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Without loss of generality the ABC systems reduce to two cases: either A = 0 and B, C >= 0, or A = 1 and 0 < B, C <= 1. In the first case it is known that the ABC system is completely integrable, here we provide its explicit first integrals. In the second case Ziglin ['Dichotomy of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom,' Izv. Akad. Nauk SSSR, Ser. Mat. 51, 1088 (1987)] proved that the ABC system with 0 < B < 1 and C > 0 sufficiently small has no real meromorphic first integrals. We improve Ziglin's result showing that there are no C1 first integrals under convenient assumptions. [ABSTRACT FROM AUTHOR]

We give a mathematically rigorous derivation of Ehrenfest’s equations for the evolution of position and momentum expectation values under general and natural assumptions which include atomic and molecular Hamiltonians with Coulomb interactions. [ABSTRACT FROM AUTHOR]

We cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo [Rep. Math. Phys. 13, 379 (1978)]. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self-adjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line. [ABSTRACT FROM AUTHOR]

We suggest a systematic method of extension of quasiexactly solvable (QES) systems. We construct finite-dimensional subspaces on the basis of special functions (hypergeometric, Airy, Bessel ones) invariant with respect to the action of differential operators of the second order with polynomial coefficients. As an example of physical applications, we show that the known two-photon Rabi Hamiltonian becomes quasiexactly solvable at certain values of parameters when it can be expressed in terms of corresponding QES operators related to the hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weakly dissipative Camassa–Holm equation. First, we try to improve other authors’ results and get some new criterion on blow up, then discuss the global existence of the solution. Finally, we intend to establish sufficient conditions on the propagation speed for the weakly dissipative Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

We present a practical algorithm for computing first integrals of motion which are polynomial in the momenta for natural Hamiltonian systems defined in a flat pseudo-Riemannian space of arbitrary dimension and signature. We then apply our algorithm to explore the integrability of two physical systems. First, we study the Holt potential in two dimensions and derive analogous potentials which admit an additional first integral quartic in the momenta. Second, we analyze a class of cylindrically symmetric potentials in three-dimensional Euclidean space and recover known families of second-order maximally superintegrable systems. [ABSTRACT FROM AUTHOR]

The idealization of a two-dimensional, ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics. Several examples are given including the integrability of the three-vortex problem, the interplay of relative equilibria of identical vortices and the roots of certain polynomials, addition formulas for the cotangent and the Weierstraß ζ function, projective geometry, and other topics. The hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The necessary and sufficient conditions are derived for the reducibility of a hydrodynamic type system into a block-diagonal form with k mutually interacting blocks. Applications to the perturbations of the Benney system and to the Hamiltonian systems of partial differential equations are presented. The algebraic identities connecting the Nijenhuis tensors NB(A)(u,v) and NA(u,v) and Haantjes tensors HB(A)(u,v) and HA(u,v) are discovered. [ABSTRACT FROM AUTHOR]

In this paper we perform an exact study of “quantum fidelity” (also called Loschmidt echo) for the time-periodic quantum harmonic oscillator of the following Hamiltonian: H⁁g(t):(P2/2)+f(t)(Q2/2)+(g2/Q2), when compared with the quantum evolution induced by H⁁0(t) (g=0), in the case where f is a T-periodic function and g a real constant. The reference (initial) state is taken to be an arbitrary “generalized coherent state” in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of t=0, this quantum fidelity may recur to its initial value 1 at an infinite sequence of times tk. We discuss the result when the classical motion induced by Hamiltonian H⁁0(t) is assumed to be stable versus unstable. [ABSTRACT FROM AUTHOR]

Consistent tensor products on auxiliary spaces, hereafter denoted “fusion procedures,” and commuting transfer matrices are defined for general quadratic algebras, nondynamical and dynamical, inspired by results on reflection algebras. Applications of these procedures then yield integer-indexed families of commuting Hamiltonians. [ABSTRACT FROM AUTHOR]

We consider the non-Hermitian eigenvalue problems -u″(z)±(iz)mu(z)=λu(z), m>=3, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z plane. We prove that the eigenvalues λ are all real and positive. [ABSTRACT FROM AUTHOR]

A bi-Hamiltonian system is a system of differential equations which can be written in Hamiltonian form in two distinct ways. The applications of Gelfand–Zakharevich bi-Hamiltonian structure, which is an extension of a Poisson–Nijenhuis structure on phase space, has been extensively explored by Falqui, Magri, and Pedroni in the context of separation of variables. It is well known that the integrable Hamiltonian systems defined by the Adler–Kostant–Symes (AKS) scheme contains bi-Hamiltonian structure. In this paper we unveil the connection between Adler–Kostant–Symes formalism applied to loop algebra and the Gelfand–Zakharevich bi-Hamiltonian structure by superposition the results of Fordy and Kulish in the AKS scheme. We also study the commuting flows of the AKS hierachy and its connection to the Zakharov–Shabat hierarchy. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Two important notions of integrability for discrete mappings, algebraic integrability and singularity confinement, have been used for discrete mappings. Algebraic integrability is related to the existence of sufficiently many conserved quantities and singularity confinement is associated with the local analysis of singularities. In this article, the relationship between these two notions is explored for birational autonomous mappings. The main result of this article is that algebraically integrable mappings are shown to have the singularity confinement property. Using this result, the proof of the nonexistence of algebraic conserved quantities for a class of discrete systems is given. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]