Orbital Stability Of Peakons And Periodic Peakons For A Quartic Nonlinear Camassa-Holm Equation

This thesis mainly studies the orbital stability of the peakon and periodic peakon of the quartic nonlinear Camassa-Holm equation.First,we verify that the obtained peakon and periodic peakon solutions are weak solutions to the equation.Then,according to the conservation amount of the equation and constructing an appropriate auxiliary function,the proof process of the orbital stability is transformed into related studies of the conservation amount and auxiliary function.It can be proved that both the peakon and periodic peakon solutions of the equation are orbitally stable.The full text is divided into five chapters,and the main content of each chapter is described as follows:In Chapter 1,we mainly introduce the research background,research status of the quartic nonlinear Camassa-Holm equation orbital stability,and the main work in this paper.In Chapter 2,we mainly introduce some related knowledge of hyperbolic function and dynamic system theory,as a preliminary knowledge.In Chapter 3,we verify that both the peakon solution and the periodic peakon solution of the quartic nonlinear Camassa-Holm equation are weak solutions of the equation.In Chapter 4,we show that the peakon and periodic peakon of the quartic nonlinear Camassa-Holm equation are orbitally stable.First,the relationship between the orbital stability of the equation and the three conserved quantities is investigated.Based on the related properties of the conserved quantities,a suitable auxiliary function is constructed.The auxiliary function is used to complete the orbital stability verification.In Chapter 5,we summarize the findings of this article and anticipate possible future research.