Lump Solution, Breather Soliton and More Soliton Solutions for a (2+1)-Dimensional Generalized Benjamin–Ono Equation

Our work is to discuss some new exact solutions of the (2+1)-dimensional generalized Benjamin–Ono (gBO) equation, which describes small amplitude long waves on the surface of shallow water. After careful consideration, lump, breather soliton and new solitons solutions of gBO equation are gained by using the Hirota bilinear method, the test function and the idea of the homogeneous balance, which have not been studied yet. By improving the expression of the test function, the quadratic function with two squares is increased to the quadratic function with three squares, and various lump solutions are obtained. Combining trigonometric function with hyperbolic function, breather soliton is derived. Obviously, sometimes the test function form of tanh–coth and tan–cot methods are so simple that it can’t get the desired result. Utilizing the improved tanh–coth and tan–cot methods, whose solutions depend on the parameter n, we can get more soliton solutions. Finally, by determining different parameters, one can draw the three-dimensional plots and density plots at different times. By observing these figures, we analyze the dynamic behavior of (2+1)-dimensional gBO equation in detail. These results can help us to understand nonlinear systems better.