtimelags whenimagedwith the incorrect velocity the apex of the cone shifts in the depth direction and along the time lag axis the characteristics of the cones are directly related to the quality of the velocity model thus their analysis provides a rich source of information for velocity model building synthetic examples verify the derived formulas characterizing the moveout surfaces the analytic formulas match the numeric experiments well demonstrating the accuracy of the formulas based on information provided by the extended imaging condition future application for velocity updates can benefit from the robustness of the depth focusing analysisandofthehighresolutionofthesemblanceanalysis

a new integrable equation valued on a cayley dickson algebra we introduce a new integrable equation valued on a cayley dickson c d algebra in the particular case in which the algebra reduces to a complex one the new interacting term in the equation cancels and the equation becomes the known korteweg de vries kdv equation for each c d algebra the equation has an infinite sequence of local conserved quantities we obtain a backlund transformation in the sense of walhquist estabrook for the equation for any cayley dickson algebra and relate it to a generalized gardner equation from this the infinite sequence of conserved quantities follows directly we give the explicit expression for the first few from the backlund transformation we get the lax pair and the one soliton and two soliton solutions generalizing the known solutions for the quaternion valued kdv equation from the gardner equation we obtain the generalized modified kdv equation which also has an infinite sequence of conserved quantities the new integrable equation is preserved under a subgroup of the automorphisms of the c d algebra in the particular case of the algebra of octonions the equation is invariant under

cylindrical cellular automata a one dimensional cellular automaton with periodic boundary conditions may be viewed as a lattice of sites on a cylinder evolving according to a local interaction rule a technique is described for finding analytically the set of attractors for such an automaton given any one dimensional automaton rule a matrixa is defined such that the number of fixed points on an arbitrary cylinder size is given by the trace ofa n where the powern depends linearly on the cylinder size more generally the number of strings of arbitrary length that appear in limit cycles of any fixed period is found as the solution of a linear recurrence relation derived from the characteristic equation of an associated matrix the technique thus makes it possible for any rule to compute the number of limit cycles of any period on any cylinder size to illustrate the technique closed form expressions are provided for the complete attractor structure of all two neighbor rules the analysis of attractors also identifies shifts as a basic mechanism underlying periodic behavior every limit cycle can be equivalently defined as a set of strings on which the action of the rule is a shift of sizes h i e each string cyclically shifts bys sites inh iterations of the rule the study of shifts provides detailed information on the structure and number of limit cycles for one dimensional automata

a bosonization procedure for hamiltonian theories with fermions hamiltonian theories with fermions are proved to be equivalent to hierarchies of ordinary hamiltonian theories the corresponding poisson brackets are defined in terms of the original super poisson structure while hamiltonian functions are simply the coefficients in the expansion of the super hamiltonian function as a formal power series in grassman generators fermion extensions of the kdv equation are considered to illustrate the general result its space supersymmetric extensions are used to show in particular how supersymmetry transformations can be recast as ordinary hamiltonian symmetries

constructing soliton solutions and super bilinear form of lattice supersymmetric kdv equation the hirota bilinear form and multisoliton solution for semidiscrete and fully discrete difference difference versions of the supersymmetric korteweg de vries kdv equation found by xue et al 2013 j phys a math theor 46 502001 are presented the solitonic interaction term displays a fermionic dressing factor as in the continuous supersymmetric case using bilinear equations it is also shown that a new integrable semidiscrete and fully discrete version of supersymmetric kdv can be constructed with a simpler bilinear form but a more complicated interaction dressing its continuum limit is also computed

n 2 supersymmetric extension of a hydrodynamic system in riemann invariants in this paper we formulate an n 2 supersymmetric extension of a hydrodynamic type system involving riemann invariants the supersymmetric version is constructed by means of a superspace and superfield formalism using bosonic superfields and consists of a system of partial differential equations involving both bosonic and fermionic variables we make use of group theoretical methods in order to analyze the extended model algebraically specifically we calculate a lie superalgebra of symmetries of our supersymmetric model and make use of a general classification method to classify the one dimensional subalgebras into conjugacy classes as a result we obtain a set of 401 one dimensional nonequivalent subalgebras for selected subalgebras we use the symmetry reduction method applied to grassmann valued equations in order to determine analytic exact solutions of our supersymmetric model these solutions include traveling waves bumps kinks double periodic solutions and solutions involving exponentials

a geometrical formulation of fermionic integrable systems a fermionic hamiltonian system is formulated on a supermanifold it is shown that if the system possesses a bi hamiltonian structure one can naturally define a lax equation associated with a 1 1 tensor on this supermanifold and this allows one to construct a set of conserved quantities furthermore if the corresponding nijenhuis tensor vanishes it is shown that all these conserved quantities would be in involution which is a sufficient condition for integrability of the system the fermionic extension of the kdv equation with a bi hamiltonian structure within this geometrical approach is studied

a fully supersymmetric akns theory we construct a fully supersymmetric bihamiltonian theory in four superfields admitting zero curvature and lax formulation this theory is an extension of the classical akns which can be recovered as a reduction other supersymmetric theories are obtained as reductions of the susy akns namely a nonlinear schrodinger a modified kdv and the manin radul kdv the susy nonlinear schrodinger hierarchy is related to the one of roelofs and kersten we determine its bihamiltonian and lax formulation finally we show that the susy kdv s mentioned before are related through a susy miura map

on the nature of the gardner transformation it is shown that every higher korteveg de vries equation can be included in a one parameter family of integrable equations

current algebra aks theorem and new super evolution equations the authors have deduced a new class of integrable super evolution equations by using a supersymmetric version of the aks adler kostant sym theorem in conjunction with the homogeneous space reduction technique of marshall 1988 and current algebra of samenov tian shansky 1983

the bi hamiltonian structure of fully supersymmetric korteweg de vries systems the bi hamiltonian structure of integrable supersymmetric extensions of the korteweg de vries kdv equation related to then 1 and then 2 superconformal algebras is found it turns out that some of these extensions admit inverse hamiltonian formulations in terms of presymplectic operators rather than in terms of poisson tensors for one extension related to then 2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical kdv they can be explained by a factorization of the corresponding lax operator all the bi hamiltonian formulations are derived in a systematic way from the lax operators

on the integrability of the super kdv equation supersymmetric analogues of the gelfand dikii polynomials are derived an alternative proof of the super kdv equation is given using identities obeyed by the polynomials they also allow the recursive generation of the infinite number of conserved quantities for the super kdv flow

a supersymmetric extension of the kadomtsev petviashvili hierarchy an extension of the kadomtsev petviashvili hierarchy by odd variables is given conservation laws and formal integrability are proved

super camassa holm type systems associated to the kuper ramond schwarz superalgebra we describe a class of super integrable systems with a local bi super hamiltonian structure including super camassa holm type systems which can be realized as euler type systems on the dual of the kuper ramond schwarz superalgebra

canonical gauge equivalences of the sakns and stb hierarchies we study the gauge transformations between the supersymmetric akns sakns and supersymmetric two boson stb hierarchies the hamiltonian nature of these gauge transformations is investigated which turns out to be canonical we also obtain the darboux backlund transformations for the sakns hierarchy from these gauge transformations

prolongation structures for supersymmetric equations the well known prolongation technique of wahlquist and estabrook 1975 for nonlinear evolution equations is generalized for supersymmetric equations and applied to the supersymmetric extension of the kdv equation of manin radul using the theory of kac moody lie superalgebras the explicit form of the resulting lie superalgebra is determined it is shown to be isomorphic to rmr cov c 2 sigma where rmr is an eight dimensional radical an auto backlund transformation is derived from the prolongation structure and the relationship with known solution methods of the skdv equation is analysed in addition it is indicated how a super position principle for the skdv equation can be obtained

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

integrable system constructed out of two interacting superconformal fields we consider an interaction between superconformal fields of the same gradation this entails the construction of a supersymmetric poisson tensor for these fields generating a new class of hamiltonian systems the lax representation is found for one of them by supersymmetrizing the lax operator for the hirota satsuma equation the supersymmetric equation is not reducible to the classical hirota satsuma case we show that our generalized system can be reduced to the the supersymmetric kdv equation a 4 surprisingly the integrals of motion are not reduced to integrals of motion of the supersymmetric kdv equation

n 2 supercovariant operators and krichever novikov algebra the n 2 superdiffeomorphism anomaly is defined on the compact n 2 super riemann surfaces by the inclusion of n 2 superprojective connection the n 2 supercovariant operators which are superconformally covariant are constructed and the relations with the n 2 super poisson operators and the krichever novikov algebra are given

on the bihamiltonian structure of the supersymmetric kdv hierarchies a lie superalgebraic approach we give a lie superalgebraic interpretation of the bihamiltonian structure of known supersymmetric kdv equations we show that the loop algebra of a lie superalgebra carries a natural poisson pencil and we subsequently deduce the bihamiltonian structure of the supersymmetric kdv hierarchies by applying to loop superalgebras an appropriate reduction technique this construction can be regarded as a superextension of the drinfeld sokolov method for building a kdv type hierarchy from a simple lie algebra

a supersymmetric version of a gaussian irrotational compressible fluid flow the lie point symmetries and corresponding invariant solutions are obtained for a gaussian irrotational compressible fluid flow a supersymmetric extension of this model is then formulated through the use of a superspace and superfield formalism the lie superalgebra of this extended model is determined and a classification of its subalgebras is performed the method of symmetry reduction is systematically applied in order to derive special classes of invariant solutions of the supersymmetric model several new types of algebraic hyperbolic multi solitonic and doubly periodic solutions are obtained in explicit form

a super hierarchy of the vector nonlinear schrodinger equations and hamiltonian structures conservation laws a super hierarchy of the vector nonlinear schrodinger equations associated with the 3 3 matrix spectral problem is proposed by using the lenard recursion equations and the zero curvature equation the corresponding super bi hamiltonian structures are established with the help of the super trace identity furthermore we derive the infinite conservation flows of the super vector nonlinear schrodinger equation and the second equation in the hierarchy by utilizing spectral parameter expansions

supersymmetric version of a hydrodynamic system in riemann invariants and its solutions in this paper a supersymmetric extension of a system of hydrodynamic type equations involving riemann invariants is formulated in terms of a superspace and superfield formalism the symmetry properties of both the classical and supersymmetric versions of this hydrodynamical model are analyzed through the use of group theoretical methods applied to partial differential equations involving both bosonic and fermionic variables more specifically we compute the lie superalgebras of both models and perform classifications of their respective subalgebras a systematic use of the subalgebra structures allows us to construct several classes of invariant solutions including traveling waves centered waves and solutions involving monomials exponentials and radicals

simultaneous inversion of full data bandwidth by tomographic full waveform inversion abstractthe convergence of full waveform inversion can be improved by extending the velocity model along either the subsurface offset axis or the time lag axis the extension of the velocity model along the time lag axis enables us to linearly model large time shifts caused by velocity perturbations this linear modeling was based on a new linearization of the scalar wave equation in which perturbation of the extended slowness squared was convolved in time with the second time derivative of the background wavefield the linearization was accurate for reflected events and transmitted events we determined that it can effectively model conventional reflection data as well as modern long offset data containing diving waves it also enabled the simultaneous inversion of reflections and diving waves even when the starting velocity model was far from being accurate we solved the optimization problem related to the inversion with a nested algorithm the inner iterations were based on the proposed linearization

global properties of cellular automata cellular automata are discrete mathematical systems that generate diverse often complicated behavior using simple deterministic rules analysis of the local structure of these rules makes possible a description of the global properties of the associated automata a class of cellular automata that generate infinitely many aperiodic temporal sequences is defined as is the set of rules for which inverses exist necessary and sufficient conditions are derived characterizing the classes of nearest neighbor rules for which arbitrary finite initial conditions i evolve to a homogeneous state ii generate at least one constant temporal sequence

multiscale seismic waveform inversion iterative inversion methods have been unsuccessful at inverting seismic data obtained from complicated earth models e g the marmousi model the primary difficulty being the presence of numerous local minima in the objective function the presence of local minima at all scales in the seismic inversion problem prevent iterative methods of inversion from attaining a reasonable degree of convergence to the neighborhood of the global minimum the multigrid method is a technique that improves the performance of iterative inversion by decomposing the problem by scale at long scales there are fewer local minima and those that remain are further apart from each other thus at long scales iterative methods can get closer to the neighborhood of the global minimum we apply the multigrid method to a subsampled low frequency version of the marmousi data set although issues of source estimation source bandwidth and noise are not treated results show that iterative inversion methods perform much better when employed with a decomposition by scale furthermore the method greatly reduces the computational burden of the inversion that will be of importance for 3 d extensions to the method

riemann liouville integrals of fractional order and extended kp hierarchy an attempt to formulate the extensions of the kp hierarchy by introducing fractional order pseudo differential operators is given in the case of the extension with the half order pseudo differential operators a system analogous to the supersymmetric extensions of the kp hierarchy is obtained unlike the supersymmetric extensions no grassmannian variable appears in the hierarchy considered here more general hierarchies constructed by the 1 nth order pseudo differential operators their integrability and the reduction procedure are also investigated in addition to finding the new extensions of the kp hierarchy a brief introduction to the riemann liouville integral is provided to yield a candidate for the fractional order pseudo differential operators

quadratic conformal superalgebras abstract in this paper we shall classify quadratic conformal superalgebras by certain compatible pairs of a lie superalgebra and a novikov superalgebra four general constructions of such pairs are given moreover we shall classify such pairs related to simple novikov algebras

cascaded wavefield tomography and inversion using extended common image point gathers a case study abstractimage domain wavefield tomography wt exploits focusing characteristics of extended images for updating the velocity field to make good use of this information one must understand how such images behave if the migration velocity is accurate this is not trivial because focusing depends on not only the model error but also on the acquisition setup the data bandwidth and illumination variation caused by the overburden we address this problem by constructing penalty functions based on the point spread functions of the imaging operator that characterize focusing in extended images moreover instead of sampling the extended images at preset distances along the surface we sample the image by constructing common image point gathers which are more economical from a computational point of view and also measure the spatial and temporal focusing of the wavefield coupled with image residuals exploiting illumination based penalty functions we construct robust wavefield tomographic updates from spar

fermionic covariant prolongation structure theory for multidimensional super nonlinear evolution equation the fermionic covariant prolongation structure theory is investigated we extend the fermionic covariant prolongation structure technique to the multidimensional super nonlinear evolution equation and present the fermionic covariant fundamental equations determining the prolongation structure furthermore we investigate a 2 1 dimensional super nonlinear schrodinger equation and analyze its integrability by means of this prolongation structure technique we derive its lax representation and backlund transformation moreover we present a solution of this multidimensional super integrable equation

an operator valued extension of the super korteweg de vries equations an extension of the super korteweg de vries kdv integrable system in terms of operator valued functions is obtained following the ideas of gardner a general algebraic approach for finding the infinitely many conserved quantities of integrable systems is presented the approach is applied to the above described system and infinitely many conserved quantities are constructed in a particular case they reduce to the corresponding conserved quantities of super kdv

lax equation lie algebra automorphism and nonlinear supersymmetric dynamical systems following an idea of bogoyavlensky 1988 it is shown that a nonlinear supersymmetric integrable system can be constructed by using the automorphisms of supersymmetric lie algebra

a novel symmetry constraint of the super ckdv system a new 1 1 dimensional integrable system i e the super coupled korteweg de vries ckdv system has been constructed by a super extension of the well known 1 1 dimensional ckdv system for this new system a novel symmetry constraint between the potential and the eigenfunction can be obtained by means of the binary nonlinearization of its lax pairs the constraints for even variables are explicit and the constraints for odd variables are implicit under the symmetry constraint the spacial part and the temporal parts of the equations associated with the lax pairs for the super ckdv system can be decomposed into the super finite dimensional integrable hamiltonian systems on the supersymmetry manifold r4n 2n 2 whose integrals of motion are explicitly given

supersymmetric exact sequence heat kernel and super korteweg de vries hierarchy we introduce the free n 1 supersymmetric derivation ring and prove the existence of an exact sequence of supersymmetric rings and linear transformations we apply necessary and sufficient conditions arising from this exact supersymmetric sequence to obtain the essential relations between conserved quantities gradients and the n 1 super korteweg de vries kdv hierarchy we combine this algebraic approach with an analytic analysis of the super heat operator we obtain the explicit expression for the green s function of the super heat operator in terms of a series expansion and discuss its properties the expansion is convergent under the assumption of bounded bosonic and fermionic potentials we show that the asymptotic expansion when t 0 of the green s function for the superheat operator evaluated over its diagonal generates all the members of the n 1 super kdv hierarchy

integrable fermionic extensions of the garnier system and the anharmonic oscillator we first derive a fermionic extension of the garnier system by applying the method of binary nonlinearization of spectral problem to the supersymmetric kdv equation of kupershmidt and then a fermionic extension of the anharmonic oscillator by a simple reduction the integrable properties of resulting systems such as lax representations corresponding r matrices and conversed integrals of motion are established

bilinearization of n 1 supersymmetric korteweg de vries equation revisited we consider the n 1 supersymmetric korteweg de vries skdv equation within the framework of hirota s bilinear method we construct a backlund transformation which may be interpreted as the modified skdv equation also we find a lax representation and a nonlinear superposition formula by direct applications of the nonlinear superposition formula we calculate soliton solutions for the skdv equation

super loop groups hamiltonian actions and super virasoro algebras the quotient widetilde lg mathord left vphantom widetilde lg g right kern nulldelimiterspace g of a super loop group widetilde lg by the subgroup of constant loops is given a supersymplectic structure and identified through a moment map embedding open image in new window with a coadjoint orbit of the centrally extended super loop algebra open image in new window the algebra widetilde diff c s 1 of super conformal vector fields on the circle is shown to have a natural representation as hamiltonian vector fields on widetilde lg mathord left vphantom widetilde lg g right kern nulldelimiterspace g generated by an equivariant moment map open image in new window this map is obtained by composition of315 8 with a super poisson map open image in new window defining a supersymmetric extension of the classical sugawara formula upon quantization this yields the corresponding formula of kac and todorov on unitary highest weight representations for any homomorphism u 1 g an associated twisted moment map open image in new window is also derived generating a super poisson bracket realization of a super virasoro subalgebra widetilde vir of the semi direct sum open image in new window the corresponding super poisson map open image in new window is interpreted as a nonabelian generalization of the super miura map and applied to two super kdv hierarchies to derive corresponding integrable generalized super mkdv hierarchies in open image in new window

quasi periodic waves of the n 1 supersymmetric modified korteweg de vries equation abstract based on the hirota bilinear method and the riemann theta function a straightforward way is shown to construct quasi periodic wave solutions of supersymmetric equations the resulting theory is applied to the supersymmetric modified korteweg de vries equation further we analyze the asymptotic properties of the solutions and give their asymptotic relations between the quasi periodic wave solutions and the soliton solutions

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

supersymmetric euler equations associated to the n 3 neveu schwarz algebra we give a systematic study about supersymmetric euler equations on the smooth dual nsreg n of the neveu schwarz algebra ns n for n 3 let a be the inertia operator and c1 c2 r we will show that the n 2 supersymmetric euler equation with a c1 c2d2 is local bi super hamiltonian with the freezing point c1 c2 nsreg 2 which is similar to that of the virasoro algebra ns 0 and the n 3 supersymmetric euler equation with a c2d3 1 is local bi super hamiltonian with the freezing point 0 c2 nsreg 3 which is similar to that of the n 1 neveu schwarz algebra ns 1

symmetries of supersymmetric integrable hierarchies of kp type this article is devoted to the systematic study of additional non isospectral symmetries of constrained reduced supersymmetric integrable hierarchies of kp type the so called skp r mb mf models the latter are supersymmetric extensions of ordinary constrained kp hierarchies which contain as special cases basic integrable systems such as m kdv akns fordy kulish yajima oikawa etc as a first main result it is shown that any skp r mb mf hierarchy possesses two different mutually anti commuting types of superloop superalgebra additional symmetries corresponding to the positive and negative grade parts of certain superloop superalgebras the second main result is the systematic construction of the full algebra of additional virasoro symmetries of skp r mb mf hierarchies which requires nontrivial modifications of the virasoro flows known from the general case of unconstrained manin radul super kp hierarchies the latter flows do not define symmetries for constrained skp r mb mf hierarchies as

the painleve analysis for n 2 super korteweg de vries equations the painleve analysis of a generic multiparameter n 2 extension of the korteweg de vries kdv equation is presented unusual aspects of the analysis pertaining to the presence of two fermionic fields are emphasized for the general class of models considered we find that the only ones which manifestly pass the test are precisely the four known integrable supersymmetric kdv equations including the skdv1 case

two super camassa holm equations reciprocal transformations and applications reciprocal transformations are introduced for two super camassa holm ch equations under these transformations and appropriate changes of dependent variables the super ch equation proposed by geng et al stud appl math 130 1 2013 is converted to a negative member of the super korteweg de vries kdv hierarchy studied by geng and wu in 2010 appl math lett 23 716 2010 while the other super ch equation due to zhang and zuo j math phys 52 073503 2011 is related to a new super kdv hierarchy in the latter case algebraic properties of this new super kdv hierarchy are established including hamiltonian operators a recursion operator and conserved quantities

group invariant solutions for the n 2 super korteweg de vries equation the method of symmetry reduction is used to solve grassmann valued differential equations the n 2 supersymmetric korteweg de vries equation is considered it admits a lie superalgebra of symmetries of dimension 5 a two dimensional subsuperalgebra is chosen to reduce the number of independent variables in this equation we are then able to give different types of exact solutions in particular soliton solutions

gardner s deformations of the boussinesq equations using the algebraic method of gardner s deformations for completely integrable systems we construct recurrence relations for densities of the hamiltonians for the boussinesq and the kaup boussinesq equations by extending the magri schemes for these equations we obtain new integrable systems adjoint with respect to the initial ones and describe their hamiltonian structures and symmetry properties in this paper we consider the most efficient way to prove the complete integrability of evolutionary pde systems namely we apply the method of gardner s deformations 1 6 t o the boussinesq and kaup boussinesq equations by construction the deformations consist of the multi hamiltonian scaling non invariant parametric extensions e e of the original systems e0 and the parameter dependent miura transformations me ee e0 inverting the miura transformations from the new systems one obtains the recurrence relations for infinite sequences of densities of the hamiltonian functionals the deformations determine the hamiltonians in the hierarchies and thus they contribute to our knowledge of the geometric integrability picture below we improve a result 3 b y kupershmidt showing that the sequences of conserved densities for the boussinesq equation satisfy two different recurrence relations simultaneously second we emphasize that the method of gardner s deformations is the most powerful instrument for proving the locality 9 of hierarchies indeed the deformations yield local hamiltonians and do not require calculation of the nijenhuis tensors for recursion operators or description of the poisson cohomology groups whose nontriviality precludes infinite motion along the magri schemes third solutions of the deformation problem determine gardner s deformations for the nth

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

wave equation tomography the relation between ray trace and diffraction tomography is usually obscured by formulation of the two methods in different domains the former in space the latter in wavenumber here diffraction tomography is reformulated in the space domain under the title of wave equation tomography with this transformation wave equation tomography projects monochromatic scattered wavefields back over source receiver wavepaths just as ray trace tomography projects traveltime delays back over source receiver raypaths derived under the born approximation these wavepaths are wave theoretic back projection patterns for reflected energy derived under the rytov approximation they are wave theoretic back projection patterns for transmitted energy differences between ray trace and wave equation tomography are examined through comparison of wavepaths and raypaths followed by their application to a transmission geometry synthetic data set rytov wave equation tomography proves superior to ray trace tomography in de

supersymmetric reciprocal transformation and its applications the supersymmetric analog of the reciprocal transformation is introduced this is used to establish a transformation between one of the supersymmetric harry dym equations and the supersymmetric modified korteweg de vries equation the reciprocal transformation as a backlund type transformation between these two equations is adopted to construct a recursion operator for the supersymmetric harry dym equation by proper factorization of the recursion operator a bi hamiltonian structure is found for the supersymmetric harry dym equation furthermore a supersymmetric kawamoto equation is proposed and is associated with the supersymmetric sawada kotera equation the recursion operator and odd bi hamiltonian structure of the supersymmetric kawamoto equation are also constructed

a supertrace identity and its applications to superintegrable systems a supertrace identity on lie superalgebras is established it provides a tool for constructing super hamiltonian structures of zero curvature equations associated with lie superalgebras applications in the case of the lie superalgebra b 0 1 present super hamiltonian structures of a super akns soliton hierarchy and a super dirac soliton hierarchy

the gardner category and nonlocal conservation laws for n 1 super kdv the nonlocal conserved quantities of the n 1 super kdv are obtained using a gardner map a fermionic substitution semigroup and the resulting gardner category are defined and several propositions concerning their algebraic structure are obtained this algebraic framework makes it possible to define general transformations between different nonlinear susy differential equations a susy ring extension is then introduced to deal with the nonlocal conserved quantities of skdv the algebraic version of the nonlocal conserved quantities is solved in terms of the exponential function applied to the d 1 of the local conserved quantities of skdv finally the same formulas are shown to work for rapidly decreasing superfields

supersymmetric extensions of the harry dym hierarchy we study the supersymmetric extensions of the harry dym hierarchy of equations we obtain the susy b extension the doubly susy b extension as well as the n 1 and the n 2 supersymmetric extensions for this system the n 2 supersymmetric extension is particularly interesting since it leads to new classical integrable systems in the bosonic limit we prove the integrability of these systems through the bi hamiltonian formulation of integrable models and through the lax description we also discuss the supersymmetric extension of the hunter zheng equation which belongs to the harry dym hierarchy of equations

supersymmetric extension of the scalar born infeld equation in this paper a supersymmetric extension of the scalar born infeld equation is constructed through a superspace formalism which involves the addition of one independent fermionic grassmann variable to the existing bosonic spacetime coordinates the bosonic scalar field is replaced by a fermionic superfield which is composed of two component fields one bosonic and one fermionic the resulting equation is invariant under a space supersymmetric transformation and in its most general form involves four arbitrary parameters for a certain specific case the lie superalgebra of symmetries was identified and the one dimensional subalgebras were systematically classified into splitting and non splitting conjugate classes a number of group invariant solutions were obtained including polynomial solutions solutions by radicals and solitary waves including bumps kinks and doubly periodic solutions

the fully supersymmetric akns equations the general fully supersymmetric approach to the akns method is presented the supersymmetric kdv and mkdv equations are considered as examples the supersymmetric backlund transformation is constructed for the supersymmetric kdv equation

invariant solutions of supersymmetric nonlinear wave equations systematic group theoretical analyses of two supersymmetric nonlinear wave equations namely the supersymmetric sinh gordon and polynomial klein gordon equations are performed in each case a generalization of the method of prolongations is used to determine the lie superalgebra of symmetries and the method of symmetry reduction is applied in order to obtain invariant solutions of the supersymmetric equations under consideration in the case of the supersymmetric sinh gordon equation the results are compared with those previously found for the supersymmetric sine gordon equation the presence of non standard invariants is discussed for the supersymmetric sinh gordon and polynomial klein gordon equations

n 4 super kdv hierarchy in n 4 and n 2 superspaces we present the results of further analysis of the integrability properties of the n 4 supersymmetric korteweg de vries kdv equation deduced earlier by two of us f delduc and e ivanov phys lett b 309 312 1993 as a hamiltonian flow on n 4 su 2 superconformal algebra in the harmonic n 4 superspace to make this equation and the relevant hamiltonian structures more tractable we reformulate it in the ordinary n 4 and further in n 2 superspaces in n 2 superspace it is represented by a coupled system of evolution equations for a general n 2 superfield and two chiral and antichiral superfields and involves two independent real parameters a and b we construct a few first bosonic conserved charges in involution of dimensions from 1 to 6 and show that they exist only for the following choices of the parameters i a 4 b 0 ii a 2 b 6 iii a 2 b 6 the same values are needed for the relevant evolution equations including n 4 kdv itself to be bi hamiltonian we demonstrate that the above three options are related via su 2 transformations and actually amount to the su 2 covariant integrability condition found in the harmonic superspace approach our results provide a strong evidence that the unique n 4 su 2 super kdv hierarchy exists upon reduction to n 2 kdv the above three possibilities cease to be equivalent they give rise to the a 4 and a 2 n 2 kdv hierarchies which thus prove to be different truncations of the single n 4 su 2 kdv one

limit cycles in a quadratic discrete iteration i study the truncated logistic equation as a map of d integers the number of limit cycles and the size of longest cycle are averaged exhaustively over many values of d for parameter values a beyond the first accumulation point fits of these quantities are compared with estimates obtained from random maps the results suggest some form of self organization i also present some analytical results on the existence and position of fixed points and a fast matrix method to enumerate limit cycles of arbitrary length for given a and d

on the gelfand dickey algebra gd sln and the wn symmetry i the bosonic case the algebra of nonlinear local and nonlocal differential operators acting on the ring of analytic functions is studied it is shown in particular that this space splits into 3 2 special subalgebras jr j 0 1 r 1 each subalgebra is completely specified by quantum numbers s and p q describing the conformal spin and the lowest and the highest degrees respectively the algebra and its dual of local pure nonlocal differential operators is used to calculate the general expression of the gelfand dickey bracket and the wn symmetry poisson one in terms of a set of spin j canonical fields uj 2 j n and a nonlinear u cubic dependent differential operator d n i j z u the explicit form of this operator is worked out other remarkable features are also discussed

the zero curvature formulation of the skdv equations the fermionic extensions of the kdv equation are derived from the zero curvature condition associated with the superalgebra osp 2 1 this derivation clarifies why there are only two such extensions possible and why only one of them is supersymmetric a lenard type of derivation of these equations is also presented

supersymmetric extensions of the nonlinear schrodinger equation symmetries and coverings a construction is proposed for a supersymmetric generalization of the cubic schrodinger equation resulting in two supersymmetric systems one of which contains a free parameter both systems are proven to admit an infinite set of higher order local and nonlocal symmetries and a seemingly infinite set of conservation laws the lowest order terms of which are given explicitly moreover the theory of coverings equivalent to the prolongation method of wahlquist and estabrook is applied to both systems both are seen to admit an infinite dimensional covering algebra the structure of which is determined explicitly resulting in a related supersymmetric system of differential equations as well as an auto backlund transformation for each equation this indicates the complete integrability of both systems

on the fokker planck equation with force term it has been suggested that the half range sturm liouville problem produced in the study of the one dimensional fokker planck equation with constant force term can be replaced by two definite sturm liouville problems the authors show that this appears not to be the case they also draw attention to some corrections to their earlier papers

a new n 2 supersymmetric korteweg de vries equation there are two known integrable n 2 space supersymmetric extensions of the kdv equation both can be written as hamiltonian systems with a common poisson structure which corresponds to the n 2 supersymmetric form of the second kdv structure by analyzing the conservation laws of the one parameter family of equations that share this hamiltonian structure one finds that a third system is singled out by the existence of nontrivial conservation laws it is conjectured to be integrable

variational calculus of supervariables and related algebraic structures abstract we establish a formal variational calculus of supervariables which is a combination of the bosonic theory of gel fand dikii and the fermionic theory in our earlier work certain interesting new algebraic structures are found in connection with hamiltonian superoperators in terms of our theory in particular we find connections between hamiltonian superoperators and novikov poisson algebras that we introduced in our earlier work in order to establish a tensor theory of novikov algebras furthermore we prove that an odd linear hamiltonian superoperator in our variational calculus induces a lie superalgebra which is a natural generalization of the super virasoro algebra under certain conditions

bi hamiltonian structure of a generalized super korteweg de vries equation the bi hamiltonian structure is constructed for a super extension of the kdv equation the related miura map is recovered by a proper gauge transformation and a new spectral problem is presented for the modified system

bilinear approach to kuperschmidt super kdv type equations hirota bilinear form and soliton solutions for the super kdv korteweg de vries equation of kuperschmidt kuper kdv are given it is shown that even though the collision of supersolitons is more complicated than in the case of the supersymmetric kdv equation of manin radul the asymptotic effect of the interaction is simpler as a physical application it is shown that the well known fpu problem having a phonon mediated interaction of some internal degrees of freedom expressed through grassmann fields transforms to the kuper kdv equation in a multiple scale approach

darboux transformations for a twisted derivation and quasideterminant solutions to the super kdv equation this paper is concerned with a generalized type of darboux transformations defined in terms of a twisted derivation d satisfying d ab d a a b where is a homomorphism such twisted derivations include regular derivations difference and q difference operators and superderivatives as special cases remarkably the formulae for the iteration of darboux transformations are identical with those in the standard case of a regular derivation and are expressed in terms of quasideterminants as an example we revisit the darboux transformations for the manin radul super kdv equation studied in liu and manas liu manas 1997 a phys lett b 396 133 140 doi 10 1016 s0370 2693 97 00134 2 the new approach we take enables us to derive a unified expression for solution formulae in terms of quasideterminants covering all cases at once rather than using several subcases then by using a known relationship between quasideterminants and superdeterminants we obtain expressions for these solutions as ratios of superdeterminants this coincides with the results of liu and manas in all the cases they considered but also deals with the one subcase in which they did not obtain such an expression finally we obtain another type of quasideterminant solution to the manin radul super kdv equation constructed from its binary darboux transformations these can also be expressed as ratios of superdeterminants and are a substantial generalization of the solutions constructed using binary darboux transformations in earlier work on this topic

on the bi hamiltonian structures of the skdv the first and the second hamiltonian structures of the supersymmetric kdv equation are derived following the conventional approach of gel fand and dickey in particular it is shown that the nonlocal nature of the first bracket is a direct consequence of the constrained nature of the lax operator

cellular automata and statistical mechanical models we elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata transient and stationary properties of probabilistic automata are linked to surface and bulk properties respectively of restricted statistical mechanical systems it is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata some exact results are derived including duality transformations exact mappings disorder and linear solutions many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties this approach is considered to be a first step toward the design of fully parallel probabilistic systems whose computational abilities rely on the cooperative behavior of their components

integrable hierarchies related to the kuper ch spectral problem in this paper from a given kuper ch spectral problem we propose two kinds of super integrable hierarchies one is the kuper ch hierarchy the other is the generalized kuper harry dym hierarchy moreover we construct their zero curvature representations and super bi hamiltonian structures

invariant strings and pattern recognizing properties of one dimensional cellular automata a cellular automaton is a discrete dynamical system whose evolution is governed by a deterministic rule involving local interactions it is shown that given an arbitrary string of values and an arbitrary neighborhood size representing the range of interaction a simple procedure can be used to find the rules of that neighborhood size under which the string is invariant the set of nearestneighbor rules for which invariant strings exist is completely specified as is the set of strings invariant under each such rule for any automaton rule an associated filtering rule is defined for which the only attractors are spatial sequences consisting of concatenations of invariant strings a result is provided defining the rule of minimum neighborhood size for which an arbitrarily chosen string is the unique invariant string the applications of filtering rules to pattern recognition problems are discussed

multi component bi super hamiltonian kdv systems it is shown that a new class of classical multi component super kdv equations is bi super hamiltonian by extending the method for the verification of graded jacobi identity the multi component extension of super mkdv equations is obtained by using the super miura transformation

on n 3 super korteweg de vries equation a one parameter family is constructed of n 3 supersymmetric and so 3 symmetric extensions of the korteweg de vries kdv equation as a hamiltonian flow on a n 3 superconformal algebra and it is argued that it is nonintegrable for any choice of the parameter then a modified n 3 super kdv equation is proposed which possesses the higher order conserved quantities and so is a candidate for an integrable system upon reduction to n 2 it yields the recently discussed would be integrable version of the n 2 super kdv equation in the bosonic core it contains a coupled system of the kdv type equation and a three component generalization of the mkdv equation a hamiltonian formulation of the new n 3 super kdv equation as a flow on some contraction of the direct sum of two n 3 superconformal algebras is given