exactly solvable potentials and the concept of shape invariance the author analyses the relationship between shape invariance and exactly solvable potentials using earlier results he shows that all actually known shape invariant potentials with a single parameter can be recovered quite simply extension to the case of shape invariant potentials with two parameters as well as to non shape invariant but exactly solvable potentials are also possible

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

a two level atom coupled to a two dimensional supersymmetric and shape invariant system models a class of bound state problems which represents the coupling of a two level atom with a two dimensional supersymmetric system involving two shape invariant potentials is introduced we study two models with different coupling hamiltonians

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

the mkdv with one half degree of nonlinearity is not integrable the authors show that the mkdv with one half degree of nonlinearity proposed by xiao 1991 does not pass the painleve test contrary to the author s claim

an algebraic approach to shape invariance with reflections we consider a realization of supersymmetric shape invariant quantum systems where generalized ladder operators are constructed through the association of the supersymmetric partner operators the parameter potential translation operator and the reflection operator the basic algebraic relations as well as the system eigenstates and energy eigenvalues are obtained as an illustration we apply the generalized formalism for two important shape invariant potential systems the scarf hyperbolic and the morse potentials

three level coupled systems and parasupersymmetric shape invariance a class of bound state problems which represents the coupling of a three level atom with a two dimensional system involving two shape invariant potentials is introduced we consider second order parasupersymmetric quantum mechanical models and using an algebraic formulation for shape invariant potential systems resolve the eigenvalue problem for these coupled systems considering two possible kinds for the coupling hamiltonian linear and nonlinear in the potential ladder operators an application is given for a couple of shape invariant potentials harmonic oscillator morse potentials

symmetrically coupled higher order nonlinear schrodinger equations singularity analysis and integrability the integrability of a system of two symmetrically coupled higher order nonlinear schrodinger equations with parameter coefficients is tested by means of singularity analysis it is proven that the system passes the painleve test for integrability only in ten distinct cases of which two are new for one of the new cases a lax pair and a multi field generalization are obtained for the other one the equations of the system are uncoupled by a nonlinear transformation

supersymmetric and shape invariant generalization for nonresonant and intensity dependent jaynes cummings systems a class of shape invariant bound state problems which represent transitions in a two level system introduced earlier are generalized to include arbitrary energy splittings between the two levels as well as intensity dependent interactions we show that the coupled channel hamiltonians obtained correspond to the generalizations of the nonresonant and intensity dependent jaynes cummings hamiltonians widely used in quantized theories of lasers in this general context we determine the eigenstates eigenvalues the time evolution matrix and the population inversion matrix factor pacs numbers 0365f 0370 0530 1130p

a generalized jaynes cummings hamiltonian and supersymmetric shape invariance a class of shape invariant bound state problems which represent two level systems are introduced it is shown that the coupled channel hamiltonians obtained correspond to the generalization of the jaynes cummings hamiltonian

completeness of coherent states associated with self similar potentials and ramanujan s integral extension of the beta function a decomposition of identity is given as a complex integral over the coherent states associated with a class of shape invariant self similar potentials there is a remarkable connection between these coherent states and ramanujan s integral extension of the beta function

exponential operators and the algebraic description of quantum confined systems we study the relations and transformations produced by exponential operators the argument of which are dependent on the basic algebraic elements of supersymmetric and shape invariant potential systems and obtain explicit expressions we apply our results to self similar potential systems and to a set of translational shape invariant systems including the morse poschl teller scarf and rosen morse potentials and obtain closed form expressions we show that our results reproduce those obtained for the harmonic oscillator in the appropriate limits

quantum dynamics of parasupersymmetric and shape invariant coupled systems a class of bound state problems which represents the coupling of a three level atom with a two dimensional system involving two shape invariant potentials was introduced in a previous paper in this paper we considered second order parasupersymmetric quantum mechanical models and two possible kinds for the coupling hamiltonian linear and nonlinear in the potential ladder operators in the present paper using an algebraic formulation for shape invariant potential systems we study the quantum dynamics of these coupled systems and obtain the temporal behaviour of some dynamical variables related with the atom and the coupling potentials an application is given for a couple of shape invariant potentials poschl teller self similar potentials

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

criteria for equidistribution of solutions of word equations on sl 2 abstract we study equidistribution of solutions of word equations of the form w x y g in the family of finite groups sl 2 q we provide criteria for equidistribution in terms of the trace polynomial of w this allows us to get an explicit description of certain classes of words possessing the equidistribution property and show that this property is generic within these classes

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

une generalisation de l inegalite de stein lorenzini soient k un corps de caracteristique quelconque k une cloture algebrique de k et p un polynome non constant de n 2 variables a coefficients dans k pour k on note n le nombre de facteurs irreductibles distincts f i d une decomposition de p sur k nous montrons l enonce suivant qui generalise des resultats de stein 1989 et lorenzini 1993 si le polynome p est non compose sur k alors n 1 est au plus egal a min ideg f i 1

on the integrability of schamel s modified korteweg de vries dequation a recent suggestion of integrability of schamel s 1973 modified korteweg de vries equation for ion acoustic waves in a two component plasma is discussed various arguments including ones based on the truncated painleve expansion are presented which support the existence of special structure for this equation however the evidence for complete integrability of schamel s equation is not yet conclusive

bounding the number of remarkable values via jouanolou s theorem abstract in this article we bound the number of remarkable values of a polynomial vector field the proof is short and based on jouanolou s theorem about rational first integrals of planar polynomial derivations our bound is given in term of the size of a newton polygon associated to the vector field we prove that this bound is almost reached

quantum dynamics of a two level system coupled to a shape invariant potential the quantum dynamics of a two level system coupled to a shape invariant potential is investigated shape invariant potentials share an integrability condition called shape invariance which identifies an underlying algebraic structure in general infinite dimensional that transforms the potential parameters such as strength range and diffuseness we determine the time evolution operator the density operator of the system and obtain the temporal behaviour of various dynamical variables by considering either pure or mixed initial states of the system constructed with the generalized coherent state of the shape invariant coupling potential we consider specific examples of shape invariant coupling potentials harmonic oscillator poschl teller and self similar potentials the results obtained for all dynamical variables exhibit rapid oscillations which periodically collapse and regenerate in different ways depending on the coupling potential nature

a new integrable modified korteweg de vries equation with one half degree of nonlinearity an alternative generalized equation which incorporates the modified korteweg de vries mkdv and kdv equations is proposed and from it a new integrable system of the mkdv type is found

the painleve property transformed noteworthy analytic properties of the dym kruskal equation such as the weak painleve property and the pogrebkov property a second order pole at infinity are proven to be exactly the image of the usual painleve property under the ibragimov transformation the transformed painleve analysis is carried out for two infinite sets of evolution equations similar to the dym kruskal equation only four of the equations pass the test being known and integrable

algebraic nature of shape invariant and self similar potentials self similar potentials generalize the concept of shape invariance which was originally introduced to explore exactly solvable potentials in quantum mechanics in this paper it is shown that the previously introduced algebraic approach to the latter can be generalized to the former the infinite lie algebras introduced in this context are shown to be closely related to the q algebras the associated coherent states are investigated

a bosonized formulation of the supersymmetric quantum mechanics for shape invariant potential systems we consider the minimal bosonization realization of supersymmetric shape invariant systems where generalized supercharge operators are constructed using the partner supersymmetric operators the parameter potential translation formalism and the reflection operator we obtain the solution of the eigenvalue equation and study the quantum dynamics of the supersymmetric system including terms in the hamiltonian which are constructed using the combination of the bosonized supercharge operators the connections between the bosonized supersymmetric formalism the bose fermi transformation and the generalization of the deformed heisenberg algebra are discussed as an illustration we apply the generalized formalism for the case of the trigonometric rosen morse potential