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A. Raouf Chouikha

Publications and source records attributed to A. Raouf Chouikha.

At least 19 recordsLinked to original sources
arXiv2018

On the Isochronous analytic motions and the quantum spectrum

The problem of the characterization of all analytic potentials which give rise to isochronous oscillatory motions still open. However, there are several approaches to highlight motions with period $T(E) \equiv T_0$ independent on the energy. It is proposed in this paper to give necessary and sufficient conditions for a center to be isochronous. The proofs produced here are self contained. As corollaries we find again lot of known conditions previously established by different authors. This allows us to produce several class of isochronous analytic motions. We are also interested in the quantum spectrum. We then use the perturbation method WKB to derive an expression for the corrections to the equally spaced valid for analytic isochronous potentials.

math-ph↗
arXiv2014

Complete Monotonicity of classical theta functions and applications

We produce trigonometric expansions for Jacobi theta functions\\ $θ_j(u,τ), j=1,2,3,4$\ where $τ=iπt, t > 0$. This permits us to prove that\ $\log \frac{θ_j(u, t)}{θ_j(0, t)}, j=2,3,4$ and $\log \frac{θ_1(u, t)}{πθ'_1(0, t)}$ as well as $\frac{\frac{δθ_j}{δu}}{θ_j}$ as functions of $t$ are completely monotonic. We also interested in the quotients $S_j(u,v,t) = \frac{θ_j(u/2,iπt)}{θ_j(u/2,iπt)}$. For fixed $u,v$ such that $0\leq u < v < 1$ we prove that the functions $\frac{(\fracδ{δt}S_j)}{S_j}$ for $j=1,4$ as well as the functions $-\frac{(\fracδ{δt}S_j)}{S_j}$ for $j=2,3$ are completely monotonic for $t \in ]0,\infty[$.\\ {\it Key words and phrases} : theta functions, elliptic functions, complete monotonicity.

math.CA↗
arXiv2013

On the period function of Newtonian systems

We study the existence of centers of planar autonomous system of the form $$(S) \quad \dot x=y,\qquad \dot y = -h(x) - g(x)y - f(x)y^2.$$ We are interested in the period function $T$ around a center 0. A sufficient condition for the isochronicity of (S) at 0 is given. Such a condition is also necessary when $f,g,h$ are analytic functions. In that case a characterization of isochronous centers of system (S) is given. Some applications will be derived. In particular, new families of isochronous centers will be described

math.CA↗
arXiv2013

Subharmonic solutions for nonautonomous sublinear first order Hamiltonian systems

In this paper, the existence of subharmonic solutions for a class of non-autonomous first-order Hamiltonian systems is investigated. We also study the minimality of periods for such solutions. Our results which extend and improve many previous results will be illustrated by specific examples. Our main tools are the minimax methods in critical point theory and the least action principle. {\bf Key words.} Hamiltonian systems. Critical point theory. Least action principle. Subharmonic solutions.

math.DS↗
arXiv2012

On the monotonicity criteria of the period function of potential systems

The purpose of this paper is to study various monotonicity conditions of the period function $T(c)$ (energy-dependent) for potential systems $\ddot x + g(x)=0$ with a center at the origin 0. We had before identified a family of new criteria noted by $(C_n)$ which are sometimes thinner than those previously known ({\it Period function and characterizations of Isochronous potentials}\quad arXiv:1109.4611). This fact will be illustrated by examples.

math.CA↗
arXiv2011

Period function and characterizations of Isochronous potentials

We are interested at first in the study of the monotonicity for the period function of the conservative equation \ $(1)\quad \ddot x + g(x) = 0.$\quad Some refinements of known criteria are brought. Moreover, we give necessary and sufficient conditions so that the analytic potential of equation $(1)$ is isochronous. These conditions which are different from those introduced firstly by Koukles and Piskounov and thereafter by Urabe appear sometime to be easier to use. We then apply these results to produce families of isochronous potentials depending on many parameters, some of them are news. Moreover, analytic isochronicity requirements of parametrized potentials will also be considered

math.DS↗
arXiv2010

Isochronicity conditions for some planar polynomial systems II

We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $$\dot x=-y+A(x,y),\;\dot y=x+B(x,y),$$ where $A,\;B\in \mathbb{R}[x,y]$, which can be reduced to the Liénard type equation. When $deg(A)\leq 4$ and $deg(B) \leq 4$, using the so-called C-algorithm we found $36$ new families of isochronous centers. When the Urabe function $h=0$ we provide an explicit general formula for linearization. This paper is a direct continuation of \cite{BoussaadaChouikhaStrelcyn2010} but can be read independantly.

math.CA↗
arXiv2009

Isochronicity conditions for some planar polynomial systems

We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $\dot x=-y+A(x,y), \dot y=x+B(x,y)$, where $A, B\in \mathbb{R}[x,y]$, which can be reduced to the Lienard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.

math.DS↗
arXiv2006

On the existence of periodic solution of perturbed generalized Liénard equations

Under conditions of Levinson-Smith type, we prove the existence of a $τ$-periodic solution for the perturbed generalized Liénard equation u''+ϕ(u,u')u'+ψ(u)=εω(\frac{t}τ,u,u') with periodic forcing term. Also we deduce sufficient condition for existence of a periodic solution for the equation u''+\sum_{k=0}^{2s+1} p_k(u){u'}^k=εω(\frac{t}τ,u,u'). Our method can be applied also to the equation u''+[u^2+(u+u')^2-1]u'+u=εω(\frac{t}τ,u,u'). The results obtained are illustrated with numerical examples.

math.CA↗
arXiv2006

Expansions of Theta Functions and Applications

We prove that the classical theta function $θ_4$ may be expressed as $$ θ_4(v,τ) = θ_4(0,τ) \exp[- \sum_{p\geq 1} \sum_{k\geq 0} \frac {1}{p} \bigg(\frac {\sin πv}{(\sin (k+{1/2})πτ)}\bigg)^{2p}].$$ We obtain an analogous expansion for the three other theta functions since they are related. \\ These results have several consequences. In particular, an expansion of the Weierstrass elliptic function will be derived. Actions of the modular group and other arithmetical properties will also be considered. Finally using a new expression for the Rogers-Ramanujan continued fraction we produce a simple proof of a Rogers identity. {\it Key words and phrases} : theta functions, elliptic functions, q-series, Fourier series, continued fractions

math.NT↗
arXiv2005

Isochronous Centers of Lienard Type Equations and Applications

In this work we study the equation $(E) \ddot x + f(x) \dot x^2 + g(x) = 0$ with a center at 0 and investigate conditions of its isochronicity. When $f$ and $g$ are analytic (not necessary odd) a necessary and sufficient condition for the isochronicity of 0 is given. This approach allows us to present an algorithm for obtained conditions for a point of (E) to be an isochronous center. In particular, we find again by another way the isochrones of the quadratic Loud systems $(L_{D,F})$. Some classes of Kukles are also considered. Moreover, we classify a 5-parameters family of reversible cubic systems with isochronous centers. Key Words and phrases: period function, monotonicity, isochronicity, center, polynomial systems.

math.DS↗
arXiv2005

On the Jacobi Elliptic functions and Applications

In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M. Proceedings and Lectures Notes, A.M.S., vol 22, Providence, 1999, 53-57) permits one establish a differential system. This system is derived from the heat equation and is satisfied by their coefficients. Several applications may be deduced. Other types of expansions for the Jacobi elliptic functions as well as for the Zeta function are examined.

math-ph↗
arXiv2005

Ricci Curvature and Singularities of Constant Scalar Curvature Metrics

In this work we consider periodic spherically symmetric metrics of constant positive scalar curvature on the n-dimensional cylinder called pseudo-cylindric metrics. These metrics belong to the conformal class $[g_0]$ of the Riemannian product $S^1\times S^{n-1}$ : a circle of length $T$ crossed with the (n-1)-dimensional standard sphere. Such metrics have a harmonic Riemannian curvature and a non parallel Ricci tensor, except for the cylindric one. Thus, it appears a natural link between them and the Derdzinski metrics which are warped product and classify a family of Riemannian manifolds. These two families actually differ by conformal transformations. Moreover, we are interested in the multiplicity problem of the pseudo-cylindric metrics in $[g_0]$. We also study the existence problem and the number of the Derdzinski metrics. Furthermore, we prove that the pseudo-cylindric metrics may be expressed in terms of elliptic functions for the dimension $n = 3,4$ and 6 only, and in terms of automorphic functions for any other dimension. This fact allows us to give new bounds for global estimates. Finally, we examine the curvature of the asymptotic pseudo-cylindric metrics which are complete singular Yamabe metrics on the standard sphere punctured of $k$ points. We show that any of such (non trivial) metric has a non parallel Ricci tensor.

math.DG↗
arXiv2004

Remark on a conjecture of conformal transformations of Riemannian manifolds

Ejiri gave a negative answer to a conjecture of Lichnerowicz concerning Riemannian manifolds with constant scalar curvature admitting an infinitesimal non isometric conformal transformation. With this aim he constructed a warped product of a circle of lenght $T$ and a compact manifold. But he omitted in his analysis the condition that $T$ must to be big enough. Here we give an explicit sharp bound $T_0 < T$ that will make the proof complete. Our presentation is self-contained and mainly uses bifurcation techniques. Moreover, we show that there are other such examples and contribute some results to the classification of these manifolds.

math.DG↗