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Abstract
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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.
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