×

Solution of the finite complex Toda lattice by the method of inverse spectral problem. (English) Zbl 1285.65044

An approximate procedure for solving initial value problems for finite-dimensional Toda lattices with complex initial conditions is proposed.
The Toda lattice is a system of particles on a line interacting pairwise with exponential forces; in this paper their motion is determined by the Hamiltonian \[ H(p,q) = (1/2) \sum_{n=0}^{N-1} p_n^2 + \sum_{n=0}^{N-2} \exp ( q_n- q_{n-1} ),\tag{1} \] where \(p_j, q_j\) are the momenta and coordinate of this \(N\)-degree of freedom system. It is shown that after the so-called Flaschka transformation the Hamilton’s equations of (1) can be written as a matrix Lax equation for a Jacobi matrix \( J = J(t)\) in the form \( \dot{J} = J A - A J \) (2), with \( J(0) \in {\mathbb C}^{N \times N}\). Now, since all solutions of this equation satisfy \( J(t) = X(t) J(0) X(t)^{-1}\) for some invertible \(N \times N\) matrix \( X(t)\), it is an isospectral flow.
The authors propose to obtain the Jacobi matrix \( J(t)\) for each \(t\) from its spectral contents and some theoretical results are given relating the matrix \(J\) with its spectral data, and an algorithm for the reconstruction of \(J\) from its spectral data is presented by means of the resolvent of \( J\).
Finally, in the case \(N=2\), explicit expressions are presented for the above general algorithm.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
06B99 Lattices
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
PDFBibTeX XMLCite
Full Text: DOI