tunnelling in multidimensional systems tunnelling is examined from the point of view of analytic continuation of lagrangian manifolds this enables an intrinsically multidimensional approach to many problems we illustrate the ideas by computing energy splittings between congruent tori in a number of integrable systems the examples include an integrable but non separable system which cannot be solved by reduction to one dimensional systems the calculations can all be viewed in a geometrical way that does not make explicit use of separation of coordinates

stability of neuronal pulses composed of concatenated unstable kinks we demonstrate that a traveling pulse solution emerging from the concatenation of two unstable kinks can be stable by means of stability analysis and numerical simulations we show the stability of neuronal pulses action potentials with increasing refractory periods which decompose into two radiationally unstable kinks in the limit these action potentials are solutions of an ultrarefractory version of the fitzhugh nagumo system

the gardner method for symmetries the gardner method traditionally used to generate conservation laws of integrable equations is generalized to generate symmetries the method is demonstrated for the kdv camassa holm and sine gordon equations the method involves identifying a generating symmetry which depends upon a parameter expansion of this symmetry in a formal power series in the parameter then gives the usual infinite hierarchy of symmetries we show that the obtained symmetries commute discuss the relation of the gardner method with lenard recursion both for symmetries and conservation laws and also the connection between the symmetries of continuous integrable equations and their discrete analogues

on the cauchy problem for the integrable modified camassa holm equation with cubic nonlinearity abstract considered in this paper is the modified camassa holm equation with cubic nonlinearity which is integrable and admits the single peaked solitons and multi peakon solutions the short wave limit of this equation is known as the short pulse equation the main investigation is the cauchy problem of the modified camassa holm equation with qualitative properties of its solutions it is firstly shown that the equation is locally well posed in a range of the besov spaces the blow up scenario and the lower bound of the maximal time of existence are then determined a blow up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated

comment on a new integrable equation with cuspons and w m shape peaks solitons j math phys 47 112701 2006 in a recent paper j math phys 47 112701 2006 qiao proposed a new completely integrable wave equation the equation was derived from the two dimensional euler equations and was proven to posses lax pair bi hamiltonian structure and new types of soliton solutions in this comment we show that qiao s equation is a subcase of an integrable equation derived by a method that was proposed by fuchssteiner and fokas physica d 4 47 1981 in the early 1980s we also discuss the existence of the new types of soliton solutions

polynomial poisson algebras for classical superintegrable systems with a third order integral of motion we present polynomial poisson algebras for the eight classical potentials in two dimensional euclidian space that separate in cartesian coordinates and allow a third order integral of motion some of the classical superintegrable potentials do not coincide with quantum ones but are their singular limits we show that all bounded trajectories in these potentials are periodic

integrable hamiltonian systems with vector potentials we investigate integrable two dimensional hamiltonian systems with scalar and vector potentials admitting second invariants which are linear or quadratic in the momenta in the case of a linear second invariant we provide some examples of weakly integrable systems in the case of a quadratic second invariant we recover the classical strongly integrable systems in cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates

existence and uniqueness of travelling waves for a neural network a one dimensional scalar neural network with two stable steady states is analysed it is shown that there exists a unique monotone travelling wave front which joins the two stable states some additional properties of the wave such as the direction of its velocity are discussed

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

darboux transformation and multi soliton solutions of the camassa holm equation and modified camassa holm equation in this paper we propose a new approach to calculate multi soliton solutions of camassa holm ch equation and modified camassa holm mch equation with the aid of darboux transformation dt the new approach simplifies the approach presented by li and zhang proc r soc london ser a 460 2617 2627 2004 we first map the ch and mch equation to a negative order kdv nkdv equation by a reciprocal transformation then we proceed to apply the dt to solve the nkdv equation in the usual way finally we invert the reciprocal transformation to recover the solutions of the ch equation and mch equation

on separable schrodinger equations we classify 1 3 dimensional schrodinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables as a result we get 11 classes of the vector potentials of the electromagnetic field a t x a0 t x a t x providing separability of the corresponding schrodinger equations it is established in particular that the necessary condition for the schrodinger equation to be separable is that the magnetic field must be independent of the spatial variables next we prove that any schrodinger equation admitting variable separation into second order ordinary differential equations can be reduced to one of the 11 separable schrodinger equations mentioned above and carry out variable separation in the latter furthermore we apply the results obtained for separating variables in the hamilton jacobi equation

general nth order integrals of motion in the euclidean plane the general form of an integral of motion that is a polynomial of order n in the momenta is presented for a hamiltonian system in two dimensional euclidean space the classical and the quantum cases are treated separately emphasizing both the similarities and the differences between the two the main application will be to study nth order superintegrable systems that allow separation of variables in the hamilton jacobi and schrodinger equations respectively

superintegrability in three dimensional euclidean space potentials for which the corresponding schrodinger equation is maximally superintegrable in three dimensional euclidean space are studied the quadratic algebra which is associated with each of these potentials is constructed and the bound state wave functions are computed in the separable coordinates

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

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

integrable heterogeneous nonlinear schrodinger equations with dielectric loss lie backlund symmetries the general cubic nonlinear schrodinger equation with heterogeneity and dielectric loss is i t 1 2 xx v x t iu x t r x t 2 all such equations are found that possess a third order lie backlund symmetry each of these equations are then transformed to the standard completely integrable cubic schrodinger equation u v 0 r 1 in addition some useful structure results are established for the time dependent symmetries of the linear 2 1 dimensional schrodinger equation with arbitrary real potential

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

on separable schrodinger maxwell equations we obtain the most general time dependent potential v t x1 x2 enabling separating of variables in the 1 2 dimensional schrodinger equation with the use of this result the four classes of separable schrodinger maxwell equations are constructed

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

conditional lie backlund symmetry and reduction of evolution equations we suggest a generalization of the notion of invariance of a given partial differential equation with respect to a lie backlund vector field such a generalization proves to be effective and enables us to construct principally new ansatz reducing evolution type equations to several ordinary differential equations in the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations ut uxx f u ux with poor lie symmetry and obtain their exact solutions it is shown that these solutions cannot be constructed by means of the symmetry reduction procedure

zero curvature formulations of dual hierarchies zero curvature formulations are given for the dual hierarchies of standard soliton equation hierarchies recently introduced by olver and rosenau including the physically interesting fuchssteiner fokas camassa holm hierarchy

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

necessary conditions for classical super integrability of a certain family of potentials in constant curvature spaces we formulate the necessary conditions for the maximal super integrability of a certain family of classical potentials defined in the constant curvature two dimensional spaces we give examples of homogeneous potentials of degree 2 on as well as their equivalents on and for which these necessary conditions are also sufficient we show explicit forms of the additional first integrals which can always be chosen as a polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta

stable propagation of activity pulses in populations of spiking neurons we investigate the propagation of pulses of spike activity in a neuronal network with feedforward couplings the neurons are of the spike response type with a firing probability that depends linearly on the membrane potential after firing neurons enter a phase of refractoriness spike packets are described in terms of the moments of the firing time distribution so as to allow for an analytical treatment of the evolution of the spike packet as it propagates from one layer to the next analytical results and simulations show that depending on the synaptic coupling strength a stable propagation of the packet with constant waveform is possible crucial for this observation is neither the existence of a firing threshold nor a sigmoidal gain function both are absent in our model but the refractory behavior of the neurons

loop soliton interaction in an integrable nonlinear telegraphy model reciprocal and backlund transformations an integrable telegraphy equation linked in turn to the geometry of pseudospherical surfaces and a class of model nonlinear elastic stress strain laws descriptive of ideally hard material response is investigated a constant coefficient ideal formalism is introduced which leads to a reciprocal transformation between this nonlinear telegraphy equation and a system compatible with the classical sine gordon equation a backlund transformation is then constructed which is used to analyse two loop soliton interaction the latter exhibits a novel exchange particle phenomenon

quasiseparation of variables in the schrodinger equation with a magnetic field we consider a two dimensional integrable hamiltonian system with a vector and scalar potential in quantum mechanics contrary to the case of a pure scalar potential the existence of a second order integral of motion does not guarantee the separation of variables in the schrodinger equation we introduce the concept of quasiseparation of variables and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

a note on traveling fronts and pulses in a firing rate model of a neuronal network we study the activity of a one dimensional synaptically coupled neural network by means of a firing rate model developed by coombes et al physica d 178 2003 their approach incorporates the biologically motivated finite conduction velocity of action potentials into a neural field equation of wilson and cowan type kybernetik 13 1973 the resulting integro differential equation with a space depending delay term under the convolution may exhibit a variety of traveling and stationary patterns in this paper we construct traveling wave solutions for the case of a firing rate given by the heaviside step function exponential synaptic kernel and exponential synaptic footprint in contrast to coombes et al where the model equation is first reduced to an equivalent system of partial differential equations we make the traveling pattern ansatz into the initial integro differential equation we analyse two types of traveling patterns fronts and pulses for which we derive shape and speed we further determine necessary conditions for the linear stability of the traveling waves

separation of variables in the hydrodynamic stability equations the problem of variable separation in the linear stability equations which govern the disturbance behaviour in viscous incompressible fluid flows is discussed the so called direct approach in which a form of the ansatz for a solution with separated variables as well as a form of reduced odes are postulated from the beginning is applied the results of application of the method are the new coordinate systems and the most general forms of basic flows which permit the postulated form of separation of variables then the basic flows are specified by the requirement that they themselves satisfy the navier stokes equations calculations are made for the 1 3 dimensional disturbance equations written in cartesian and cylindrical coordinates the fluid dynamics interpretation and stability properties of some classes of the exact solutions of the navier stokes equations defined as flows for which the stability analysis can be reduced via separation of variables to the eigenvalue problems of ordinary differential equations are discussed the eigenvalue problems are solved numerically with the help of the spectral collocation method based on chebyshev polynomials for some classes of perturbations the eigenvalue problems can be solved analytically those unique examples of exact explicit solution of the nonparallel unsteady flow stability problems provide a very useful test for numerical methods of solution of eigenvalue problems and for methods used in the hydrodynamic stability theory in general

on a classical constrained system and its quantum mechanical counterpart the authors discuss a nonintegrable system of three coupled anharmonic oscillators with certain constraints so that it is transformed into an integrable one after having quantized the system with constraints and the classically equivalent unconstrained integrable system using the bfv algorithm and the ebk quantization method respectively the equivalence of the two quantization procedures is shown by explicitly performing the reduction to the physical subspace on the quantum level

the proper quantum analogue of the birkhoff gustavson method of normal forms a deformation approach resolves problems concerning the ordering of operators and the commutation of quantization with canonical transformation this completes the programme of robnik 1984 to develop a purely algebraic method for the quantization of non integrable hamiltonian systems about a point of stable equilibrium

n loop solitons and their link with the complex harry dym equation n loop solitons are constructed by means of n cusp soliton solutions of the harry dym equation the proposed approach is based on a novel link between the loop soliton equation and the complex harry dym equation considered on a special curve

orbital stability of the train of peakons for an integrable modified camassa holm equation abstract in this paper we consider the modified camassa holm equation with cubic nonlinearity which is integrable and admits the single peakons and multi peakons using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm we prove that the sum of n sufficiently decoupled peakons is orbitally stable in the energy space

superintegrable 3d systems in a magnetic field corresponding to cartesian separation of variables we consider three dimensional superintegrable systems in a magnetic field we study the class of such systems which separate in cartesian coordinates in the limit when the magnetic field vanishes i e possess two second order integrals of motion of the cartesian type for such systems we look for additional integrals up to second order in momenta which make these systems minimally or maximally superintegrable and construct their polynomial algebras of integrals and their trajectories we observe that the structure of the leading order terms of the cartesian type integrals should be considered in a more general form than for the case without magnetic field

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

absolute and convective instabilities of waves on unbounded and large bounded domains abstract instabilities of nonlinear waves on unbounded domains manifest themselves in different ways an absolute instability occurs if the amplitude of localized wave packets grows in time at each fixed point in the domain in contrast convective instabilities are characterized by the fact that even though the overall norm of wave packets grows in time perturbations decay locally at each given point in the unbounded domain wave packets are convected towards infinity in experiments as well as in numerical simulations bounded domains are often more relevant we are interested in the effects that the truncation of the unbounded to a large but bounded domain has on the aforementioned in stability properties of a wave these effects depend upon the boundary conditions that are imposed on the bounded domain we compare the spectra of the linearized evolution operators on unbounded and bounded domains for two classes of boundary conditions it is proved that periodic boundary conditions reproduce the point and essential spectrum on the unbounded domain accurately spectra for separated boundary conditions behave in quite a different way firstly separated boundary conditions may generate additional isolated eigenvalues secondly the essential spectrum on the unbounded domain is in general not approximated by the spectrum on the bounded domain instead the so called absolute spectrum is approximated that corresponds to the essential spectrum on the unbounded domain calculated with certain optimally chosen exponential weights we interpret the difference between the absolute and the essential spectrum in terms of the convective behavior of the wave on the unbounded domain in particular it is demonstrated that the stability of the absolute spectrum implies convective instability of the wave but that convectively unstable waves can destabilize under domain truncation the theoretical predictions are compared with numerical computations

backlund transformation and smooth multisoliton solutions for a modified camassa holm equation with cubic nonlinearity we present a compact parametric representation of the smooth bright multisoliton solutions for the modified camassa holm mch equation with cubic nonlinearity we first transform the mch equation to an associated mch equation through a reciprocal transformation and then find a novel backlund transformation between solutions of the associated mch equation and a model equation for shallow water waves sww introduced by ablowitz et al we combine this result with the expressions of the multisoliton solutions for the sww and modified korteweg de vries equations to obtain the multisoliton solutions of the mch equation subsequently we investigate the properties of the one and two soliton solutions as well as the general multisoliton solutions we show that the smoothness of the solutions is assured only if the amplitude parameters of solitons satisfy certain conditions we also find that at a critical value of the parameter beyond which the solution becomes singular the soliton solution exhibits a different feature from that of the peakon solution of the ch equation then by performing an asymptotic analysis for large time we obtain the formula for the phase shift and confirm the solitonic nature of the multisoliton solutions finally we use the backlund transformation to derive an infinite number of conservation laws of the mch equation

smooth and singular multisoliton solutions of a modified camassa holm equation with cubic nonlinearity and linear dispersion we develop a direct method for solving a modified camassa holm equation with cubic nonlinearity and linear dispersion under the rapidly decreasing boundary condition we obtain a compact parametric representation for the multisoliton solutions and investigate their properties we show that the introduction of a linear dispersive term exhibits various new features in the structure of solutions in particular we find the smooth solitons whose characteristics are different from those of the camassa holm equation as well as the novel types of singular solitons a remarkable feature of the soliton solutions is that the underlying structure of the associated tau functions is the same as that of a model equation for shallow water waves introduced by ablowitz et al 1974 stud appl math 53 249 315 finally we demonstrate that the short wave limit of the soliton solutions recovers the soliton solutions of the short pulse equation which describes the propagation of ultra short optical pulses in nonlinear media

orthogonal and non orthogonal separation of variables in the wave equation utt uxx v x u 0utt uxx v x u 0 we develop a direct approach to the separation of variables in partial differential equations within the framework of this approach the problem of the separation of variables in the wave equation with time independent potential reduces to solving an over determined system of nonlinear differential equations we have succeeded in constructing its general solution and as a result all potentials v x permitting variable separation have been found for each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study it should be noted that the above approach yields both orthogonal and non orthogonal systems of coordinates

traveling waves in a spring block chain sliding down a slope travelling waves in a spring block chain sliding down an inclined plane are studied for a piecewise linear spinodal friction force we construct analytically front waves pulse waves are obtained as the matching of two travelling fronts with identical speeds explicit formulas are obtained for the wavespeed and the wave form in the anti continuum limit the link with propagating phenomena in the burridge knopoff model is briefly discussed

integrable hamiltonian systems with velocity dependent potentials the integrability of a two dimensional hamiltonian in which the potential depends explicitly on the momenta is investigated hamiltonians of this kind are encountered in the description of the motion of a particle in a magnetic field two integrable classes of potentials are identified and the second integral of motion is constructed for each of them the singularity analysis of the equations of motion is also performed confirming once more the relation between the weak painleve property and integrability

on separable pauli equations we classify 1 3 dimensional pauli equations for a spin 12 particle interacting with the electro magnetic field that are solvable by the method of separation of variables as a result we obtain the 11 classes of vector potentials of the electro magnetic field a t x a0 t x a t x providing separability of the corresponding pauli equations it is established in particular that the necessary condition for the pauli equation to be separable into second order matrix ordinary differential equations is its equivalence to the system of two uncoupled schrodinger equations in addition the magnetic field has to be independent of spatial variables we prove that coordinate systems and the vector potentials of the electro magnetic field providing the separability of the corresponding pauli equations coincide with those for the schrodinger equations furthermore an efficient algorithm for constructing all coordinate systems providing the separability of pauli equation with a fixed vector potential of t

stabilization of pulse waves through inhibition in a feedforward neural network we consider a firing rate model of a neural network of excitatory and inhibitory populations with an excitatory feedforward connectivity we analyze traveling wave solutions and determine the conditions for their existence and stability our study demonstrates the role of inhibition in stable pulse propagation in a purely excitatory network pulse waves are unstable because of the existence of stable front wave and back wave with different velocities pulse waves can propagate stably in the network where excitation is appropriately balanced by inhibition analytical results on the wave speeds and the shape of waves are obtained

the evolution of synaptically generated waves in one and two dimensional domains we study the onset and evolution of waves in an integrate and fire network of synaptically coupled neurons in one and two dimensional domains which are restricted to firing one spike we determine the critical size of initial excitation of the network necessary for the onset of propagation we also determine the ignition time we derive an integro differential equation for the evolution of the firing time as a function of spatial position we use the evolution equation to understand propagation failure we compare the behavior of the integrate and fire model with a biophysically based model finally we show that weak heterogeneity has similarly weak effects on the firing times

nonlinear differential difference equations as backlund transformations shows that the best known nonlinear differential difference equations associated with the discrete schrodinger spectral problem and also with the discrete zakharov shabat spectral problem can be interpreted as backlund transformations for some continuous nonlinear evolution equations

hamiltonians separable in cartesian coordinates and third order integrals of motion we present in this article all hamiltonian systems in e 2 that are separable in cartesian coordinates and that admit a third order integral both in quantum and in classical mechanics many of these superintegrable systems are new and it is seen that there exists a relation between quantum superintegrable potentials invariant solutions of the korteweg de vries equation and the painleve transcendents

classical versus quantum integrability classical integrability and quantum integrability are compared for two degrees of freedom hamiltonian systems we use c number representatives for quantum operators and the moyal bracket for the commutator three different cases are found i the c number representative of the quantum mechanical second invariant is identical to the classical second invariant ii o 2 corrections are needed in the classical second invariant to obtain the quantum invariant and iii also the potential must be deformed by an o 2 term several examples from the henon heiles and holt families of integrable potentials are included

remarks on the connection between the additive separation of the hamilton jacobi equation and the multiplicative separation of the schrodinger equation i the completeness and robertson conditions the fundamental elements of the variable separation theory are revisited including the eisenhart and robertson theorems kalnins miller theory and the intrinsic characterization of the separation of the hamilton jacobi equation in a unitary and geometrical perspective the general notion of complete integrability of first order normal systems of pdes leads in a natural way to completeness conditions for separated solutions of the schrodinger equation and to the robertson condition two general types of multiplicative separation for the schrodinger equation are defined and analyzed they are called free and reduced separation respectively in the free separation the coordinates are necessarily orthogonal while the reduced separation may occur in nonorthogonal coordinates but only in the presence of symmetries killing vectors

superintegrable systems in darboux spaces almost all research on superintegrable potentials concerns spaces of constant curvature in this paper we find by exhaustive calculation all superintegrable potentials in the four darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta in addition to the hamiltonian these are two dimensional spaces of nonconstant curvature it turns out that all of these potentials are equivalent to superintegrable potentials in complex euclidean 2 space or on the complex 2 sphere via coupling constant metamorphosis or equivalently via stackel multiplier transformations we present a table of the results

lax integrability and the peakon problem for the modified camassa holm equation peakons are special weak solutions of a class of nonlinear partial differential equations modelling non linear phenomena such as the breakdown of regularity and the onset of shocks we show that the natural concept of weak solutions in the case of the modified camassa holm equation studied in this paper is dictated by the distributional compatibility of its lax pair and as a result it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the burgers type subsequently we give a complete construction of peakon solutions satisfying the modified camassa holm equation in the sense of distributions our approach is based on solving certain inverse boundary value problem the solution of which hinges on a combination of classical techniques of analysis involving stieltjes continued fractions and multi point pade approximations we propose sufficient conditions needed to ensure the global existence of peakon solutions and analyze the large time asymptotic behaviour whose special features include a formation of pairs of peakons that share asymptotic speeds as well as toda like sorting property

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

integrable and superintegrable quantum systems in a magnetic field integrable quantum mechanical systems with magnetic fields are constructed in two dimensional euclidean space the integral of motion is assumed to be a first or second order hermitian operator contrary to the case of purely scalar potentials quadratic integrability does not imply the separation of variables in the schrodinger equation moreover quantum and classical integrable systems do not necessarily coincide the hamiltonian can depend on the planck constant in a nontrivial manner

the modified camassa holm equation backlund transformation and nonlinear superposition formula this paper is concerned with the modified camassa holm equation and its backlund transformation with the aid of reciprocal transformation and the associated modified camassa holm equation a backlund transformation which involves both dependent and independent variables is constructed for the modified camassa holm equation the related nonlinear superposition formula is worked out and applications of both backlund transformation and nonlinear superposition formula are considered

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a riemann hilbert approach to the modified camassa holm equation with nonzero boundary conditions this paper aims at developing the riemann hilbert problem approach to the modified camassa holm mch equation in the case when the solution is assumed to approach a non zero constant at both infinities of the space variable in this case the spectral problem for the associated lax pair equation has a continuous spectrum which allows formulating the inverse spectral problem as a riemann hilbert factorization problem with jump conditions across the real axis we obtain a representation for the solution of the cauchy problem for the mch equation and also a description of certain soliton type solutions both regular and non regular

wave breaking and peakons for a modified camassa holm equation in this paper we investigate the formation of singularities and the existence of peaked traveling wave solutions for a modified camassa holm equation with cubic nonlinearity the equation is known to be integrable and is shown to admit a single peaked soliton and multi peakon solutions of a different character than those of the camassa holm equation singularities of the solutions can occur only in the form of wave breaking and a new wave breaking mechanism for solutions with certain initial profiles is described in detail

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

construction of new integrable hamiltonians in two degrees of freedom a new procedure for deriving integrable hamiltonians and their constants of the motion is introduced we term this procedure the truncation program integrable hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter e the relationship between this program and the whittaker program in two degrees of freedom is discussed integrable hamiltonians occurring in the whittaker program a generalization of whittaker s work possess constants of the motion which are polynomials in the momentum coordinates many previously known integrable hamiltonians are derived a new family of integrable double resonance hamiltonians and a new family of integrable hamiltonians of the form p21 p22 2 v q1 q2 are derived

generalized micz kepler system duality polynomial and deformed oscillator algebras we present the quadratic algebra of the generalized micz kepler system in three dimensional euclidean space e3 and its dual the four dimensional singular oscillator in four dimensional euclidean space e4 we present their realization in terms of a deformed oscillator algebra using the daskaloyannis construction the structure constants are in these cases functions not only of the hamiltonian but also of other integrals commuting with all generators of the quadratic algebra we also present a new algebraic derivation of the energy spectrum of the micz kepler system on the three sphere s3 using a quadratic algebra these results point out also that results and explicit formula for structure functions obtained for quadratic cubic and higher order polynomial algebras in the context of two dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles

variable separation for natural hamiltonians with scalar and vector potentials on riemannian manifolds the additive variable separation in the hamilton jacobi equation is studied for a natural hamiltonian with scalar and vector potentials on a riemannian manifold with positive definite metric the separation of this hamiltonian is related to the separation of a suitable geodesic hamiltonian over an extended riemannian manifold thus the geometrical theory of the geodesic separation is applied and the geometrical characterization of the separation is given in terms of killing webs killing tensors and killing vectors the results are applicable to the case of a nondegenarate separation on a manifold with indefinite metric where no null essential separable coordinates occur

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

new integrable hierarchy its parametric solutions cuspons one peak solitons and m w shape peak solitons in this paper we propose a new completely integrable hierarchy particularly in the hierarchy we draw two new soliton equations 1 mt 12 1 m2 xxx 12 1 m2 x 2 mt mx u2 ux2 2m2ux 0 m u uxx the first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy both equations can be derived from the two dimensional euler equation by using the approximation procedure all equations in the hierarchy are proven to have bi hamiltonian operators and lax pairs through solving a crucial matrix equation moreover we develop parametric solutions of the entire hierarchy through constructing two kinds of constraints one is for all negative members of the hierarchy on a symplectic submanifold and the other is for all positive members in the standard symplectic space the most interesting things are both equations possess new type of peaked solitons continuous and piecewise smooth w m shape peak soliton solutions in addition we find new cusp solit

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

generalized rate code model for neuron ensembles with finite populations we have proposed a generalized langevin type rate code model subjected to multiplicative noise in order to study stationary and dynamical properties of an ensemble containing a finite number n of neurons calculations using the fokker planck equation have shown that owing to the multiplicative noise our rate model yields various kinds of stationary non gaussian distributions such as ensuremath gamma inverse gaussian like and log normal like distributions which have been experimentally observed the dynamical properties of the rate model have been studied with the use of the augmented moment method amm which was previously proposed by the author from a macroscopic point of view for finite unit stochastic systems in the amm the original n dimensional stochastic differential equations des are transformed into three dimensional deterministic des for the means and fluctuations of local and global variables the dynamical responses of the neuron ensemble to pulse and sinusoidal inputs calculated by the amm are in good agreement with those obtained by direct simulation the synchronization in the neuronal ensemble is discussed the variabilities of the firing rate and of the interspike interval are shown to increase with increasing magnitude of multiplicative noise which may be a conceivable origin of the observed large variability in cortical neurons

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases

on superintegrable symmetry breaking potentials in n dimensional euclidean space we give a graphical prescription for obtaining and characterizing all separable coordinates for which the schr dinger equation admits separable solutions for one of the superintegrable potentials here xn 1 is a distinguished cartesian variable the algebra of second order symmetries of the resulting schr dinger equation is given and for the first potential the closure relations of the corresponding quadratic algebra these potentials are particularly interesting because they occur in all dimensions n 1 the separation of variables problem is highly nontrivial for them and many other potentials are limiting cases

the dual modified korteweg de vries fokas qiao equation geometry and local analysis we study a bi hamiltonian equation with cubic nonlinearity shown to appear in the theory of water waves by fokas derived by qiao using the two dimensional euler equation and also known to arise as the dual of the modified korteweg de vries equation thanks to work by fokas fuchssteiner olver and rosenau we present a quadratic pseudo potential we compute infinite sequences of local and nonlocal conservation laws and we construct an infinite dimensional lie algebra of symmetries which contains a semi direct sum of the sl 2 r loop algebra and the centerless virasoro algebra as an application we prove a theorem on the existence of smooth solutions and we construct some explicit examples moreover we consider the cauchy problem and we prove existence and uniqueness of weak solutions in the sobolev space hq 2 r q 1 2

a new integrable equation with cuspons and w m shape peaks solitons in this paper we propose a new completely integrable wave equation mt mx u2 ux2 2m2ux 0 m u uxx the equation is derived from the two dimensional euler equation and is proven to have lax pair and bi hamiltonian structures this equation possesses new cusp solitons cuspons instead of regular peakons ce x ct with speed c through investigating the equation we develop a new kind of soliton solutions w m shape peaks solitons there exist no smooth solitons for this integrable water wave equation

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators