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In this paper, we investigate the invariant properties of the coupled time-fractional Boussinesq-Burgers system. The coupled time-fractional Boussinesq-Burgers system is established to study the fluid flow in the power system and describe the propagation of shallow water waves. Firstly, the Lie symmetry analysis method is used to consider the Lie point symmetry, similarity transformation. Using the obtained symmetries, then the coupled time-fractional Boussinesq-Burgers system is reduced to nonlinear fractional ordinary differential equations (FODEs), with E r d e ´ l y i - K o b e r fractional differential operator. Secondly, we solve the reduced system of FODEs by using a power series expansion method. Meanwhile, the convergence of the power series solution is analyzed. Thirdly, by using the new conservation theorem, the conservation laws of the coupled time-fractional Boussinesq-Burgers system is constructed. In particular, the presentation of the numerical simulations of q-homotopy analysis method of coupled time fractional Boussinesq-Burgers system is dedicated.

In this paper, we consider the Riemann–Hilbert (RH) method for a fourth-order nonlinear Schrödinger (NLS) equation, which is reduced on the basis of the generalized Davydov’s model by selecting some special parameters. On the basis of the spectral analysis for the Lax pair of the equation, the RH problem is presented. Through a specific RH problem in the sense of irregularity, the multi-soliton solutions are also obtained. In addition, dynamic behaviors of these soliton solutions are given to illustrate the soliton characteristics.

In this paper, we investigate a (2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. The lump waves, lumpoff waves, and rogue waves are presented based on the Hirota bilinear form of this equation. It is worth noting that the moving path as well as the appearance time and place of the lump waves are given. Moreover, the special rogue waves are considered when lump solution is swallowed by double solitons. Finally, the corresponding characteristics of the dynamical behavior are displayed.