Painlevé Analysis, Bäcklund Transformation, Lax Pair, Periodic- and Travelling-Wave Solutions for a Generalized (2+1)-Dimensional Hirota–Satsuma–Ito Equation in Fluid Mechanics

In this paper, under investigation is a generalized (2+1)-dimensional Hirota–Satsuma–Ito (HSI) equation in fluid mechanics. Motivated by its application in simulating the propagation of small-amplitude surface waves and shallow water waves, we focus on the Painlevé integrability, commonly used transformation forms and analytical solutions of the HSI equation. Via the Painlevé analysis, it is found that the HSI equation is Painlevé integrable under certain condition. Bilinear form, Bell-polynomial-type Bäcklund transformation and Lax pair are constructed with the binary Bell polynomials. One-periodic-wave solutions are derived via the Hirota–Riemann method and displayed graphically. Through the polynomial-expansion method, travelling-wave solutions are obtained.