Dynamic behaviors and soliton solutions of the modified Zakharov-Kuznetsov equation in the electrical transmission line. (English) Zbl 1362.35275
Summary: The modified Zakharov-Kuznetsov (mZK) equation in the electrical transmission line is investigated in this paper. Different expressions on the parameters in the mZK equation are given. By means of the Hirota method, bilinear forms and soliton solutions of the mZK equation are obtained. Linear-stability analysis yields the instability condition for such soliton solutions. We find that the soliton amplitude becomes larger when the inductance \(L\) and capacitance \(C_0\) decrease. Phase-plane analysis is conducted on the mZK equation for the properties at equilibrium points. Then, we investigate the perturbed mZK equation, which can be proposed when the external periodic force is considered. Both the weak and developed chaotic motions are observed. Our results indicate that the two chaotic motions can be manipulated with certain relation between the absolute values of nonlinear terms and the perturbed one. We also find that the chaotic motions can be weakened with the absolute values of \(L\) and \(C_0\) decreased.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35C08 | Soliton solutions |
| 35B35 | Stability in context of PDEs |
| 37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |
Keywords:
electrical transmission line; modified Zakharov-Kuznetsov equation; soliton propagation; symbolic computation; chaotic motionsReferences:
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