30 research outputs found
Harmonic Balance for Non-Periodic Hyperbolic Solutions of Nonlinear Ordinary Differential Equations
In this paper, we propose a new approach for obtaining explicit analytical approximations to the homoclinic or heteroclinic solutions of a general class of strongly nonlinear ordinary differential equations describing conservative singledegree-of-freedom systems. Through a simple and explicit change of the independent variable that we introduce, these equations are transformed to others for which the original homoclinic or heteroclinic solutions are mapped into periodic solutions that satisfy some boundary conditions. Recent simplified methods of harmonic balance can then be exploited to construct highly accurate analytic approximations to these solutions. Here, we adopt the combination of Newton linearization with the harmonic balance to construct the approximates in incremental steps, thereby proposing both appropriate initial approximates and increments that together satisfy the required boundary conditions. Three examples including a septic Duffing oscillator, a controlled mechanical pendulum and a perturbed KdV equations are presented to illustrate the great accuracy and simplicity of the new approach
Modulation instability gain and localized waves by modified Frenkel-Kontorova model of higher order nonlinearity
In this paper, modulation instability and nonlinear supratransmission are
investigated in a one-dimensional chain of atoms using cubic-quartic
nonlinearity coefficients. As a result, we establish the discrete nonlinear
evolution equation by using the multi-scale scheme. To calculate the modulation
instability gain, we use the linearizing scheme. Particular attention is given
to the impact of the higher nonlinear term on the modulation instability.
Following that, full numerical integration was performed to identify modulated
wave patterns, as well as the appearance of a rogue wave. Through the nonlinear
supratransmission phenomenon, one end of the discrete model is driven into the
forbidden bandgap. As a result, for driving amplitudes above the
supratransmission threshold, the solitonic bright soliton and modulated wave
patterns are satisfied. An important behavior is observed in the transient
range of time of propagation when the bright solitonic wave turns into a
chaotic solitonic wave. These results corroborate our analytical investigations
on the modulation instability and show that the one-dimensional chain of atoms
is a fruitful medium to generate long-lived modulated waves