连续与离散SG方程及相关孤子族的精确解析解
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摘要
本文研究几个可积的偏微分方程,包括连续和离散的sine-Gordon方程(SG),变形Korteweg-de Vries方程(mKdV)等。为计算这些可积模型的精确解析解,应用了代数曲线方法,并与特征值问题非线性化相结合,求解分为“分解,拉直,反演”三个步骤。
首先,从与SG方程和mKdV方程相关的谱问题出发,借助基本恒等式和Lenard分析等强有力工具,得到三族可积模型(孤子族)及相应的零曲率表示。它们分别是:mKdV族,algebraic SG族和2+1 SG族。
在Bargmann约束下,通过将两个谱问题非线性化,得到两个有限维Liouville可积系统。有趣的是这两个系统具有相同的Lax-Moser矩阵和相同的N-守恒积分系,这些守恒积分两两对合,在相空间的某开集上函数独立。孤子方程(可积的偏微分方程)被分解为几个Liouville可积的常微分方程,其相容解生成可积偏微分方程的特解。由Lax-Moser矩阵决定一条代数曲线,在其Jacobi簇上,这些常微分方程的Hamiliton相流被拉直,于是可以直接积出。最后通过Jacobi反演,借助多元theta函数,得到这些偏微分方程的精确解析解。
Darboux变换(DT)是构造孤子方程精确解的十分有效的方法。本文除用此方法解决两个与Toda链相关的离散孤子方程的求解问题之外,还成功地从mKdV-SG族的Darboux变换中找到一个离散谱问题。从连续和离散谱问题组成的Lax对的相容条件出发,导出离散的SG方程。Lax对的离散部分由Darboux变换得到。文中找到相应的一个包含双叶黎曼面分支点的Bargmann约束。利用这个非常特殊的Bargmann约束,将由Darboux变换产生的离散谱问题非线性化,成功得到一个可积辛映射。利用类似连续系统的程序,先在Jacobi簇上将离散流拉直,然后算出离散SG方程的精确解析解。
In the present paper,some integrable partial differential equations(PDEs),including the continuous and the discrete sine-Gordon equations(SG),together with the modified Korteweg-de Vries equations(mKdV),are investigated.Algebraic curve methods,combined with the nonlinearization of the eigenvalue problems,are applied to calculate the exact analytic solutions of these integrable models.They are solved through three steps: decomposition,straightening out and inversion.
Firstly,from two spectral problems associated with SG and mKdV,resorting to the powerful tools of the fundamental identities and the Lenard analysis,three families of integrable models(soliton equations),the mKdV,the algebraic SG and the(2+1)-dimensional SG hierarchies,together with their zero-curvature representations,are derived.
Two finite-dimensional Liouville-integrable systems are obtained from the nonlinearization procedures of the two spectral problems under two Bargmann constraints.It is interesting that they share the same Lax-Moser matrix and the same N-set of integrals on motions,involutive with each other and functionally independent in an open set of the phase space.The soliton equations are decomposed into these Liouville integrable ODEs(ordinary differential equations),whose compatible solutions yield special solutions of the integrable PDEs.The Hamiltonian flows of these ODEs are straightened out in the Jacobi variety of an algebraic curve,determined by the Lax-Moser matrix.Thus they are integrated by quadrature and the final exact analytic solution for these PDEs are obtained through the Jacobi inversion,expressed by means of the multi-variable theta functions.
Two discrete equations related to Toda lattice are studied with Darboux transformation(DT), in addition,a discrete spectral problem is obtained.The discrete SG equation is derived as a compatible condition of the Lax pair,composed of a continuous part and a discrete one.The discrete part is obtained from the DT of mKdV-SG hierarchy.It is succeeded in finding the associated Bargmann's constraint,and in nonlinearizing the DT into an integrable symplectic map,which is also straightened out in the same Jacobi variety.Exact analytic solution for the discrete SG is calculated in a similar procedure.
首先,从与SG方程和mKdV方程相关的谱问题出发,借助基本恒等式和Lenard分析等强有力工具,得到三族可积模型(孤子族)及相应的零曲率表示。它们分别是:mKdV族,algebraic SG族和2+1 SG族。
在Bargmann约束下,通过将两个谱问题非线性化,得到两个有限维Liouville可积系统。有趣的是这两个系统具有相同的Lax-Moser矩阵和相同的N-守恒积分系,这些守恒积分两两对合,在相空间的某开集上函数独立。孤子方程(可积的偏微分方程)被分解为几个Liouville可积的常微分方程,其相容解生成可积偏微分方程的特解。由Lax-Moser矩阵决定一条代数曲线,在其Jacobi簇上,这些常微分方程的Hamiliton相流被拉直,于是可以直接积出。最后通过Jacobi反演,借助多元theta函数,得到这些偏微分方程的精确解析解。
Darboux变换(DT)是构造孤子方程精确解的十分有效的方法。本文除用此方法解决两个与Toda链相关的离散孤子方程的求解问题之外,还成功地从mKdV-SG族的Darboux变换中找到一个离散谱问题。从连续和离散谱问题组成的Lax对的相容条件出发,导出离散的SG方程。Lax对的离散部分由Darboux变换得到。文中找到相应的一个包含双叶黎曼面分支点的Bargmann约束。利用这个非常特殊的Bargmann约束,将由Darboux变换产生的离散谱问题非线性化,成功得到一个可积辛映射。利用类似连续系统的程序,先在Jacobi簇上将离散流拉直,然后算出离散SG方程的精确解析解。
In the present paper,some integrable partial differential equations(PDEs),including the continuous and the discrete sine-Gordon equations(SG),together with the modified Korteweg-de Vries equations(mKdV),are investigated.Algebraic curve methods,combined with the nonlinearization of the eigenvalue problems,are applied to calculate the exact analytic solutions of these integrable models.They are solved through three steps: decomposition,straightening out and inversion.
Firstly,from two spectral problems associated with SG and mKdV,resorting to the powerful tools of the fundamental identities and the Lenard analysis,three families of integrable models(soliton equations),the mKdV,the algebraic SG and the(2+1)-dimensional SG hierarchies,together with their zero-curvature representations,are derived.
Two finite-dimensional Liouville-integrable systems are obtained from the nonlinearization procedures of the two spectral problems under two Bargmann constraints.It is interesting that they share the same Lax-Moser matrix and the same N-set of integrals on motions,involutive with each other and functionally independent in an open set of the phase space.The soliton equations are decomposed into these Liouville integrable ODEs(ordinary differential equations),whose compatible solutions yield special solutions of the integrable PDEs.The Hamiltonian flows of these ODEs are straightened out in the Jacobi variety of an algebraic curve,determined by the Lax-Moser matrix.Thus they are integrated by quadrature and the final exact analytic solution for these PDEs are obtained through the Jacobi inversion,expressed by means of the multi-variable theta functions.
Two discrete equations related to Toda lattice are studied with Darboux transformation(DT), in addition,a discrete spectral problem is obtained.The discrete SG equation is derived as a compatible condition of the Lax pair,composed of a continuous part and a discrete one.The discrete part is obtained from the DT of mKdV-SG hierarchy.It is succeeded in finding the associated Bargmann's constraint,and in nonlinearizing the DT into an integrable symplectic map,which is also straightened out in the same Jacobi variety.Exact analytic solution for the discrete SG is calculated in a similar procedure.
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