Bilinear Bäcklund transformation, breather- and travelling-wave solutions for a (2+1)-dimensional extended Kadomtsev–Petviashvili II equation in fluid mechanics

Fluid-mechanics studies are applied in mechanical engineering, biomedical engineering, oceanography, meteorology and astrophysics. In this paper, we investigate a (2+1)-dimensional extended Kadomtsev–Petviashvili II equation in fluid mechanics. Based on the Hirota bilinear method, we give a bilinear Bäcklund transformation. Via the extended homoclinic test technique, we construct the breather-wave solutions under certain constraints. We obtain the velocities of the breather waves, which depend on the coefficients in that equation. Besides, we derive the lump solutions with the periods of the breather-wave solutions tending to the infinity. Based on the polynomial-expansion method, travelling-wave solutions are constructed. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. We graphically discuss the effects of those coefficients on the breather wave and lump.

Ablowitz–Kaup–Newell–Segur system, conservation laws and Bäcklund transformation of a variable-coefficient Korteweg–de Vries equation in plasma physics, fluid dynamics or atmospheric science

For a variable-coefficient Korteweg–de Vries equation in a lake/sea, two-layer liquid, atmospheric flow, cylindrical plasma or interactionless plasma, in this paper, we derive the bilinear Bäcklund transformation, non-isospectral Ablowitz–Kaup–Newell–Segur system and infinite conservation laws for the wave amplitude under certain constraints among the external force, dissipation, nonlinearity, dispersion and perturbation.

The integrability of the coupled Ramani equation with binary Bell polynomials

Binary Bell polynomial approach is applied to study the coupled Ramani equation, which is the generalization of the Ramani equation. Based on the concept of scale invariance, the coupled Ramani equation is written in terms of binary Bell polynomials of two dimensionless field variables, which leads to the bilinear coupled Ramani equation directly. As a consequence, the bilinear Bäcklund transformation, Lax pair and conservation laws are systematically constructed by virtue of binary Bell polynomials.