He's fractional derivative is adopted in this paper, and analytical methods for fractional differential equations are briefly reviewed, two modifications of the exp-function method (the generalized Kudryashov method and generalized exponential rational function method) are emphasized, and fractional Benjamin-Bona-Mahony equation with He's fractional derivative is used as an example to elucidate the solution process.

A fractional Whitham-Broer-Kaup equation is suggested using He's fractional derivative to model solitary waves in shallow water in porous medium near a dam. A modification of the exp-function method, the generalized exponential rational function method, is adopted to elucidate the basic solution properties of the equation, revealing that the value of the fractional order can be used effectively to control the wave velocity, the wave height, and the wave morphology. This theoretical result can be used for possible tsunami prevention.

In this paper, the Hirota’s bilinear form is employed to investigate the lump, periodic lump and interaction lump stripe solutions of the (2+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation. Many results are obtained by dynamic process of figures. We analyze the propagation direction and horizontal velocity of lump solutions to find some constraint conditions which include positiveness and localization. In the process of the travel of the periodic lump solutions, it appears that the energy distribution is not symmetrical. The interaction lump stripe solutions of non-elastic indicate that the lump solitons are dropped and swallowed by the stripe soliton.

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We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov-Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical system, heat equation, and so on, are followed to be generated. Then we expand some differential operators presented in the paper to deduce two types of expanding dynamical models. By taking the generalized Burgers dynamical system as an example, we deform its expanding model to get a half-expanding system, whose recurrence operator is derived from Lax representation, and its Hamiltonian structure is also obtained by adopting a new way. Finally, we expand the generalized Burgers dynamical system to the (2+1)-dimensional case whose Hamiltonian structure is derived by Poisson tensor and gradient of the Casimir function. Besides, a kind of (2+1)-dimensional expanding dynamical model of the (2+1)-dimensional dynamical system is generated as well.