24 research outputs found
Continued fractions and irrationality exponents for modified engel and pierce series
An Engel series is a sum of reciprocals of a non-decreasing
sequence (xn) of positive integers, which is such that each term is divisible
by the previous one, and a Pierce series is an alternating sum of the
reciprocals of a sequence with the same property. Given an arbitrary rational
number, we show that there is a family of Engel series which when
added to it produces a transcendental number ? whose continued fraction
expansion is determined explicitly by the corresponding sequence
(xn), where the latter is generated by a certain nonlinear recurrence of
second order. We also present an analogous result for a rational number
with a Pierce series added to or subtracted from it. In both situations (a
rational number combined with either an Engel or a Pierce series), the
irrationality exponent is bounded below by (3 + ?5)/2, and we further
identify infinite families of transcendental numbers ? whose irrationality
exponent can be computed precisely. In addition, we construct the
continued fraction expansion for an arbitrary rational number added to
an Engel series with the stronger property that x2j divides xj+1 for all
j
Group Analysis of the Novikov Equation
We find the Lie point symmetries of the Novikov equation and demonstrate that
it is strictly self-adjoint. Using the self-adjointness and the recent
technique for constructing conserved vectors associated with symmetries of
differential equations, we find the conservation law corresponding to the
dilations symmetry and show that other symmetries do not provide nontrivial
conservation laws. Then we investigat the invariant solutions
Spherical geometry and integrable systems
We prove that the cosine law for spherical triangles and spherical tetrahedra
defines integrable systems, both in the sense of multidimensional consistency
and in the sense of dynamical systems.Comment: 15 pages, 5 figure
D'Alembertâtype solution of the Cauchy problem for the BoussinesqâKleinâGordon equation
In this paper, we construct a weaklyânonlinear d'Alembertâtype solution of the Cauchy problem for the BoussinesqâKleinâGordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multipleâscales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembertâtype solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the rightâ and leftâpropagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leadingâorder Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values
Analogues of Kahan's method for higher order equations of higher degree
Kahan introduced an explicit method of discretization for systems of first order differential equations with nonlinearities of degree at most two (quadratic vector fields). Kahan's method has attracted much interest due to the fact that it preserves many of the geometrical properties of the original continuous system. In particular, a large number of Hamiltonian systems of quadratic vector fields are known for which their Kahan discretization is a discrete integrable system. In this note, we introduce a special class of explicit order-preserving discretization schemes that are appropriate for certain systems of ordinary differential equations of higher order and higher degree
Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics
We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on \u21022
7 S 2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S 2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W N algebrae, thus providing a gauge theoretical proof of AGT correspondence. \ua9 2014 The Author(s)
Generalizations of the short pulse equation
We classify integrable scalar polynomial partial differential equations of second order generalizing the short pulse equation
A generic travelling wave solution in dissipative laser cavity
A large family of cosh-Gaussian travelling wave solution of a complex GinzburgâLandau equation (CGLE), that describes dissipative semiconductor laser cavity is derived. Using perturbation method, the stability region is identified. Bifurcation analysis is done by smoothly varying the cavity loss coefficient to provide insight of the system dynamics. Heâs variational method is adopted to obtain the standard sech-type and the notso-explored but promising cosh-Gaussian type, travelling wave solutions. For a given set of system parameters, only one sech solution is obtained, whereas several distinct solution points are derived for cosh-Gaussian case. These solutions yield a wide variety of travelling wave profiles, namely Gaussian, near-sech, flat-top and a cosh-Gaussianwith variable central dip. A split-step Fourier method and pseudospectral method have been used for direct numerical solution of the CGLE and travelling wave profiles identical to the analytical profiles have been obtained. We also identified the parametric zone that promises an extremely large family of cosh-Gaussian travelling wave solutions with tunable shape. This suggests that the cosh-Gaussian profile is quite generic and would be helpful for further theoretical as well as experimental investigation on pattern formation, pulse dynamics andlocalization in semiconductor laser cavity
Properties of the series solution for Painleve I
We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painleve equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented