In this paper, the solitary solutions of the [Formula: see text]-dimensional Benjamin–Bona–Mahony and strain wave equations are set up by employing the simplest equation method for the first time. By determining the value of parameters, the images of the traveling wave solutions are displayed, and they intuitively reflect that choosing different Bernoulli equations will affect the form of solutions. The validity and wider applicability of the method for solving known wave phenomena models are verified, moreover, the solution structure is enriched.

The aim of this paper is on the propagations of barotropic–baroclinic coherent structures based on the two-layer quasi-geostrophic model (2LQG) through a Fourier spectrum compliant approach. First, by introducing the barotropic and baroclinic stream functions starting from the 2LQG model, a new coupled Gardner-type evolution equations, representing the interaction processes between the barotropic flow and baroclinic one, are obtained by combining the multi-scale method and the perturbation expansion method. Second, based on the obtained coupled model equations, the physical mechanisms of the nonlinear barotropic–baroclinic interaction are analyzed qualitatively. Within the range of parameters chosen in this paper, quantitative results show that the basic flow, the β effect, and the bottom topography are necessary factors to excite the nonlinear Rossby isolated waves. The results also declare that the dipole-like blockings are readily excited in the flow field and move slowly eastward in both barotropic and baroclinic flow fields.

In the text, rational solutions to a (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in determinant form are solved by means of the Kadomtsev-Petviashvili hierarchy reduction method and Hirota’s bilinear form. The first-order, higher-order, multi-lump waves and multi-rogue waves on the basis of the solution in Gramian are presented in graphs. Periodic and parabolic line rogue waves are obtained by choosing different dispersion coefficient functions. The second-order rogue waves are also depicted, which can be seen as the composition of two first-order rogue waves. For multiple rogue waves, taking the periodic and parabolic line rogue waves as examples, it can be observed the change tendency of amplitudes of the intersection region is different from that in branches of line rogue waves. Besides, propagation of the first-order lump wave, interaction between one-lump-soliton and one-rogue-wave, and two kinds of collisions of two lump waves, i.e., elastic and inelastic, are also shown in the x - t plane.