Showing papers in "Journal of Nonlinear Mathematical Physics in 2005"
TL;DR: In this article, the authors present results of a search for CAC lattice equations assuming also the same symmetry properties, but not the tetrahedron property, and show that the results of this search are consistent around a cube.
Abstract: For two-dimensional lattice equations the standard definition of integrability is that it should be possible to extend the map consistently to three dimensions, i.e., that it is “consistent around a cube” (CAC). Recently Adler, Bobenko and Suris conducted a search based on this principle, together with the additional assumptions of symmetry and “the tetrahedron property”. We present here results of a search for CAC lattice equations assuming also the same symmetry properties, but not the tetrahedron property.
117 citations
TL;DR: In this paper, the Jacobi last multiplier was used for finding Lie symmetries of first-order systems, and several illustrative examples are given, as well as a brief account of the relationship between Lie symmetric equations and differential equations.
Abstract: After giving a brief account of the Jacobi last multiplier for ordinary differential equations and its known relationship with Lie symmetries, we present a novel application which exploits the Jacobi last multiplier to the purpose of finding Lie symmetries of first-order systems. Several illustrative examples are given.
105 citations
TL;DR: A large part of the theory of classical Bernoulli polynomials follows from their reflection symmetry around x = 1/2: B n (1 − x) = (−1) n B n(x) as mentioned in this paper.
Abstract: A large part of the theory of classical Bernoulli polynomials B n (x)’s follows from their reflection symmetry around x = 1/2: B n (1 − x) = (−1) n B n (x). This symmetry not only survives quantization but has two equivalent forms, classical and quantum, depending upon whether one reflects around 1/2 the classical x or quantum [x] q .
92 citations
TL;DR: In this article, the q,k-generalized Pochhammer symbol was introduced, and integral representations for the generalized gamma and beta functions were provided for Γ q, k and B q, K, respectively.
Abstract: We introduce the q,k-generalized Pochhammer symbol. We construct Γ q,k and B q,k , the q,k-generalized gamma and beta functions, and show that they satisfy properties that generalize those satisfied by the classical gamma and beta functions. Moreover, we provide integral representations for Γ q,k and B q,k .
91 citations
TL;DR: In this paper, it was shown that for any non-zero initial data, the solution of the CamassaHolm equation loses the property of being compactly supported, and for any initial data set, it is shown that the solution is not compact.
Abstract: We give a simple proof that for any non-zero initial data, the solution of the CamassaHolm equation loses instantly the property of being compactly supported.
87 citations
TL;DR: In this paper, it was shown that smooth solutions of the Degasperis-Procesi equation have infinite propagation speed, and that they loose the property of having compact support.
Abstract: We prove that smooth solutions of the Degasperis-Procesi equation have infinite propagation speed: they loose instantly the property of having compact support.
81 citations
TL;DR: In this article, the integrability of a class of 1+1 dimensional models describing non-linear dispersive waves in continuous media was investigated, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc.
Abstract: We investigate the integrability of a class of 1+1 dimensional models describing non-linear dispersive waves in continuous media, e.g. cylindrical compressible hyperelastic rods, shallow water waves, etc. The only completely integrable cases coincide with the Camassa-Holm and Degasperis-Procesi equations.
67 citations
TL;DR: In this article, the rational potentials of the one-dimensional mechanical systems with periodic solutions with the same period (isochronous potentials) have been studied and it was shown that up to a shift and adding a constant all such potentials have the form or.
Abstract: We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form or .
62 citations
TL;DR: Shape invariance is an important ingredient of many exactly solvable quantum mechanics and several examples of shape invariant "discrete quantum mechanical systems" are introduced and discussed in some detail as mentioned in this paper.
Abstract: Shape invariance is an important ingredient of many exactly solvable quantum mechanics. Several examples of shape invariant “discrete quantum mechanical systems” are introduced and discussed in some detail. They arise in the problem of describing the equilibrium positions of Ruijsenaars-Schneider type systems, which are “discrete” counterparts of Calogero and Sutherland systems, the celebrated exactly solvable multi-particle dynamics. Deformed Hermite and Laguerre polynomials are the typical examples of the eigenfunctions of the above shape invariant discrete quantum mechanical systems.
60 citations
TL;DR: In this article, a theory of bidirectional solitons on water is developed by using the classical Boussinesq equation and analytical multi-solitons of Camassa-Holm equation are presented.
Abstract: A theory of bidirectional solitons on water is developed by using the classical Boussinesq equation. Moreover, analytical multi-solitons of Camassa-Holm equation are presented.
59 citations
TL;DR: In this paper, the authors investigated the possible periods of the system for fixed point particles subject to mutual interaction with velocity-dependent forces under the action of a constant magnetic field transverse to the plane of motion.
Abstract: Calogero's goldfish N-body problem describes the motion of N point particles subject to mutual interaction with velocity-dependent forces under the action of a constant magnetic field transverse to the plane of motion. When all coupling constants are equal to one, the model has the property that for generic initial data, all motions of the system are periodic. In this paper we investigate which are the possible periods of the system for fixed N, and we show that there exist initial data that realize each of these possible periods. We then discuss the asymptotic behaviour of the maximal period for large particle number N.
TL;DR: In this article, the authors considered a hard-core model with three states on a homogeneous Cayley tree of order k and showed that in this model, ∀ λ > 0 and k ≥ 1, there are reversible equilibrium distributions for the above process.
Abstract: We consider a nearest-neighbor hard-core model, with three states , on a homogeneous Cayley tree of order k (with k + 1 neighbors). This model arises as a simple example of a loss network with nearest-neighbor exclusion. The state σ(x) at each node x of the Cayley tree can be 0, 1 and 2. We have Poisson flow of calls of rate λ at each site x, each call has an exponential duration of mean 1. If a call finds the node in state 1 or 2 it is lost. If it finds the node in state 0 then things depend on the state of the neighboring sites. If all neighbors are in state 0, the call is accepted and the state of the node becomes 1 or 2 with equal probability 1/2. If at least one neighbor is in state 1, and there is no neighbor in state 2 then the state of the node becomes 1. If at least one neighbor is in state 2 the call is lost. We focus on ‘splitting’ Gibbs measures for this model, which are reversible equilibrium distributions for the above process. We prove that in this model, ∀ λ > 0 and k ≥ 1, there e...
TL;DR: In this paper, a method for the reduction of integrable two-dimensional discrete systems to one-dimensional mappings is presented, which allows for the derivation of non-autonomous systems, which are typically discrete (difference or q) Painleve equations, or of autonomous ones.
Abstract: We present a novel method for the reduction of integrable two-dimensional discrete systems to one-dimensional mappings. The procedure allows for the derivation of nonautonomous systems, which are typically discrete (difference or q) Painleve equations, or of autonomous ones. In the latter case we produce the discrete analogue of an integrable subcase of the Henon-Heiles system.
TL;DR: In this article, an algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed, and conditions for global existence are isolated and convergence of the method is proved in the limit of zero spatial step size and infinite number of particles.
Abstract: An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each “particle” in this method travels along a characteristic curve of the shallow water equation. The resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time. Conditions for global existence are isolated and convergence of the method is proved in the limit of zero spatial step size and infinite number of particles. A fast summation algorithm is introduced to evaluate integrals in the particle method so as to reduce computational cost from O(N 2 ) to O(N), where N is the number of particles. Accuracy tests based on exact solutions and invariants of motion assess the global properties of the method. Finally, results on the study of the nonlinear equation posed in the quarter (space-time) plane are presented. The minimum number of boundary conditions required for solution uniqueness and the complete integrability are discussed in this case, while a modified particle scheme illustrates the evolution of solutions with numerical examples.
TL;DR: In this article, a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x) was considered, where g is either the peakon kernel or one of its degenerations.
Abstract: We consider a family of integro-differential equations depending upon a parameter b as well as a symmetric integral kernel g(x) When b = 2 and g is the peakon kernel (ie g(x) = exp(−|x|) up to rescaling) the dispersionless Camassa-Holm equation results, while the Degasperis-Procesi equation is obtained from the peakon kernel with b = 3 Although these two cases are integrable, generically the corresponding integro-PDE is non-integrable However, for b = 2 the family restricts to the pulson family of Fringer & Holm, which is Hamiltonian and numerically displays elastic scattering of pulses On the other hand, for arbitrary b it is still possible to construct a nonlocal Hamiltonian structure provided that g is the peakon kernel or one of its degenerations: we present a proof of this fact using an associated functional equation for the skew-symmetric antiderivative of g The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon dynamical system, for any value of b 6 1
TL;DR: In this article, the authors present a representation of the complete symmetry group of the Ermakov-Pinney equation using the reduction of order method described in Nucci, J. Nonlin. Math. Phys.
Abstract: The Ermakov-Pinney equation possesses three Lie point symmetries with the algebra sl(2, R). This algebra does not provide a representation of the complete symmetry group of the Ermakov-Pinney equation. We show how the representation of the group can be obtained with the use of the method described in Nucci, J. Nonlin. Math. Phys. 12 (2005) (this issue), which is based on the properties of Jacobi’s last multiplier (Bianchi L, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Enrico Spoerri, Pisa, 1918), the method of reduction of order (Nucci,J. Math. Phys 37 (1996), 1772–1775) and an interactive code for calculating symmetries (Nucci, Interactive REDUCE programs for calcuating classical, non-classical and Lie-Backlund symmetries for differential equations (preprint: Georgia Institute of Technology, Math 062090-051, 1990, and CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 3: New Trends in Theoretical Developments and Computational Methods, Editor: Ibragimov N H...
TL;DR: In this article, the authors consider the cubic and quartic Henon-Heiles Hamiltonians with additional inverse square terms, which pass the Painleve test for only seven sets of coefficients.
Abstract: We consider the cubic and quartic Henon-Heiles Hamiltonians with additional inverse square terms, which pass the Painleve test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties: meromorphy of the general solution, which is hyperelliptic with genus two and completeness in the Painleve sense (impossibility to add any term to the Hamiltonian without destroying the Painleve property).
TL;DR: In this article, a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations are described, where the authors restrict themselves to considering only first order operators, and do not make any assumption as to their symmetry.
Abstract: We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in [7] (Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Research Notices, 2004 : 10 (2004), 461–484) and [15] (Matrix valued orthogonal polynomials of the Jacobi type, Indag. Math. 14 nrs. 3, 4 (2003), 353–366). While we restrict ourselves to considering only first order operators, we do not make any assumption as to their symmetry. For simplicity we restrict to the case N = 2. We draw a few lessons from these examples; many of them serve to illustrate the fundamental difference between the scalar and the matrix valued case.
TL;DR: In this article, the authors determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition.
Abstract: We determine the solutions of a nonlinear Hamilton-Jacobi-Bellman equation which arises in the modelling of mean-variance hedging subject to a terminal condition. Firstly we establish those forms of the equation which admit the maximal number of Lie point symmetries and then examine each in turn. We show that the Lie method is only suitable for an equation of maximal symmetry. We indicate the applicability of the method to cases in which the parametric function depends also upon the time.
TL;DR: In this paper, an integral dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is considered and symmetry constraints for the dBKP hierarchy are studied.
Abstract: Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is considered. Addition formula for the τ-function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry constraints for the dBKP hierarchy are studied.
TL;DR: In this article, several O(N)-invariant classes of hyperbolic equations U tx =f(U, U t, U x ) for an N -component vector U(t, x) are considered.
Abstract: Motivated by recent work on integrable flows of curves and 1+1 dimensional sigma models, several O(N)-invariant classes of hyperbolic equations U tx =f(U, U t , U x ) for an N -component vector U(t, x) are considered. In each class we find all scalinghomogeneous equations admitting a higher symmetry of least possible scaling weight. Sigma model interpretations of these equations are presented.
TL;DR: In this article, the correlation functions of the one-dimensional Asymmetric Simple Exclusion Process (ASEP) with open boundaries are investigated and the conditions for the boundaries are made most general.
Abstract: We investigate the correlation functions of the one-dimensional Asymmetric Simple Exclusion Process (ASEP) with open boundaries. The conditions for the boundaries are made most general. The correlation function is expressed in a multifold integral whose behavior we study in detail. We present a phase diagram of the correlation length. For the case the correlation length diverges, we further give the leading terms of the finite-size correction.
TL;DR: In this article, it was shown that any decomposition of the loop algebra over a simple Lie algebra into a direct sum of the Taylor series and a complementary subalgebra is defined by a pair of compatible Lie brackets.
Abstract: It is shown that any decomposition of the loop algebra over a simple Lie algebra into a direct sum of the Taylor series and a complementary subalgebra is defined by a pair of compatible Lie brackets.
TL;DR: In this paper, the Fokas transform was used to analyze boundary value problems for the sine-Gordon equation posed on a finite interval, and it was shown that the solution corresponding to constant boundary data is dominated for large times by the underlying similarity solution.
Abstract: In this article we use the Fokas transform method to analyze boundary value problems for the sine-Gordon equation posed on a finite interval. The representation of the solution of this problem has already been derived using this transform method. We interchange the role of the independent variables to obtain an equivalent representation which can be used to study the asymptotic behavior for large times. We use this analysis to prove that the solution corresponding to constant boundary data is dominated for large times by the underlying similarity solution. Dedicated to Francesco Calogero in occasion of his 70th birthday
TL;DR: In this paper, a soliton cellular automaton on a one dimensional semi-infinite lattice with a reflecting end is presented, which extends a box-ball system on an infinite lattice associated with the crystal base of the crystal.
Abstract: A soliton cellular automaton on a one dimensional semi-infinite lattice with a reflecting end is presented. It extends a box-ball system on an infinite lattice associated with the crystal base of ....
TL;DR: In this paper, a method of solving n th order scalar ordinary differential equations by extending the ideas based on the Prelle-Singer (PS) procedure for second order ODEs is discussed.
Abstract: We discuss a method of solving n th order scalar ordinary differential equations by extending the ideas based on the Prelle-Singer (PS) procedure for second order ordinary differential equations. We also introduce a novel way of generating additional integrals of motion from a single integral. We illustrate the theory for both second and third order equations with suitable examples. Further, we extend the method to two coupled second order equations and apply the theory to two-dimensional Kepler problem and deduce the constants of motion including Runge-Lenz integral.
TL;DR: It is shown that some effects concerning a Fuzzy Difference Equation of a rational form can be proved.
Abstract: In this paper, we prove some effects concerning a Fuzzy Difference Equation of a rational form.
TL;DR: In this article, two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed, in order to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear.
Abstract: Materials with memory are here considered. The introduction of the dependence on time not only via the present, but also, via the past time represents a way, alternative to the introduction of possible non linearities, when the physical problem under investigation cannot be suitably described by any linear model. Specifically, the two different models of a rigid heat conductor, on one side, and of a viscoelastic body, on the other one, are analyzed. In them both, to evaluate the quantities of physical interest a key role is played by the past history of the material and, accordingly, the behaviour of such materials is characterized by suitable constitutive equations where Volterra type kernels appear. Specifically, in the heat conduction problem, the heat flux is related to the history of the temperature-gradient while, in isothermal viscoelasticity, the stress tensor is related to the strain history. Then, the notion of equivalence is considered to single out and associate together all those dif...
TL;DR: In this article, the Singular Manifold Method is presented as an excellent tool to study a 2 + 1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1 + 1 reductions of the same equation.
Abstract: The Singular Manifold Method is presented as an excellent tool to study a 2 + 1 dimensional equation in despite of the fact that the same method presents several problems when applied to 1 + 1 reductions of the same equation. Nevertheless these problems are solved when the number of dimensions of the equation is increased.
TL;DR: The Somos 4 sequences as mentioned in this paper are a family of sequences satisfying a fourth order bilinear recurrence relation for genus two hyperelliptic curves of genus two defined by the affine model y 2 = 4x 5 +c4x 4 +...+c1x+c0.
Abstract: The Somos 4 sequences are a family of sequences satisfying a fourth order bilinear recurrence relation. In recent work, one of us has proved that the general term in such sequences can be expressed in terms of the Weierstrass sigma function for an associated elliptic curve. Here we derive the analogous family of sequences associated with an hyperelliptic curve of genus two defined by the affine model y 2 = 4x 5 +c4x 4 + ...+c1x+c0. We show that the sequences associated with such curves satisfy bilinear recurrences of order 8. The proof requires an addition formula which involves the genus two Kleinian sigma function with its argument shifted by the Abelian image of the reduced divisor of a single point on the curve. The genus two recurrences are related