Peakons And Periodic Cusp Wave Solutions Of A μ-type Combination Equation

With the development of modern mathematics,the study of nonlinearpartial diferential equation are now attracting great attention on account oftheir potential applications in nonlinear physical geography,fluid mechanic-s,optoelectronics,natural disaster warning and so on.Althought it is very dif-ficult to solve nonlinear partial diferential equation thoroughly, sometime itis even impossible, mathematician find many efective methods,such as Dar-boux transformation,the inverse scattering method,B¨acklund transformation,etc.Among them,the most efective method is Hirota direct method.Consideration here is a generalized-type integrable equation,whichcan be regarded as a generalization to the modified-Camassa-Holm equa-tion,-Camassa-Holm equation and-Degasperis-Procesi equation.By theapplication of the property of Green’s function,as well as seeking a PDEweak solution approach,the existence of single peaked solution and periodicpeaked solution for the combination equation are investigated.