Exact solutions of the Zakharov equations by using the first integral method

Ünal, A.

doi:10.1501/commua1_0000000676

Cited by 3 publications

(1 citation statement)

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“…The role of these first integrals is often crucial in the study of solutions and properties of differential equations, including questions related to stability [9,10] or integrability [11][12][13][14]. The so-called first integral method for finding exact solutions of nonlinear partial differential equations has also been successfully applied to a wide range of equations modeling various phenomena in physics (see [15] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

First Integrals and Hamiltonians of Some Classes of ODEs of Maximal Symmetry

Ndogmo

2017

Journal of Applied Mathematics

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Section: Introductionmentioning

confidence: 99%

First Integrals and Hamiltonians of Some Classes of ODEs of Maximal Symmetry

Ndogmo

2017

Journal of Applied Mathematics

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Soliton solutions and chaotic motions of the Zakharov equations for the Langmuir wave in the plasma

et al. 2015

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For the interaction between the high-frequency Langmuir waves and low-frequency ion-acoustic waves in the plasma, the Zakharov equations are studied in this paper. Via the Hirota method, we obtain the soliton solutions, based on which the soliton propagation is presented. It is found that with λ increasing, the amplitude of u decreases, whereas that of v remains unchanged, where λ is the ion-acoustic speed, u is the slowly-varying envelope of the Langmuir wave, and v is the fluctuation of the equilibrium ion density. Both the head-on and bound-state interactions between the two solitons are displayed. We observe that with λ decreasing, the interaction period of u decreases, while that of v keeps unchanged. It is found that the Zakharov equations cannot admit any chaotic motions. With the external perturbations taken into consideration, the perturbed Zakharov equations are studied for us to see the associated chaotic motions. Both the weak and developed chaotic motions are investigated, and the difference between them roots in the relative magnitude of the nonlinearities and perturbations. The chaotic motions are weakened with λ increasing, or else, strengthened. Periodic motion appears when the nonlinear terms and external perturbations are balanced. With such a balance kept, one period increases with λ increasing.

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