Solitons and dynamic analysis for a (2Â +Â 1)-dimensional breaking soliton equation

In this paper, we study a (2 + 1)-dimensional breaking soliton equation, which describes the (2 + 1)-dimensional interaction of a Riemann wave propagating along the y axis with a long wave along the x axis, where x and y are the scaled space coordinates. Grammian N-soliton solutions for the equation are derived. With N = 1 and 2, the one- and two-soliton solutions are given. Graphic analysis shows that the soliton amplitude and velocity are related to the dispersion. An overtaking interaction between the two parallel solitons is shown. We find that the two solitons always have the same soliton direction. Then, we investigate the equation from a planar-dynamic-system viewpoint. That equation is reduced to a two-dimensional planar dynamic system, which is proved to be a Hamiltonian system. Through the qualitative analysis, we give the phase portraits of the dynamic system, based on which the relation among the Hamiltonian, orbits of the dynamic system and types of the analytic solutions are discussed. The analysis shows that the solitary- and periodic-wave solutions for that equation correspond to the homoclinic and periodic orbits of the dynamic system, respectively.