The Hamiltonian Theory Of The Landau-Lifshitz Equations

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 choosingthe 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 twoequations 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.