推广的mKP方程族
|
摘要
孤子方程最基本的性质是它们可以写成一对线性问题的可积条件。如KdV方程,若假设本征函数随时间的发展由特定的微分算子N实现,再结合一维定态Schr¨odinger方程,就得到Schr¨odinger算子L与N算子的相容性表现为L随时变化由其本身与微分算子对易给出。从算子变化与对易角度来研究看待各种可积系统是很重要的;可积理论的一个基本问题就是寻找非线性偏微分方程与算子对L、N,使得该方程是L、N的相容性条件。此时称该方程是Lax意义下的可积系,算子对L、N称为Lax对。
带自相容源(以下简称为“带源”)的可积系统在物理学中有广泛的应用。对于如何构造带源的方程,近来流行各种推广算子的办法。我们知道,标准的算子法构建方程族[1]的首要假设是存在一系列所谓的时间流,微分代数元沿着特定流的变化是通过相应的微分算子与给定的拟微分算子L的对易来给出的。从Gelfand-Dickey方程族到KP方程族,以及mKP方程族的推广则是令给定的“L”算子一步步地选取更一般的形式。
另一方面,如文献[2]表明,对于2+1维(两个时间变量和一个空间变量)的情形,利用对称生成函数以及将约化方程看作带源二维方程族的静态方程,得到了带源二维方程族及其Lax表示。出于启发,我们可以采取如下观点:各种约束流虽然脱胎于原有的时间流,但在可能的约束条件下,它又是标准假定的时间流之变异;这使得我们干脆可以把种种约束流看成是一种新的时间流,并可假设微分代数元沿着新流的变化是刚好是通过相应的约束算子与给定的拟微分算子的对易来给出的。
在这个思想的指导下,我们按照相似逻辑,在拟微分算子环的理论框架内分析了约束算子的运算特性,通过证明了新旧两种时间流的可交换性,最终建立了所谓的推广的mKP方程族(emKP),其间自然地得到相应的Lax表示。例子表明,有某些重要的已知方程族如可由我们的emKP取特定参数条件得到。
Soliton equations can be written as the integrable conditions of a couple of linear problems, which is the most fundamental properties. For example, in the 1-dimensional stationary Schr¨odinger equation, if we suppose that the eigen function’s divergence can be given by a pure differential operator N, then the consistent relation between the Schr¨odinger operator L and the operator N will be written as the time evolution of L equivalent to the commutation of L and N. So it’s important to study the integrable systems in a similar way. One of the fundamental problems in the theory of integrable systems is to look for the nonlinear PDE and a couple of operators Land N, such that the nonlinear PDE is just the consistent relation between the operators L and N; in which condition we call the nonlinear PDE is a integrable systems in Lax’s meaning, while L and N are called a Lax pair.
Integrable systems with self-consistent sources have been widely used in physics. In order to construct equations of this type, we need generalize the corresponding operators. It’s known that in a given system characterized by L, the supposed existence of the so-called time-flows is of the first importance; the time evolution of every element in the differential algebra is given by the commutation of some corresponding operator and the common one L. The generalization from GD hierarchies to KP hierarchies, and then to mKP hierarchies is realized by extending the operator L.
What’s more, for 2+1 dimensional(two temporal variables and one discrete spatial variable) case in[2], by using symmetry generating functions and treating the constrained equation as the stationary equation for 2-dimensional hierarchy with sources, we obtained 2-d hierarchy with sources and its Lax representation. Enlightened by the idea reflectded in that paper, we now take a view of the constrained flows as exactly some new time flows; at the same time, we may suppose operators in constraint conditions respectively.
With the guidance of the idea just described above, we then analyze the properties of the operators in constraint in the frame of a ring made up of pseudodifferential operators, prove the commutable relation of the new and the original time flows, finally construct the so-called extended mKP hierarchy(emKP), the Lax representation is also obtained. As the examples show, some important known hierarchies can be achieved via choosing certain parameters in the emKP hierarchy.
带自相容源(以下简称为“带源”)的可积系统在物理学中有广泛的应用。对于如何构造带源的方程,近来流行各种推广算子的办法。我们知道,标准的算子法构建方程族[1]的首要假设是存在一系列所谓的时间流,微分代数元沿着特定流的变化是通过相应的微分算子与给定的拟微分算子L的对易来给出的。从Gelfand-Dickey方程族到KP方程族,以及mKP方程族的推广则是令给定的“L”算子一步步地选取更一般的形式。
另一方面,如文献[2]表明,对于2+1维(两个时间变量和一个空间变量)的情形,利用对称生成函数以及将约化方程看作带源二维方程族的静态方程,得到了带源二维方程族及其Lax表示。出于启发,我们可以采取如下观点:各种约束流虽然脱胎于原有的时间流,但在可能的约束条件下,它又是标准假定的时间流之变异;这使得我们干脆可以把种种约束流看成是一种新的时间流,并可假设微分代数元沿着新流的变化是刚好是通过相应的约束算子与给定的拟微分算子的对易来给出的。
在这个思想的指导下,我们按照相似逻辑,在拟微分算子环的理论框架内分析了约束算子的运算特性,通过证明了新旧两种时间流的可交换性,最终建立了所谓的推广的mKP方程族(emKP),其间自然地得到相应的Lax表示。例子表明,有某些重要的已知方程族如可由我们的emKP取特定参数条件得到。
Soliton equations can be written as the integrable conditions of a couple of linear problems, which is the most fundamental properties. For example, in the 1-dimensional stationary Schr¨odinger equation, if we suppose that the eigen function’s divergence can be given by a pure differential operator N, then the consistent relation between the Schr¨odinger operator L and the operator N will be written as the time evolution of L equivalent to the commutation of L and N. So it’s important to study the integrable systems in a similar way. One of the fundamental problems in the theory of integrable systems is to look for the nonlinear PDE and a couple of operators Land N, such that the nonlinear PDE is just the consistent relation between the operators L and N; in which condition we call the nonlinear PDE is a integrable systems in Lax’s meaning, while L and N are called a Lax pair.
Integrable systems with self-consistent sources have been widely used in physics. In order to construct equations of this type, we need generalize the corresponding operators. It’s known that in a given system characterized by L, the supposed existence of the so-called time-flows is of the first importance; the time evolution of every element in the differential algebra is given by the commutation of some corresponding operator and the common one L. The generalization from GD hierarchies to KP hierarchies, and then to mKP hierarchies is realized by extending the operator L.
What’s more, for 2+1 dimensional(two temporal variables and one discrete spatial variable) case in[2], by using symmetry generating functions and treating the constrained equation as the stationary equation for 2-dimensional hierarchy with sources, we obtained 2-d hierarchy with sources and its Lax representation. Enlightened by the idea reflectded in that paper, we now take a view of the constrained flows as exactly some new time flows; at the same time, we may suppose operators in constraint conditions respectively.
With the guidance of the idea just described above, we then analyze the properties of the operators in constraint in the frame of a ring made up of pseudodifferential operators, prove the commutable relation of the new and the original time flows, finally construct the so-called extended mKP hierarchy(emKP), the Lax representation is also obtained. As the examples show, some important known hierarchies can be achieved via choosing certain parameters in the emKP hierarchy.
引文
[1] L.A. Dickey. Soliton equations and Hamiltonian Systems. 2nd Edition. World Scientific. 2003.
[2] Ting Xiao and Yunbo Zeng. A New Constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources. Physica A, 2005, 353:38-60.
[3] Xiaojun Liu, Yunbo Zeng and Runliang Lin. A new extended KP Hierarchy. Phys. Lett. A, 2008, 372:3819-3823.
[4] W. Oevel, W. Strampp, Constrained KP hierarchy and bi-Hamiltonian Structures. Commun. Math. Phys. 1993, 157:51-81.
[5] I. Loris. Dimensional reductions of BKP and CKP hierarchies. J. Phys A: Math. Gen.2001, 34:3447-3459.
[6] W. Strampp. The Kaup-Broer system and trilinear odes of Painleve type. JMP. 1994, 35: 5850-5861.
[7] Xiaojun Liu, Runliang Lin, Bo Jin, and Yunbo Zeng. A generalized dressing approach for solving the extended KP and the extended mKP hierarchy. J.Math.Phys.2009, 50: 053506.
[8] Dianlou Du, Cewen Cao, YongTang Wu. The nonlinearized eigenvalue problem of the Toda hierarchy in the Lie-Poisson framework Physica A, 2000, 285:332-350.
[9] V. K. Mel’nikov. New method for deriving nonlinear integrable systems. JMP. 1990, 31:1106.
[10]J. Sidorenko, W. Strampp. Symmetry constraints of the KP hierarchy. Inverse Problems. 1991, 7: L37-L43.
[11] M. Adler. On a trace functional for formal pseudo-differential operators and thesymplectic structure of the Korteweg-devries type equations. Inventions math. 1979, 50:219-248.
[12]A. N. Parshin. Ring of Formal Pseudo-differential Operators.Proc. Steklov Math. Inst.1999, 224:266-280.
[13]N. J.Zabusky , M. D. Kruskal . Interaction of”Solitons”in a Collisionless Plasma and the Recurrence of Initial States. Phys. Rev. Lett., 1965, 15 (6):240–243.
[14]C. S. Gardner, J. M. Greene, M. D. Kruskal, et al. Method for Solving the Korteweg-de VriesEquation. Phys. Rev. Lett., 1967, 19 (19):1095–1097.
[15]P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math., 1968, 21:467–490.
[16]V. E. Zakharov, L. D. Faddeev. Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Appl., 1971, 5 (4):280–287.
[17]M. Antonowicz. Gel0fand-Diki?? hierarchies with sources and Lax representations for restricted flows. Phys. Lett. A, 1992, 165 (1):47–52.
[18]M. Antonowicz, Rauch-Wojciechowski S. Lax representation for restricted flows of the KdV hierarchy and for the Kepler problem. Phys. Lett. A, 1992, 171 (5-6):303–310.
[19]M. Antonowicz, Rauch-Wojciechowski S. Soliton hierarchies with sources and Lax representation for restricted flows. Inverse Problems, 1993, 9 (2):201–215.
[20]Zeng YB, Li YS. The deduction of the Lax representation for constrained flows from the adjoint representation. J. Phys. A, 1993, 26 (5):L273–L278.
[21]Zeng YB. New factorization of the Kaup-Newell hierarchy. Phys. D, 1994, 73(3):171–188.
[22]Zeng YB, Lin RL, Wang X. Bi-Hamiltonian structure of the KdV hierarchy with self-consistent sources. Adv. Math. (China), 1998, 27 (5):451–463.
[23]Zeng YB. Bi-Hamiltonian structure of JM hierarchy with self-consistent sources. Phys. A, 1999, 262 (3-4):405–419.
[24]肖霆. 2+1维带自相容源的孤立子方程簇及对应的无色散系统[博士学位论文].北京清华大学数学科学系, 2006.
[25]V. E. Zakharov, E. A. Kuznetsov. Multiscale expansions in the theory of systems integrable by the inverse scattering transform. Phys. D, 1986, 18 (1-3):455–463.
[26]Cao CW. Nonlinearization of the Lax system for AKNS hierarchy. Sci. China Ser. A, 1990, 33 (5):528–536.
[27]Zeng YB, Li YS. The constraints of potentials and the finite-dimensional integrable systems. J. Math. Phys., 1989, 30 (8):1679–1689.
[28]Zeng YB, Li YS. An approach to the integrability of Hamiltonian systems obtained by reduction. J. Phys. A, 1990, 23 (3):L89–L94.
[29]Lou SY, Hu XB. Infinitely many Lax pairs and symmetry constraints of the KP equation. J. Math. Phys., 1997, 38 (12):6401–6427
[2] Ting Xiao and Yunbo Zeng. A New Constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources. Physica A, 2005, 353:38-60.
[3] Xiaojun Liu, Yunbo Zeng and Runliang Lin. A new extended KP Hierarchy. Phys. Lett. A, 2008, 372:3819-3823.
[4] W. Oevel, W. Strampp, Constrained KP hierarchy and bi-Hamiltonian Structures. Commun. Math. Phys. 1993, 157:51-81.
[5] I. Loris. Dimensional reductions of BKP and CKP hierarchies. J. Phys A: Math. Gen.2001, 34:3447-3459.
[6] W. Strampp. The Kaup-Broer system and trilinear odes of Painleve type. JMP. 1994, 35: 5850-5861.
[7] Xiaojun Liu, Runliang Lin, Bo Jin, and Yunbo Zeng. A generalized dressing approach for solving the extended KP and the extended mKP hierarchy. J.Math.Phys.2009, 50: 053506.
[8] Dianlou Du, Cewen Cao, YongTang Wu. The nonlinearized eigenvalue problem of the Toda hierarchy in the Lie-Poisson framework Physica A, 2000, 285:332-350.
[9] V. K. Mel’nikov. New method for deriving nonlinear integrable systems. JMP. 1990, 31:1106.
[10]J. Sidorenko, W. Strampp. Symmetry constraints of the KP hierarchy. Inverse Problems. 1991, 7: L37-L43.
[11] M. Adler. On a trace functional for formal pseudo-differential operators and thesymplectic structure of the Korteweg-devries type equations. Inventions math. 1979, 50:219-248.
[12]A. N. Parshin. Ring of Formal Pseudo-differential Operators.Proc. Steklov Math. Inst.1999, 224:266-280.
[13]N. J.Zabusky , M. D. Kruskal . Interaction of”Solitons”in a Collisionless Plasma and the Recurrence of Initial States. Phys. Rev. Lett., 1965, 15 (6):240–243.
[14]C. S. Gardner, J. M. Greene, M. D. Kruskal, et al. Method for Solving the Korteweg-de VriesEquation. Phys. Rev. Lett., 1967, 19 (19):1095–1097.
[15]P. D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math., 1968, 21:467–490.
[16]V. E. Zakharov, L. D. Faddeev. Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Appl., 1971, 5 (4):280–287.
[17]M. Antonowicz. Gel0fand-Diki?? hierarchies with sources and Lax representations for restricted flows. Phys. Lett. A, 1992, 165 (1):47–52.
[18]M. Antonowicz, Rauch-Wojciechowski S. Lax representation for restricted flows of the KdV hierarchy and for the Kepler problem. Phys. Lett. A, 1992, 171 (5-6):303–310.
[19]M. Antonowicz, Rauch-Wojciechowski S. Soliton hierarchies with sources and Lax representation for restricted flows. Inverse Problems, 1993, 9 (2):201–215.
[20]Zeng YB, Li YS. The deduction of the Lax representation for constrained flows from the adjoint representation. J. Phys. A, 1993, 26 (5):L273–L278.
[21]Zeng YB. New factorization of the Kaup-Newell hierarchy. Phys. D, 1994, 73(3):171–188.
[22]Zeng YB, Lin RL, Wang X. Bi-Hamiltonian structure of the KdV hierarchy with self-consistent sources. Adv. Math. (China), 1998, 27 (5):451–463.
[23]Zeng YB. Bi-Hamiltonian structure of JM hierarchy with self-consistent sources. Phys. A, 1999, 262 (3-4):405–419.
[24]肖霆. 2+1维带自相容源的孤立子方程簇及对应的无色散系统[博士学位论文].北京清华大学数学科学系, 2006.
[25]V. E. Zakharov, E. A. Kuznetsov. Multiscale expansions in the theory of systems integrable by the inverse scattering transform. Phys. D, 1986, 18 (1-3):455–463.
[26]Cao CW. Nonlinearization of the Lax system for AKNS hierarchy. Sci. China Ser. A, 1990, 33 (5):528–536.
[27]Zeng YB, Li YS. The constraints of potentials and the finite-dimensional integrable systems. J. Math. Phys., 1989, 30 (8):1679–1689.
[28]Zeng YB, Li YS. An approach to the integrability of Hamiltonian systems obtained by reduction. J. Phys. A, 1990, 23 (3):L89–L94.
[29]Lou SY, Hu XB. Infinitely many Lax pairs and symmetry constraints of the KP equation. J. Math. Phys., 1997, 38 (12):6401–6427