The Investigation Of The Traveling Wave Solutions To Two Nonlinear Evolutionary Equations

In many fields, such as physics, mechanics, chemistry, biology and economics,etc, many models are nonlinear evolutionary equations. So it is more and more important to solve them in order to facilitate the description and understanding the discribed physical phenomena and other phenomena. The osmosis K?3,2? equation and the modified Fornberg-Whitham equation are two physical nonlinear evolutionary equations. It is of great practical value to investigate their traveling wave solutions.Firstly, this paper studies the traveling wave solutions of the osmosis K)2,3(equation. By traveling wave transformation, this equation is transformed into a dynamical system and then the bifurcation curves are obtained by using the bifurcation method of the plane dynamic system. Furthermore, the phase portraits of the corresponding traveling wave system is given. And with the phase portraits, we find the existence of smooth solitary wave solutions and obtains the exact expressions of the compacton, peakon and periodic wave solutions. On the same time, using Maple, we simulate the smooth solitary wave solution and compacton solution, and the results show that the theoretical analysis agrees well with the simulations.Secondly, this paper invetigats the traveling wave solutions of the modified Fornberg-Whitham equation. Under a new traveling wave transformation and using the bifurcation method of plane dynamic system, we get the bifurcation curves.Meanwhile, according to the phase portraits of the corresponding traveling wave system, we derive new smooth solitary wave solutions and peakon solutions of this equation. The obtained results generalizes the ones in the literature. At the same time,it is proved that the new tpye of peakon, which is in the form of rational funciton, is the weak solution of the modified Fornberg-Whitham equation.The obtained expressions of the traveling wave solutons can help people deeply know the decribed physical process in these two nonlinear evolutionary equations. On the same time, the expressions of the traveling wave solutons can be used to test the validity of some numerical methods.