In this paper, we construct global solutions to the Cauchy problem on a nonlinear Fokker-Planck equation near Maxwellian with small-amplitude initial data in Sobolev space $H_x^2{L}_v^2$ by a refined nonlinear energy method. Compared with the results of Liao et al. (Global existence and decay rates of the solutions near Maxwellian for non-linear Fokker-Planck equations, J. Stat. Phys., 173 (2018), 222–241.), the regularity assumption on the initial data is much weaker.

In this paper, a fully discrete local discontinuous Galerkin finite element method is proposed to solve the KdV-Burgers-Kuramoto equation with variable-order Riemann-Liouville time fractional derivative. The method proposed in this paper is based on the finite difference method in time and local discontinuous Galerkin method in space. For all $ϵ(t)\in (0,1)$ with variable order, we prove the scheme is unconditional stable and convergent. Finally, numerical examples are provided to verify the theoretical analysis and the order of convergence for the proposed method.

This paper focuses on the new (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa (DJKM) equation with some variable coefficients and derives its new analytical solutions. Based on the (2+1)-dimensional DJKM equation, this equation adds an additional spatial dimension (z) and introduces time-dependent coefficients. Therefore, it is more suitable for describing dynamic wave behaviors in inhomogeneous media. Firstly, by virtue of the Hirota method, we derive the Hirota bilinear form of the equation. Secondly, based on the bilinear form, we present three types of solutions, including the three wave solution and lump-type solution. Finally, by choosing appropriate parameters, we plot some graphs to intuitively display the physical characteristics of these solutions.