Energy spectrum of oscillations in generalized Sagdeev potential

Abstract

In this paper, the full energy spectrum of nonlinear oscillations, known as the cnoidal waves, is studied in the framework of small-amplitude Korteweg de Vries and modified Korteweg de Vries (mKdV) theories based on the pseudoparticle motion in Helmholtz and Duffing potentials by employing the newly introduced pseudoenergy concept. The pseudoenergy dependence of various cnoidal oscillation parameters is then studied, and it is shown that superposition of cnoidal waves leads to familiar beating and Lissajous profiles. One of the most important aspects of the nonlinear oscillation is found to be the frequency dependence of the oscillation amplitude which mainly characterizes the nature of oscillations. It is shown that the developed method can be used to study the spectrum of oscillations and shock waves in the fully nonlinear Sagdeev pseudopotential and to directly calculate many dynamic parameters of the given nonlinear system. Current research may be helpful in understanding of basic excitations and interaction of nonlinear oscillation in various hydrodynamic systems including plasmas. It is also shown that nonlinear excitations in a hydrodynamic fluid can be effectively investigated by close inspection of shock waves which contain the full nonlinear spectrum of dynamical systems.

Exact solutions of the cubic Boussinesq and the coupled Higgs system

“…We observed that there are so many classes of NLEE arising in physics, fluid mechanics and engineering fields transferred to the following ODE: 3 0 αθ βθ γθ ′′ + + = (4) see [9,10,19,[22][23][24][25][26][27][28][29]31] and so on. This equation is often referred to the pseudoptential or sagdeev potential [30].…”

Quantum Faraday excitations in degenerate electron-ion plasma

“…The Jacobi-cn elliptic function has the limiting forms cn(x, 0)=cos(x) and cn(x, 1)=sech(x). Also, In the sinusoidal excitation limit we have Ψ 2 ;Ψ 3 , while in the soliton limit Ψ 1 ; Ψ 2 [68]. The weakly nonlinear driven excitations studied in this section may also be extended to the case with the hard wall rectangular potential well of length l similar to the case of linear excitations.…”

“…in which ¢ E is the energy eigenvalue of the plasma-phonon [68] and the pseudo-potential corresponding to this Hamiltonian reads…”

Pseudoquatum Features of Parametrically Driven Classical Fluids

“…The later describes many interesting features of the nonlinear electrostatic waves in an electron-ion plasma such as energy spectrum, nonlinear resonance, harmonic generation, chaotic excitations and autoresonance effects which has been recently revisited in Refs. [32][33][34][35][36][37] using the recently developed pseudoparticle dynamics method. In the stationary frame ξ = x − Mt (M being the Mach number), the dissipationless hydrodynamic equations can be reduced to the following pseudoforce (Newton) equation of motion [38]…”

“…The pseudoparticle formalism is particularly interesting because various collective excitations in a fluid even in the presence of dissipation [36] and external forcing effect can be described in terms of dynamics of a single particle-like motion in the generalized Sagdeev pseudopotential. In the small-amplitude weakly nonlinear potential limit of form V (φ) = −(c/2)Φ 2 + (d/3)Φ 3 or V (φ) = −(c/2)Φ 2 + (d/3)Φ 4 which are known as the Helmholtz and Duffing potentials, respectively, the energy equation ( 2) can be fully integrated to give Φ(ξ) in terms of Jacobielliptic functions [32]. The small-amplitude weakly nonlinear waves in a fluid are described by the celebrated Korteweg de Vries equation (KdVE) given as…”