非线性可积系统及其相关问题
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摘要
本文的主要内容包括:
1.从一个3×3矩阵谱问题出发,推导出广义MKdV方程族,构造此方程族Hamilton结构,证明在Liouville意义下是可积的.通过对称约束得到有限维Hamilton系统。通过Lie代数半直和构造可积耦合系统,利用变分恒等式得到可积耦合系统的Hamilton结构。由拟微分算子技术构造非等谱非交换的KP方程族。
2.首次给出两类变系数非线性演化方程的Frobenius可积分解,包括变系数KdV方程,势KdV方程,Boussinesq方程,Camassa-Holm方程等。把(2+1)维广义KP,cKP,mKP方程分解为(1+1)维可积方程,研究2阶复AKNS方程和3阶复AKNS方程的相容解与广义(2+1)维KP,cKP,mKP方程的解之间的关系,并利用Darboux变换得到它们的孤子解,进而将解表示为双Wronskian行列式形式。
3.分别利用Hirota方法与Wronskian技术给出五阶KdV方程及其约束方程的精确解,并证明两种解的一致性.将双Wronskian元素满足的条件推广到矩阵情形,导出等谱Levi方程广义双Wronskian行列式解,其中包括孤子解、有理解、Matveev解、complexiton解及混合解。给出非等谱Levi方程的双Wronskian行列式解。研究等谱与非等谱Levi方程孤子解的动力学行为包括单孤子的特征以及双孤子的散射。
The major contents in this dissertation consist of:
1.The generalized MKdV equation hierarchy is constrcted from a 3×3 matrix spectral problem,then,its Hamilton structure is presented and the Liouville integrability is demonstrated.By establishing binary symmetric constraints,the constrained flows of the hierarchy are presented,which are then reduced to finite-dimension Hamilton systems. Then,an integrable couplings system is obtained by applying a semi-direct sum of Lie algebras.Then the Hamilton structure of the integrable couplings is constructed by the variational identity.The nonisospectral noncommutative KP hierarchy is presented by pseudo-difference operator technique.
2.Frobenius integrable decompositions are introduced for two classes nonlinear evolution equations with variable coefficients for the first time,including the KdV equation and potential KdV equation,the Boussinesq equation and the Camassa-Holm equation with variable coefficients etc.The generalized(2+1)-dimensional KP,cKP and mKP were decomposed into the(1+1)-dimensional integrable equations.We investigate the relations between the consistent solutions of the 2-order and 3-order complex AKNS equations and the solutions of three(2+1)-dimensional soliton equations.With the help of the Darboux transformation,we get the explicit solutions of the generalized KP equation,cKP equation and mKP equation,and they are presented in double Wronskian form.
3.The first one is to derive the exact solutions for the fifth-order KdV equation and its constraint equation through Hirota method and Wronskian technique,respectively.Further, the uniformity of these two kinds of exact solutions is proved.The second one is to obtain the general double Wronskian solution of the isospectral Levi equations by generalizing the equation satisfied by Wronskian entries to the matrix equation,including soliton solutions,rational solutions,Matveev solutions,complexiton solutions and interaction solutions. The last part,we derive the double Wronskian solution of the non-isospectral Levi equations.The dynamics including one-soliton characteristics and two-solitons scattering for the isospectral and non-isospectral Levi equations are investingated analytically.
1.从一个3×3矩阵谱问题出发,推导出广义MKdV方程族,构造此方程族Hamilton结构,证明在Liouville意义下是可积的.通过对称约束得到有限维Hamilton系统。通过Lie代数半直和构造可积耦合系统,利用变分恒等式得到可积耦合系统的Hamilton结构。由拟微分算子技术构造非等谱非交换的KP方程族。
2.首次给出两类变系数非线性演化方程的Frobenius可积分解,包括变系数KdV方程,势KdV方程,Boussinesq方程,Camassa-Holm方程等。把(2+1)维广义KP,cKP,mKP方程分解为(1+1)维可积方程,研究2阶复AKNS方程和3阶复AKNS方程的相容解与广义(2+1)维KP,cKP,mKP方程的解之间的关系,并利用Darboux变换得到它们的孤子解,进而将解表示为双Wronskian行列式形式。
3.分别利用Hirota方法与Wronskian技术给出五阶KdV方程及其约束方程的精确解,并证明两种解的一致性.将双Wronskian元素满足的条件推广到矩阵情形,导出等谱Levi方程广义双Wronskian行列式解,其中包括孤子解、有理解、Matveev解、complexiton解及混合解。给出非等谱Levi方程的双Wronskian行列式解。研究等谱与非等谱Levi方程孤子解的动力学行为包括单孤子的特征以及双孤子的散射。
The major contents in this dissertation consist of:
1.The generalized MKdV equation hierarchy is constrcted from a 3×3 matrix spectral problem,then,its Hamilton structure is presented and the Liouville integrability is demonstrated.By establishing binary symmetric constraints,the constrained flows of the hierarchy are presented,which are then reduced to finite-dimension Hamilton systems. Then,an integrable couplings system is obtained by applying a semi-direct sum of Lie algebras.Then the Hamilton structure of the integrable couplings is constructed by the variational identity.The nonisospectral noncommutative KP hierarchy is presented by pseudo-difference operator technique.
2.Frobenius integrable decompositions are introduced for two classes nonlinear evolution equations with variable coefficients for the first time,including the KdV equation and potential KdV equation,the Boussinesq equation and the Camassa-Holm equation with variable coefficients etc.The generalized(2+1)-dimensional KP,cKP and mKP were decomposed into the(1+1)-dimensional integrable equations.We investigate the relations between the consistent solutions of the 2-order and 3-order complex AKNS equations and the solutions of three(2+1)-dimensional soliton equations.With the help of the Darboux transformation,we get the explicit solutions of the generalized KP equation,cKP equation and mKP equation,and they are presented in double Wronskian form.
3.The first one is to derive the exact solutions for the fifth-order KdV equation and its constraint equation through Hirota method and Wronskian technique,respectively.Further, the uniformity of these two kinds of exact solutions is proved.The second one is to obtain the general double Wronskian solution of the isospectral Levi equations by generalizing the equation satisfied by Wronskian entries to the matrix equation,including soliton solutions,rational solutions,Matveev solutions,complexiton solutions and interaction solutions. The last part,we derive the double Wronskian solution of the non-isospectral Levi equations.The dynamics including one-soliton characteristics and two-solitons scattering for the isospectral and non-isospectral Levi equations are investingated analytically.
引文
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