The Zakharov-Kuznetsov equation plays a key role in the plasma physics, and its solution has always been a hot topic, however its variational principle has not been studied. So this paper develops a variational formulation for the (3+1)-dimensional extended quantum Zakharov-Kuznetsov equation by the semi-inverse method, which shows an energy conservation law for the dust-ion-acoustic solitary waves in a magnetized two-ion-temperature dusty plasma. The whole derivation process is presented in detail. The obtained results in this paper are expected to shed a bright light on the study of theoretical basis for the magnetized plasma.

中文翻译：

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等离子体物理学中（3 + 1）维扩展量子Zakharov-Kuznetsov方程的变分原理

Zakharov-Kuznetsov方程在等离子体物理学中起着关键作用，其解一直是一个热门话题，但尚未研究其变分原理。因此，本文通过半逆方法为（3 + 1）维扩展量子Zakharrov-Kuznetsov方程开发了一个变分公式，表明了磁化两离子中粉尘离子声孤立波的能量守恒定律。温度多尘的等离子体。详细介绍了整个推导过程。本文的研究结果有望为磁化等离子体的理论研究提供一个亮点。