Several Dynamical Systems With Peakons And Their Integrability

In this paper,four soliton equations with peakons are constructed and their integrability are studied.They are the Degasperis-Procesi equation II,which is associated with a 3 × 3 matrix spectral problem with one potential,the multicomponent Geng-Xue equation related to a(7)+2)×(7)+2)matrix spectral problem(7)is any positive integer)with two vector potentials,the multi-component super CH equation associated with a(9)+ 2)×(9)+ 2)matrix spectral problem(9)is any positive integer)with three potentials(two of which are vector potentials),the multi-component generalization of the three-component CH equation,which is related to a(9)+ 2)×(9)+ 2)matrix spectral problem(9)is any positive integer)with three potentials(two of which are vector potentials).In addition,the peakon wavefunction of a integrable evolution equation with peakon and the peakon wavefunction of the coupled CH equation are studied.In chapter 2,based on a 3 × 3 matrix spectral problem,a new hierarchy of nonlinear evolution equations is obtained by means of the zero-curvature equation and the Lenard recursion equation,and then the Hamilton structures of this hierarchy are established by using the trace identity.Then the Degasperis-Procesi equation II is derived from a negative flow related to the spectral problem,which is a new generalized equation of the Degasperis-Procesi equation with -peakon.The equation is of fifth order and has 2-peakon.In this paper,the Lax pair and 2-peakon of the Degasperis-Procesi equation II are given,and the 2-peakon is studied with the aid of the properties of distribution function ,and the dynamical system with which the 2-peakon evolve is deduced.Finally,the infinite conserved quantities of Degasperis-Procesi equation II are obtained with the help of the spectral parameter expansions.In chapter 3,4 and 5,three different multi-component soliton equations with peakons are studied respectively.In chapter 3,starting from a(7)+ 2)×(7)+ 2)matrix spectral problem,a new nonlinear vector equation is obtained.Due to no commutativity between vectors and matrices,there are certain limitations in calculation,and it is difficult to solve the Hamilton structures of the vector equation.However,in order to learn more about the integrability of vector equation,we thought about a special case,that is,7)= 2,then a new hierarchy of nonlinear evolution equations is obtained and its Hamilton structures are derived.Similar to the chapter 2,the multi-component Geng-Xue equation is obtained from a negative flow,and the Lax pair and -peakon of the multi-component Geng-Xue equation are given.Whereafter,the dynamical system with which the -peakon evolve and infinite conserved quantities of the multi-component Geng-Xue equation are deduced.Finally,a special reduction of the multi-component Geng-Xue equation is given.In chapter 4,we study the multi-component super CH equation,which is different from the chapter 2 and 3,and involves some knowledge of super integrable and supersymmetry.At first,we introduce a(9)+ 2)×(9)+ 2)matrix spectral problem,and obtain a new nonlinear super integrable vector equation from the compatibility condition.Similarly,due to no commutativity between vectors and matrices,and the properties of super integrable and supersymmetry,the construction of Hamilton structures of super integrable vector equation is quite tedious.Hereon,two special cases,9)= 1 and 9)= 2,are selected,and the corresponding hierarchies of equations and their Hamilton structures are derived respectively.Then a new super integrable vector equation is derived from a negative flow.When a variable takes a particular value,the super integrable vector equation is reduced to the multi-component super CH equation.After that the-peakon of the multi-component super CH equation and the super dynamical system with which the -peakon evolve are deduced.In addition,the infinite conserved quantities of the new super integrable vector equation and the multicomponent super CH equation are derived respectively.In chapter 5,based on a(9)+ 2)×(9)+ 2)matrix spectral problem,a new nonlinear vector equation is obtained.The vector equation is not discussed in this paper,because when9)= 2,unsolvable higher order differential equations appear in the calculation.Similar to chapter 3,the multi-component generalization of the three-component CH equation is obtained from a negative flow,and its -peakon and infinite conserved quantities are deduced,as well as the dynamical system with which the-peakon evolve.In chapter 6,there are two parts: one is to derive a new integrable evolution equation with peakon and its the peakon wavefunction,then the conditions satisfied by the evolution of peakon wavefunction’s coefficients are obtained.The other part is to derive the peakon wavefunction of the coupled CH equation and the conditions satisfied by the evolution of peakon wavefunction’s coefficients,and then we turn the spectral problem into the string problem.