Stochastic exact solutions and averaging dynamical behaviors of the Hirota-Maccari system driven by multiplicative white noise

The stochastic exact solutions and qualitative analysis of the Hirota-Maccari system driven by a multiplicative white noise is studied in details. This model describes the phenomena of different propagations of waves in the different spatial scales. Bifurcation theory is used to give qualitative information on the existence of some special solutions. Furthermore, the classification of stochastic exact traveling wave solutions is given, and topological stability of solutions is analyzed under the change of parameters. In particular, for the model, we consider the averaging values of solutions under Brown motion and show a delay factor of amplitude which means that the white noise not only affects the phase factor, but also affects the amplitude of the solutions. This result recovers a new aspect of dynamical behavior of the stochastic Hirota-Maccari model in large time under the action of noise.