Landau-Lifschitz方程的Hamilton理论
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摘要
对非线性微分方程的研究长期以来是数学和物理学中的热门领域。在非线性理论的发展过程中,出现了一些最简单的“典型”的非线性波动方程,在某种意义上这些方程具有普适性质,就象经典的线性D'alamber方程那样,在各种物理现象中都可以遇到它们。例如熟知的KdV方程、非线性Schr(o|¨)dinger方程、sin-Gordon方程和Landau-Lifshitz铁磁方程,都属于这类方程。这些方程(至少在一维情形下)都具有突出的数学性质,它们含有一种隐蔽的代数对称性一利用对辅助线性算子的逆问题方法(即反散射法)可导至“可积性”,我们称之为完全可积性。非线性方程的完全可积性,就是说,该方程描述的是一个多周期系统,即,它是一个Hamilton系统,且可以引入作为正则共轭变量的作用变量和角变量,而Hamilton量可以表为作用变量的函数。因此,由作用变量与角变量的对易关系,作用变量是守恒量;角变量是周期地依赖时间的量,要能引入这样的量,我们自然会特别着眼于在完全可积方程反散射方法求解时得到的那些与时间有关的量的性质。因此,将符合上述条件的作用变量显式地表达出来,我们才能认为建立了完全可积方程的Hamilton理论。
铁磁链的Landau-Lifshitz方程是一类有物理背景的重要的非线性方程组,其重要性在于,Landau-Lifshitz方程是一类典型的1+1维完全可积方程。然而,对此类方程Hamilton理论的研究还存在许多问题未解决。我们来看最简单的情况一完全各向同性的Landau-Lifshitz方程,也称为Heisenberg铁磁方程,在用修正过的反散射法得到其孤子解后,L.D.Faddeev等开始着手建立此类方程的Hamilton理论。在引入自旋变量的Lie-Poisson括号后,将Heisenberg铁磁方程写成以Lie-Poisson括号表出的自旋变量的Hamilton方程的形式,Hamilton量作为自旋变量对x的积分,即坐标积分表达式就无歧义的定出。另一方面,用标准的操作过程,可以求出单式矩阵元的Lie-Poisson括号,因而在连续谱情况下,也唯一地定出系统的角变量和作用变量。由反散射法给出的角变量的时间相依关系,就决定了连续谱情况下Hamilton量的谱参数积分表示,即被积函数是作用变量乘以谱参数的确定函数。于是Hamilton量的两种积分表示,坐标积分表示和谱参数积分表示,都是确定的。下面的任务是从Jost解的渐近行为导出守恒律,得到某一守恒量,其坐标积分表达式正比于Hamilton量的坐标积分表达式,其谱参数积分表达式也正比于Hamilton量的谱参数积分表达式。这种处理是数学物理中的夹逼法。
There have been a very hot field in mathematics and physics that studied nonlinear PDE for a long time. In the development of nonlinear theory, there emerge some simplest typical nonlinear wave equations that possess universality to some extent that we encounter frequently them in a great variety of physical phenomena, as the classic linear D'alamber equation. For example, the well-known KdV equation, nonlinear Schro|¨dinger equation, sin-Gordon equation, and Landau-Lifshitz ferromagnetic equation are this case. These equations have prominent mathematic poverty that they imply a covert algebraic symmetry that the " integrability" can be deduced through the inverse method for the auxiliary linear operator, which is called complete integrability. complete integrability of the nonlinear equations mean that they are multi-periodic systems and imply that they are Hamiltonian systems. Moreover, we can introduce the conjugate action-angle variables while the Hamiltonian can be represented by action variables. Hence, by the hierarchy between the action and angle variables, the action variables are conservation quantities; and angle variable depend periodically on the time variables, therefore, we concentrate obviously on the poverties of the variables depending on the time variables as solving the complete integrability equations through the inverse scattering method, consequently, we ensure to establish the Hamiltonian theory of the completely integrable equations as long as we obtain the explicit representative of the action variables in the conditions mentioned above.Landau-Lifshitz ferromagnetic equations with spin chain are a sort of nonlinear PDEs with important physical background, which are a typical class of completely integrable equations with 1 + 1 dimension, but the problem of researching on the Hamiltonian theory is far from being solved. As for the simplest case that is isotropic Landau-Lifshitz equations and that is also called Heisenberg ferromagnetic equations, L. D. Faddeev and other men attend to establish the Hamiltonian theory after getting the soliton solution using the revised inverse scattering method. The method used to study the Hamiltonian theory of the Landau-Lifshitz equation is as follows. Firstly, introduce Lie-Poisson bracket for spin variables, then the Hamiltonian equation of spin is obtained by suitably choosing
the coordinate representation of Hamiltonian. The Lie-Poisson bracket for elements of the monodromy matrix can be obtained by a standard procedure. Next, the action and angle variables are constructed. The spectral parameter representation of Hamiltonian must be determined from the form of the time dependence of the angle variables obtained from the inverse scattering method, whose integrand is the product of the action variables and the deciding function of the spectral parameter. Thus, these two kinds of representations are determined. The next step is to look for a conserved quantity whose coordinate representation is directly proportional to that of the Hamiltonian of Landau-Lifshitz equation and whose spectral parameter representation is exactly that of the Hamiltonian.It is on the base of the asymptotic behavior of Jost solution that we derive the asymptotic behavior of the transmission coefficient a(k). As for the Heisenberg ferromagnetic equations, the asymptotic behavior of the Jost solution as x —>? oo is determined by L —> Lq = —iha$, and so is that of the Jost solution as k —)■ oo. Because variable x is arbitrary when we discuss the asymptotic behavior of the Jost solution as k —> oo, L = —ikSaaa. The forms of L and L$ are different. Hence, the analysis of the asymptotic behavior of Jost solution as k —> oo using the first compatibility equation shows that the zeroth order term does not vanish which resulted in missing the connection between the 1st order term and the coordinate integral representation of the Hamiltonian. Fogedby gave the spectral representation, but he considers this problem only from the physical point of view, using the fact that Hamiltonian is just the energy of the system. Indeed, he says nothing about the connection between the two representations of the Hamiltonian. To avoid this difficulty Faddeev introduced a constant phase ip as a factor of the transmission coefficient. a(k) was replaced by a(k)e~iip. Although now Iq can vanish, the phase as introduced by Faddeev is not justified, and the coordinate integral representation of 1st order term is not equal to that of the Hamiltonian H. In fact Takhtajan pointed out that the introduction of this additional phase is not correct in the sense that the gauge equivalence between the Heisenberg ferromagnetic equation and the nonlinear Schrodinger equation(NLS) is not maintained. We know that the monodromy matrix is invariant under a gauge transformation, hence, the transmission coefficient should be invariant under a gauge transformation. Hence, the spectral representations of the Hamiltonian of these two
equations mentioned above are equal, whereas the coordinate integral representations of the Hamiltonian of them are gauge equivalent, namely, the coordinate integral representation of the Hamiltonian of the Heisenberg ferromagnetic equations can be attained from that of the NLS. However, this project using gauge transformation is not perfect. Firstly, Takhtajan did not compute any conserved quantities involving the spin variable Sa from the conserved quantities of NLS. Secondly, it is a misconception that in order to establish the Hamiltonian theory for Landau-Lifshitz equations one should find out a known equation, e.g. certain the NLS transformed, which is gauge equivalent to the Landau-Lifshitz equations. This misleading idea delayed establishing a Hamiltonian theory for Landau-Lifshitz equations with an easy axis and with an easy plane recently. Therefore, the misplay of Faddeev result in the state that a series of problems in the Hamiltonian theory of Landau-Lifshitz equations can not be solved. In fact, gauge transformation is a transformation for the compatibility pair of the same equation. From the new compatibility pair, the original equation can be recovered from the same compatible condition, without using any other equations.Hence, we consider the conserved quantities again but from another view point. The essence of the gauge transformation is that for arbitrary x, t, the spin direct which is directly proportional to the spectral parameter k in the compatibility pair is rotated formally to the 3rd axis in the spin space whereas the real change of the spin variables is involved in the gauge transformation independent of the spectral parameter. It is necessary to point out that new physical variables and hypotheses are not introduced in the application of the gauge transformation. When we succeed in introducing a gauge transformation to transform the first operator of compatibility pair into the asymptotic form — ika$ + O(l), the conserved quantities are derived, the zeroth term vanishes, and the first one has the desired form of Hamiltonian. This is how our procedure will overcome the difficulty in deriving of conservation quantities. We also point out that the idea can be extended to establish the Hamiltonian theory of other equations, for example, Landau-Lifshitz equation with an easy axis and with an easy plane.
铁磁链的Landau-Lifshitz方程是一类有物理背景的重要的非线性方程组,其重要性在于,Landau-Lifshitz方程是一类典型的1+1维完全可积方程。然而,对此类方程Hamilton理论的研究还存在许多问题未解决。我们来看最简单的情况一完全各向同性的Landau-Lifshitz方程,也称为Heisenberg铁磁方程,在用修正过的反散射法得到其孤子解后,L.D.Faddeev等开始着手建立此类方程的Hamilton理论。在引入自旋变量的Lie-Poisson括号后,将Heisenberg铁磁方程写成以Lie-Poisson括号表出的自旋变量的Hamilton方程的形式,Hamilton量作为自旋变量对x的积分,即坐标积分表达式就无歧义的定出。另一方面,用标准的操作过程,可以求出单式矩阵元的Lie-Poisson括号,因而在连续谱情况下,也唯一地定出系统的角变量和作用变量。由反散射法给出的角变量的时间相依关系,就决定了连续谱情况下Hamilton量的谱参数积分表示,即被积函数是作用变量乘以谱参数的确定函数。于是Hamilton量的两种积分表示,坐标积分表示和谱参数积分表示,都是确定的。下面的任务是从Jost解的渐近行为导出守恒律,得到某一守恒量,其坐标积分表达式正比于Hamilton量的坐标积分表达式,其谱参数积分表达式也正比于Hamilton量的谱参数积分表达式。这种处理是数学物理中的夹逼法。
There have been a very hot field in mathematics and physics that studied nonlinear PDE for a long time. In the development of nonlinear theory, there emerge some simplest typical nonlinear wave equations that possess universality to some extent that we encounter frequently them in a great variety of physical phenomena, as the classic linear D'alamber equation. For example, the well-known KdV equation, nonlinear Schro|¨dinger equation, sin-Gordon equation, and Landau-Lifshitz ferromagnetic equation are this case. These equations have prominent mathematic poverty that they imply a covert algebraic symmetry that the " integrability" can be deduced through the inverse method for the auxiliary linear operator, which is called complete integrability. complete integrability of the nonlinear equations mean that they are multi-periodic systems and imply that they are Hamiltonian systems. Moreover, we can introduce the conjugate action-angle variables while the Hamiltonian can be represented by action variables. Hence, by the hierarchy between the action and angle variables, the action variables are conservation quantities; and angle variable depend periodically on the time variables, therefore, we concentrate obviously on the poverties of the variables depending on the time variables as solving the complete integrability equations through the inverse scattering method, consequently, we ensure to establish the Hamiltonian theory of the completely integrable equations as long as we obtain the explicit representative of the action variables in the conditions mentioned above.Landau-Lifshitz ferromagnetic equations with spin chain are a sort of nonlinear PDEs with important physical background, which are a typical class of completely integrable equations with 1 + 1 dimension, but the problem of researching on the Hamiltonian theory is far from being solved. As for the simplest case that is isotropic Landau-Lifshitz equations and that is also called Heisenberg ferromagnetic equations, L. D. Faddeev and other men attend to establish the Hamiltonian theory after getting the soliton solution using the revised inverse scattering method. The method used to study the Hamiltonian theory of the Landau-Lifshitz equation is as follows. Firstly, introduce Lie-Poisson bracket for spin variables, then the Hamiltonian equation of spin is obtained by suitably choosing
the coordinate representation of Hamiltonian. The Lie-Poisson bracket for elements of the monodromy matrix can be obtained by a standard procedure. Next, the action and angle variables are constructed. The spectral parameter representation of Hamiltonian must be determined from the form of the time dependence of the angle variables obtained from the inverse scattering method, whose integrand is the product of the action variables and the deciding function of the spectral parameter. Thus, these two kinds of representations are determined. The next step is to look for a conserved quantity whose coordinate representation is directly proportional to that of the Hamiltonian of Landau-Lifshitz equation and whose spectral parameter representation is exactly that of the Hamiltonian.It is on the base of the asymptotic behavior of Jost solution that we derive the asymptotic behavior of the transmission coefficient a(k). As for the Heisenberg ferromagnetic equations, the asymptotic behavior of the Jost solution as x —>? oo is determined by L —> Lq = —iha$, and so is that of the Jost solution as k —)■ oo. Because variable x is arbitrary when we discuss the asymptotic behavior of the Jost solution as k —> oo, L = —ikSaaa. The forms of L and L$ are different. Hence, the analysis of the asymptotic behavior of Jost solution as k —> oo using the first compatibility equation shows that the zeroth order term does not vanish which resulted in missing the connection between the 1st order term and the coordinate integral representation of the Hamiltonian. Fogedby gave the spectral representation, but he considers this problem only from the physical point of view, using the fact that Hamiltonian is just the energy of the system. Indeed, he says nothing about the connection between the two representations of the Hamiltonian. To avoid this difficulty Faddeev introduced a constant phase ip as a factor of the transmission coefficient. a(k) was replaced by a(k)e~iip. Although now Iq can vanish, the phase as introduced by Faddeev is not justified, and the coordinate integral representation of 1st order term is not equal to that of the Hamiltonian H. In fact Takhtajan pointed out that the introduction of this additional phase is not correct in the sense that the gauge equivalence between the Heisenberg ferromagnetic equation and the nonlinear Schrodinger equation(NLS) is not maintained. We know that the monodromy matrix is invariant under a gauge transformation, hence, the transmission coefficient should be invariant under a gauge transformation. Hence, the spectral representations of the Hamiltonian of these two
equations mentioned above are equal, whereas the coordinate integral representations of the Hamiltonian of them are gauge equivalent, namely, the coordinate integral representation of the Hamiltonian of the Heisenberg ferromagnetic equations can be attained from that of the NLS. However, this project using gauge transformation is not perfect. Firstly, Takhtajan did not compute any conserved quantities involving the spin variable Sa from the conserved quantities of NLS. Secondly, it is a misconception that in order to establish the Hamiltonian theory for Landau-Lifshitz equations one should find out a known equation, e.g. certain the NLS transformed, which is gauge equivalent to the Landau-Lifshitz equations. This misleading idea delayed establishing a Hamiltonian theory for Landau-Lifshitz equations with an easy axis and with an easy plane recently. Therefore, the misplay of Faddeev result in the state that a series of problems in the Hamiltonian theory of Landau-Lifshitz equations can not be solved. In fact, gauge transformation is a transformation for the compatibility pair of the same equation. From the new compatibility pair, the original equation can be recovered from the same compatible condition, without using any other equations.Hence, we consider the conserved quantities again but from another view point. The essence of the gauge transformation is that for arbitrary x, t, the spin direct which is directly proportional to the spectral parameter k in the compatibility pair is rotated formally to the 3rd axis in the spin space whereas the real change of the spin variables is involved in the gauge transformation independent of the spectral parameter. It is necessary to point out that new physical variables and hypotheses are not introduced in the application of the gauge transformation. When we succeed in introducing a gauge transformation to transform the first operator of compatibility pair into the asymptotic form — ika$ + O(l), the conserved quantities are derived, the zeroth term vanishes, and the first one has the desired form of Hamiltonian. This is how our procedure will overcome the difficulty in deriving of conservation quantities. We also point out that the idea can be extended to establish the Hamiltonian theory of other equations, for example, Landau-Lifshitz equation with an easy axis and with an easy plane.
引文
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