The (3+1)-dimensional generalized mKdV-ZK equation for ion-acoustic waves in quantum plasmas as well as its non-resonant multiwave solution* *Project supported by the National Natural Science Foundation of China (Grant No. 11975143), the Natural Science Foundation of Shandong Province of China (Grant No. ZR2018MA017), the Taishan Scholars Program of Shandong Province, China (Grant No. ts20190936), and the Shandong University of Science

Abstract

The quantum hydrodynamic model for ion-acoustic waves in plasmas is studied. First, we design a new disturbance expansion to describe the ion fluid velocity and electric field potential. It should be emphasized that the piecewise function perturbation form is new with great difference from the previous perturbation. Then, based on the piecewise function perturbation, a (3+1)-dimensional generalized modified Korteweg–de Vries Zakharov–Kuznetsov (mKdV-ZK) equation is derived for the first time, which is an extended form of the classical mKdV equation and the ZK equation. The (3+1)-dimensional generalized time-space fractional mKdV-ZK equation is constructed using the semi-inverse method and the fractional variational principle. Obviously, it is more accurate to depict some complex plasma processes and phenomena. Further, the conservation laws of the generalized time-space fractional mKdV-ZK equation are discussed. Finally, using the multi-exponential function method, the non-resonant multiwave solutions are constructed, and the characteristics of ion-acoustic waves are well described.