Anharmonic Oscillator Equation Research Articles

Reconstruction of f(R) Lagrangian from a massive scalar field

We present a reconstruction of the Lagrangian for $f(R)$ gravity by using a massive scalar field. The scalar field is minimally coupled to the action of $f(R)$ gravity. We demonstrate the use of a theorem based on invertible point transformation of anharmonic oscillator equation that has only recently been applied to gravitational physics. This avenue gives us a direct way to solve the field equations for a few well-known self-interaction potential of the scalar field. A few cases which do not fall in the regime of the theorem are also discussed. The $f(R)$ models are discussed in the context of recent cosmological observations and viability issues.

Exact solutions and accelerating universe in modified Brans\u2013Dicke theories

Exact solutions are studied in the context of modified Brans–Dicke theory. The non-linearity of the modified Brans–Dicke field equations is treated with the Euler–Duarte–Moreira method of integrability of anharmonic oscillator equation. While some solutions show a forever accelerating nature, in some cases there is a signature flip in the evolution of deceleration parameter in recent past. Importance of these latter models are studied in the context of late time acceleration of the universe. Constraints on the model parameters are obtained from Markov Chain Monte Carlo (MCMC) analysis using the Supernova distance modulus data, observational measurements of Hubble parameter, Baryon acoustic oscillation data and the CMB Shift parameter data.

Collapsing spherical star in Scalar-Einstein-Gauss-Bonnet gravity with a quadratic coupling

We study the evolution of a self interacting scalar field in Einstein-Gauss-Bonnet theory in four dimension where the scalar field couples non minimally with the Gauss-Bonnet term. Considering a polynomial coupling of the scalar field with the Gauss-Bonnet term, a self-interaction potential and an additional perfect fluid distribution alongwith the scalar field, we investigate different possibilities regarding the outcome of the collapsing scalar field. The strength of the coupling and choice of the self-interaction potential serves as the pivotal initial conditions of the models presented. The high degree of non-linearity in the equation system is taken care off by using a method of invertibe point transformation of anharmonic oscillator equation, which has proven itself very useful in recent past while investigating dynamics of minimally coupled scalar fields.

Intrinsic noise induced coherence resonance in a glow discharge plasma.

Experimental evidence of intrinsic noise induced coherence resonance in a glow discharge plasma is being reported. Initially the system is started at a discharge voltage (DV) where it exhibited fixed point dynamics, and then with the subsequent increase in the DV spikes were excited which were few in number and with further increase of DV the number of spikes as well as their regularity increased. The regularity in the interspike interval of the spikes is estimated using normalized variance. Coherence resonance was determined using normalized variance curve and also corroborated by Hurst exponent and power spectrum plots. We show that the regularity of the excitable spikes in the floating potential fluctuation increases with the increase in the DV, up to a particular value of DV. Using a Wiener filter, we separated the noise component which was observed to increase with DV and hence conjectured that noise can play an important role in the generation of the coherence resonance. From an anharmonic oscillator equation describing ion acoustic oscillations, we have been able to obtain a FitzHugh-Nagumo like model which has been used to understand the excitable dynamics of glow discharge plasma in the presence of noise. The numerical results agree quite well with the experimental results.

Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with self-steepening and self-frequency shift effects

In this paper, we have obtained the optical solitary wave solutions for the nonlinear Schrödinger equation which describes the propagation of femtosecond light pulses in optical fibers in the presence of self-steepening and a self-frequency shift terms. The solitary wave ansatz method was used to carry out the derivations of the solitons. The parametric conditions for the formation of soliton pulses were determined. Using the 1-soliton solution, a number of conserved quantities have been calculated for Hirota and Sasa-Satsuma cases. We have also constructed some periodic wave solutions of the higher order nonlinear Schrödinger equation by reducing it to quartic anharmonic oscillator equation and by using projective Ricatti equations. Moreover by using He's semi-inverse method, variational formulation was established to obtain exact soliton solutions. The 1-soliton solutions of time dependent form of this equation was also obtained. To visualize the propagation characteristics of dark-bright soliton solutions, few numerical simulations are given.

DARK AND BRIGHT SOLITARY WAVE SOLUTIONS OF THE HIGHER ORDER NONLINEAR SCHRÖDINGER EQUATION WITH SELF-STEEPENING AND SELF-FREQUENCY SHIFT EFFECTS

In this paper, we obtain the exact bright and dark soliton solutions for the nonlinear Schrödinger equation (NLSE) which describes the propagation of femtosecond light pulses in optical fibers in the presence of self-steepening and a self-frequency shift terms. The solitary wave ansatz method is used to carry out the derivations of the solitons. The parametric conditions for the formation of soliton pulses are determined. Using the one-soliton solution, a number of conserved quantities have been calculated for Hirota and Sasa–Satsuma cases and finally, we have constructed some periodic wave solutions by reducing the higher order nonlinear Schrödinger equation (HNLS) to quartic anharmonic oscillator equation. The obtained exact solutions may be useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a HNLS model equation.

Cnoidal and solitary wave solutions of the coupled higher order nonlinear Schrödinger equation in nonlinear optics

By using coupled amplitude-phase formulation, we construct the quartic anharmonic oscillator equation from the coupled higher order nonlinear Schrödinger equation. For arbitrary physical parameter values, we discuss the construction of new cnoidal wave solutions and the exact solutions of both bright (GVD < 0) and dark (GVD < 0) solitary waves. In addition, we investigate the pairs of dark–dark and bright–bright solitary waves.

Dynamics of nonlinear pulse propagation through a fiber Bragg grating with linear coupling

We consider nonlinear pulse propagation through a fiber Bragg grating with Kerr nonlinearity and linear coupling effects. Using the Stokes parameters, we obtain a set of four equations from the nonlinear coupled-mode equations. From these equations we find the relation for energy density of a gap soliton and construct a quartic anharmonic oscillator equation after finding the appropriate conserved quantities. With the help of the anharmonic oscillator equation we investigate the bright gap soliton solution and as a special case we also obtain another bright gap soliton solution that resembles the solutions of two coupled nonlinear equations for the envelopes of the fundamental and the second-harmonic field components. Moreover, we analyze the motion of the photon in two different potential wells. Finally, we carry out a linear stability analysis to study the stability of the system.

Painlevé analysis and integrability of the damped anharmonic oscillator equation

The Painleve analysis is applied to the anharmonic oscillator equation $$\ddot x + d\dot x + Ax + Bx^2 + Cx^3 = 0$$ . The following three integrable cases are identified: (i)C=0,d 2=25A/6,A>0,B arbitrary, (ii)d 2=9A/2,B=0,A>0,C arbitrary and (iii)d 2=−9A/4,C=2B 2/(9A),A<0,C<0,B arbitrary. The first two integrable choices are already reported in the literature. For the third integrable case the general solution is found involving elliptic function with exponential amplitude and argument.

Special analytical solutions of the damped-anharmonic-oscillator equation.

The anharmonic-oscillator equation with dissipation has been widely used in various areas of science, and surprisingly little has been known about its analytical solutions. In this paper, we survey the known classes of solutions and the methods employed to obtain them including the Painlev\'e Test and its results. We propose several approaches, one of which is based on compatibility with a Bernoulli equation. Another method investigated explores the many solutions that take the form of a rational fraction of linear combinations of exponentials. In addition to elementary function solutions, those involving elliptic functions are examined. The way the restriction on the parameters of the equation may be relaxed somewhat is demonstrated, leading to an extension of elliptic-type solutions.

Classical anharmonic oscillators: Rescaling the perturbation series

A solution to the classical anharmonic-oscillator equation of motion $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{x}=\ensuremath{-}x\ensuremath{-}\ensuremath{\lambda}{x}^{2n\ensuremath{-}1}$ is obtained by rescaling the perturbation series. The resulting series involves a coupling constant that remains finite for $\ensuremath{\lambda}\ensuremath{\gg}1$ and thus converges rapidly for all $\ensuremath{\lambda}$.

Combined natural and forced convection between parallel walls: Developing flow at higher Rayleigh numbers

The well-known, fourth-order, ordinary differential equation, governing vertical fully developed convection between parallel vertical walls, is discussed in the light of numerical solutions and perturbation expansions in terms of the Eckert number. The existence of multiple solutions is reaffirmed and the manner in which the various solutions grow and merge into each other with increasing Rayleigh number is established. Understanding of the structure of the solutions is aided by the study of a second-order anharmonic oscillator equation.

Nonlinear equilibrium and stability analysis of rippled, partially neutralized, magnetically focused electron beams

In the first part of this work a higher-order solution of the anharmonic oscillator equation describing the nonlinear rippled equilibrium state of magnetically focused, partially neutralized electron beams is given. Thus, using the method of harmonic balance, we derive a ripple-&amp;litude solution of the formwhere ø=ω1t+β0,ω1 being the nonlinear proper frequency and β0 a phase shift depending on the initial conditions. In the second part of the work we carry out a stability analysis of the nonlinear equilibrium state found in the first part with respect to long- and short-wavelength surface space-charge perturbations. In the framework of a local approximation the wave equation for the rippled beam is found to be a Hill type of equation which contains harmonic terms up to cos 3kSz (ks is the wavenumber of the ripple). This equation is solved by a resonant-mode coupling method; coupling of fast-fast, slow-slow and slow-fast waves is considered. The growth rates and band widths for different possible wave couplings are derived and compared.

Asymptotic solutions of time dependent anharmonic oscillator equation

A one-dimensional time dependent anharmonic oscillator is considered in the frame of transformation group techniques. Asymptotic solutions are obtained for the class of time decreasing coefficients of the equation.

A nonlinear oscillator model for an acoustic resonance chamber

The exact nonlinear acoustic wave equation, derived from conservation of momentum, mass, and the equation of state, was used to average out the spatial dependence and obtain a forced anharmonic oscillator equation for the time-dependent part. The non-linearity is expressed by a resistive-restoring coupling term ɑxẋ, rather than the restoring term αx2 + βx3 in the modified Duffing equation. A solution is sought in the form of a Fourier series, including all harmonics, and the equation is solved by a perturbation technique, retaining the dominant term in each harmonic. This results in an infinite system of coupled, nonlinear algebraic equations and leads to non-Lorentzian response curves for a sinusoidally driven resonator. These response curves are evaluated numerically and compared with predictions of the Duffing equation and with recent measurements of response curves and harmonic generation in an acoustic resonance chamber (M. Barmatz, N. Jacobi, and J. Stoneburner, 10th International Congress on Acoustics, Sydney, 1980). [Work supported by NASA.]

Connection between perturbation theory, projection-operator techniques, and statistical linearization for nonlinear systems

We employ the equivalence between Zwanzig's projection-operator formalism and perturbation theory to demonstrate that the approximate-solution technique of statistical linearization for nonlinear stochastic differential equations corresponds to the lowest-order $\ensuremath{\beta}$ truncation in both the consolidated perturbation expansions and in the "mass operator" of a renormalized Green's function equation. Other consolidated equations can be obtained by selectively modifying this mass operator. We particularize the results of this paper to the Duffing anharmonic oscillator equation.

Continued-Franction Method for Perturbation Theory

A direct operational approach to the solution of the Schr\"odinger equation has led to an iterative nonperturbative method for its solution. The method, when applied to the Mathieu equation and the anharmonic-oscillator equation, is superior to the perturbation-iteration method and to Rayleigh-Schr\"odinger perturbation theory both in terms of rate of convergence and range of coupling constant allowed.