Qualitative Analysis And Solutionto The Traveling Wave Solutions For Two Kinds Of Nonlinear Wave Equations

In this paper, we study the existence, behavior and solution of the bounded travelling wave solutions of the following two nonlinear wave equationsFirstly, we employ the theory of planar dynamical systems to convert the equations which the bounded travelling wave solutions of equation (Ⅰ) and (Ⅱ) correspond to, and then make comprehensive qualitative analysis to these two planar dynamical systems. According to the results of the qualitative analysis, we show all global phase portraits in the different conditions which the coefficients of equation (Ⅰ) and (Ⅱ) satisfy, and find the existent conditions of all the bounded travelling wave solutions. We discuss how the behavior of the bounded travelling wave solutions of equation (Ⅰ) and (Ⅱ) changes, respectively, when the dissipation coefficient y varies. The critical values or intervals which can be characterized the dissipation effect are represented, and the dissipation coefficient’s range are pointed out when the bell profile solitary wave solutions, kink profile solitary solutions, or damped oscillatory solutions appear. Furthermore, the exact expressions of bell profile solitary wave solutions and kink profile solitary solutions in various conditions are obtained by using undetermined coefficients method. For damped oscillatory solutions, it is too hard to find out their exact expressions because of their complex structure. However, in this paper, according to the evolution relations of orbits in the global phase portraits, and by the qualitative analysis results and undetermined coefficients method, we obtain the approximate expressions of damped oscillatory solutions. Finally, we establish the integral equation reflecting the relations between exact damped oscillatory solution and approximate solution by the idea of homogenization principle. Moreover, we give the error estimate for these approximate solutions. From the expression of error estimate, we can see the error we obtain by the method in this paper is infinitesimal decreasing in the exponential form.