Orbital Stability Of Periodic Peakons For A Higher Nonlinear Modified Camassa-Holm Equation

In mathematical physics equations,the discovery of the soliton has played a very important role in this field of research.These solutions have a sharp shape at the peak of the wave and are therefore also called peakons.They have attracted the concern and interest of many mathematicians and physicists for more than a decade,especially because their size and velocity do not change due to collisions,and therefore these peakons are said to be stable,the concept of stability here is orbital stability.That is,a wave that starts out close to an individual wave will always remain close to some translation of it later.Thus,the wave shape remains approximately the same at all times.In this paper,the orbital stability of periodic peakons for a higher Camassa-Holm equation is proved mainly based on the orbital stability of the lower Camassa-Holm e-quation,the equation calls it a higher nonlinear modified Camassa-Holm equation.The specific work of this paper mainly consists of the following two points:First,it is shown that the periodic solutions of the higher nonlinear modified Camassa-Holm equation are weak solutions;and second,by studying the relationship between the stability and the conservation law of the equation,it is obtained that if the energy and height of a wave are close to the energy and height of those peakons,the overall shape of the wave is close to the shape of those peakons.In order to prove the orbital stability of the periodic peakons of the higher nonlinear modified Camassa-Holm equation,on the one hand,we construct a ap-propriate coefficient of auxiliary function?（）by using the divisional coefficient method,and then estimate the maximum value of the coefficient of auxiliary function?（）by Y-oung inequality,so as to establish a function（,8）)consisting of two real variables（,8）)by auxiliary function2)（）,the wave height is close to the peakons height by the properties of the（,8）)function on the peak;On the other hand,the wave energy is close to the peakons energy indirectly by proving the continuity of the conservation law at¹parametrization,mainly by using some basic inequalities.