Bäcklund transformation, infinitely-many conservation laws, solitary and periodic waves of an extended (3 + 1)-dimensional Jimbo–Miwa equation with time-dependent coefficients

In this paper, an extended (3+1)-dimensional Jimbo–Miwa equation with time-dependent coefficients is investigated, which comes from the second member of the Kadomtsev–Petviashvili hierarchy and is shown to be conditionally integrable. Bilinear form, Bäcklund transformation, Lax pair and infinitely-many conservation laws are derived via the binary Bell polynomials and symbolic computation. With the help of the bilinear form, one-, two- and three-soliton solutions are obtained via the Hirota method, one-periodic wave solutions are constructed via the Riemann theta function. Additionally, propagation and interaction of the solitons are investigated analytically and graphically, from which we find that the interaction between the solitons is elastic and the time-dependent coefficients can affect the soliton velocities, but the soliton amplitudes remain unchanged. One-periodic waves approach the one-solitary waves with the amplitudes vanishing and can be viewed as a superposition of the overlapping solitary waves, placed one period apart.