The Travelling Wave Solutions And Bifurcations Of The Generalized Camassa-holm Equation And The Modified Fornberg-whitham Equation

In this paper, we study the exact traveling wave solutions for two types of famous nonlinear equations. The first one is the generalized Camassa-Holm equation ut+2kux-uxxt+3u2ux=2uxuxx+uuxxx.(1)The second is the modified Fornberg-Whitham equation ut-uxxt+ux+u2ux=3uxuxx+uuxxx.(2)For Eq.(1) we obtain the following results:(1°) It is verified that k=3/8is a bifurcation parametric value for several types of explicit nonlinear wave solutions.(i) When k<3/8, there are five types of the explicit nonlinear wave solutions, which are(1) hyperbolic peakon wave solution,(2) fractional peakon wave solution,(3) fractional singular wave solution,(4) hyperbolic singular wave solution,(5) hyperbolic smooth solitary wave solution.(ⅱ) When k=3/8, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution.(ⅲ) When k>3/8, there is not any type of explicit nonlinear wave solutions.(2°) It is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately.(3°) It is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution.For Eq.(2) we get the following results:(1°) Some explicit expressions of solutions are obtained by using auxiliary equation method, which include peakon wave solution, solitary wave solution, trigonometric func-tion solutions, elliptic function periodic solutions, and fractal type blow-up solutions. The latter three types of solution are new and the expressions of the second,the third and the forth types of solution are valid for any wave velocity c, which extend the results by other authors (He et. al just gave the explicit expressions of solitary wave solution for special wave velocity). (2°) We use the software Mathematica to test the correctness of these solutions. That is, it is confirmed that these functions indeed satisfy Eq.(2) by computer. The testing programs and the testing results are given, and numerical simulation is also shown.