The (3+1)-dimensional generalized mKdV-ZK equation for ion-acoustic waves in quantum plasmas as well as its non-resonant multiwave solution

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.

(3+1)维时间分数阶KdV-Zakharov-Kuznetsov方程的分支分析及其行波解

首先,运用拟设方法和动力系统分支方法,获得了(3+1)维时间分数阶KdV-Zakharov-Kuznetsov方程的奇异孤子解、亮孤子解、拓扑孤子解、周期爆破波解、孤立波解等.再利用MAPLE软件画出了KdV-Zakharov-Kuznetsov方程在不同条件下的分支相图.最后,讨论了行波解之间的联系.

Exact Solutions to (3+1) Conformable Time Fractional Jimbo–Miwa,Zakharov–Kuznetsov and Modified Zakharov–Kuznetsov Equations

Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.

(3+1)维时空分数阶mKdV-ZK方程的精确解

借助一个分数阶子方程和修正的Riemann-Liouville分数阶导数,基于扩展的(G′/G)-展开法,介绍了求解分数阶微分方程精确解的一种新方法,并利用该方法求解了(3+1)维时空分数阶mKdV-ZK方程,获得了该方程用双曲函数和三角函数等表示的精确解.

扩展的辅助函数法求一类非线性分数阶偏微分方程的精确解

为进一步扩大解的范围,丰富解的结构.文章在前人运用的辅助函数法的基础上做推广,将辅助函数满足的方程扩展到满足一般的Riccati方程上,并借助分数阶复变换和整合的分数阶导数的性质,将该方法运用到求解时间分数阶modified Benjamin-Bona-Mahony(简称mBBM)方程以及(3+1)维非线性分数阶Jimbo-Miwa方程,获得这2个方程的许多新精确行波解.