Time-dependent linear Hamiltonian systems and quantum mechanics

In this Letter, a theorem on time-dependent linear Hamiltonian systems is recalled and its connection with the Schrödinger equation is discussed. The kernel of the evolution operator of such quantum systems is computed. Furthermore, the Lewis and Riesenfeld theory for systems with many degrees of freedom is generalized.     

Symmetric functions in the Kontsevich-Witten intersection theory of the moduli space of curves

We show that symmetric functions introduced by Di Francesco and co-workers, to prove Kontsevich's formula and Witten's conjecture in the intersection theory of the moduli space of curves are, up to sign, SchurQ-functions.On leave from Institute of Mathematics, Polish Academy of Sciences, Toru, Poland     

Explicit integration of the full symmetric Toda hierarchy and the sorting property

We give an explicit formula for the solution to the initial-value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szegö, and is also interpreted as a consequence of the QR factorization method of Symes. The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridagonal formulae are given for the case of matrices with 2M+1 nonzero diagonals.     

Quadratic transformations for the sixth Painlevé equation

A new type of transformation is found for the Painlevé six equation which can be considered as an analog of the well-known quadratic transformations for hypergeometric functions.     

Tropical Weyl Group Action via Point Configurations and τ-Functions of the q-Painlevé Equations

Starting from certain point sets in the projective plane, we construct a tropical (or subtraction-free birational) representation of Weyl groups over the field of τ-functions. In particular, our construction includes E ₈ ⁽¹⁾ , E ₇ ⁽¹⁾ , E ₆ ⁽¹⁾ and D ₅ ⁽¹⁾ as affine cases; each of them gives rise to the q-difference Painlevé equation.     

Lax representation with spectral parameter on a torus for integrable particle systems

Complete integrability is proved for the most general class of systems of interacting particles on a straight line with the Hamiltonian including elliptic functions of coordinates, depending on seven arbitrary parameters and having the structure defined by the root systems of the classical Lie algebras. The Lax representation for them depends on the spectral parameter given on a complex torus /, where is the lattice of periods of the Jacobi functions dependent on the Hamiltonian parameters. The possibility of constructing explicit solutions to the equations of motion is discussed.     

Relative equilibria of Lagrangian systems with symmetry

We discuss the characterization of relative equilibria of Lagrangian systems with symmetry.     

Variational iteration method for solving an inverse parabolic equation

The variational iteration method is applied to solving an inverse problem of determining an unknown parameter in a linear parabolic equation. This method is based on the use of Lagrange multipliers for identification of optimal values of parameters in a functional. We get a rapid convergent sequence tending to the exact solution of the inverse problem. To show the efficiency of the present method, some interesting examples are presented.     

Nonlinear Grassmannσ-models,Toda equations,and self-dual Einstein equations: supplements to previous papers

We review recent results on the relations between classical solutions of nonlinear-models and Toda equations, and point out some relations among Toda equations (two-dimensions), the continuum Toda equation (three-dimensions), and self-dual Einstein equations (four-dimensions).Partially supported by the Grant-in-Aid for Scientific Research, No. 04640088.     

One-dimensional model of a quantum nonlinear harmonic oscillator

In this paper we study the quantization of the nonlinear oscillator introduced by Mathews and Lakshmanan. This system with position-dependent mass allows a natural quantization procedure and is shown to display shape invariance. Its energy spectrum is found by factorization. The linear harmonic oscillator appears as the λ → 0 limit of this nonlinear oscillator, whose energy spectrum and eigenfunctions are compared to the linear ones.     

Apparent singularities of Fuchsian equations and the Painlevé property for Garnier systems

We study movable singularities of Garnier systems using the connection of the latter with Schlesinger isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered.     

Orbital stability of peakons with nonvanishing boundary for CH and CH-γ equations

In this Letter, we consider the general expressions of peaked traveling wave solutions for CH and CH-γ equations. The orbital stability of these peakons are directly proved in the H¹ norm. Some previous results are extended.     

On pseudo-continuities in the eigenvalue continuum of one-speed neutron transport theory

We report on a new closed-form approximant to the singular eigenfunction transient solution of the one-speed, one-dimensional neutron transport equation with an anisotropic scattering kernel. It is proved that the associated eigenvalue continuum must be pointwise perforated.     

Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations

In this Letter, homotopy perturbation method (HPM) is directly applied to modified Camassa-Holm and Degasperis-Procesi equations to obtain their solitary wave solutions. The results show the applicability, accuracy and efficiency of HPM in solving nonlinear differential equations with fully nonlinear dispersion term. It is predicted that HPM can be widely applied in engineering problems.     

Formulae for null curves deriving from elliptic curves

Any elliptic curve can be realised in the tangent bundle of the complex projective line as a double cover branched at four distinct points on the zero section. Such a curve generates, via classical osculation duality, a null curve in C³${\mathbb{C}}^3$ and thus an algebraic minimal surface in R³${\mathbb{R}}^3$. We derive simple formulae for the coordinate functions of such a null curve.     

Global attractors and invariant measures for non-invertible planar piecewise isometric maps

The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A.