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Soliton solutions and chaotic motions for the \((2+1)\)-dimensional Zakharov equations in a laser-induced plasma. (English) Zbl 1443.82016
Summary: The \((2+1)\)-dimensional Zakharov equations arising from the propagation of a laser beam in a plasma are studied in this paper. Analytic soliton solutions are obtained by means of the symbolic computation, based on which we find that \(|E|\) is inversely related to \(\omega_{pe}\), but positively related to \(m_i\) and \(c_s\), while \(n\) is inversely related to \(\omega_{pe}\) and \(\omega_L\), but positively related to \(n_0\), with \(E\) as the envelope of the high-frequency electric field, \(n\) as the plasma density, while \(\omega_{pe}\), \(\omega_L\), \(n_0\), \(m_i\) and \(c_s\) as the plasma electronic frequency, frequency of the laser beam, mean density of the plasma, mass of an ion and ion-sound velocity in the plasma, respectively. Head-on interaction is found to be transformed into an overtaking one with \(\omega_{pe}\) increasing or \(n_0\) decreasing. Also, period of the bound-state interaction decreases with \(\omega_L\) decreasing. Considering the driving forces in the laser-induced plasma, we explore the associated chaotic motions as well as the effects of \(\omega_L\), \(\omega_{pe}\), \(k_L\), \(n_0\), \(m_i\), \(c_s\), \(\omega_{F_1}\) and \(\omega_{F_2}\), where \(k_L\) is the wave number of the laser beam, \(\omega_{F_1}\) and \(\omega_{F_1}\) represent the frequencies of driving forces, respectively. It is found that the chaotic motions can be weakened with \(\omega_{pe}\), \(c_s\) and \(\omega_{F_1}\) increasing, or with \(n_0\), \(m_i\) and \(\omega_{F_2}\) decreasing, and the periodic motion can occur when \(\omega_{F_1}\) reaches the critical value \(2\pi\), while the chaotic motions are independent of \(\omega_L\) and \(k_L\).
MSC:
82D10 Statistical mechanics of plasmas
35C08 Soliton solutions
35L70 Second-order nonlinear hyperbolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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