Bifurcations Of Solitary Waves For A Class Of Higher Order Nonlinear Equations

In this paper,we consider the solitary wave bifurcation of a class of higher order nonlinear equations with parameter k,whose traveling wave system has a singular line.We study the solitary wave bifurcation of this kind of equation by using qualitative analysis theory and dynamical system bifurcation method.The main contents are as follows:The first chapter is the introduction,which mainly introduces the previous research results on the equation,and introduces the main content and results of this paper.The second chapter studies the case of the degree of 3.It is found that the traveling wave system has two bifurcation curves:(?)and(?)There are at most two singular points in the traveling wave system:(?1,0)and(?2,0).For the parameter k,there are two bifurcation values of the system:0 and(?):(?)when k?0 or k=(?),there is only one bifurcation wave velocity;(?)when 0<k<(?),there is only one bifurcation wave velocities;(?)when k>(?),there is no bifurcation wave velocity.The third chapter studies the case of the degree of 4.It is found that the traveling wave system has three bifurcation curves:±g3(c)=±4/5(?)and g4(c)=1/5(5c2-10kc-c5).There are at most three singular points in the traveling wave system:(?3,0),(?4,0)and(?5,0).For the parameter k,there are three bifurcation values fo hte system:0,(?)and k*:(?)when k?U,there is one bifurcation wave velocity between g3(c)and g4(c),adn the same as-g3(c)and g4(c);(?)when 0<k<(?),there are two bifurcation wave velocities between g3(c)and g4(c),and the same as-93(c)and 94(c);(?)when k=(?),there is one bifurcation wave velocity between g3(c)and g4(c)and two bifurcation wave veiocites between-g3(c)and g4(c);(?)when(?)<k<k*,there are two bifurcation wave velocities between-g3(c)and g4(c);(?)when k=k*,there is one bifurcation wave velocity between-g3(c)and g4(c);(vi)when k>k*,there is no bifurcation wave velocity.In the fourth chapter,we discuss the case of the general n.We find that when n is odd,the traveling wave system has at most two singular points and two bifurcation curves.When n is even,the traveling wave system has at most three singular points and three bifurcation curves.Because of the high degree,we can not make a specific bifurcation graph,but we guess that when n is odd,its bifurcation graph is similar to the case of the degree of 3,and when n is even,it is similar to the case of the degree of 4.