Action-angle maps and scattering theory for some finite-dimensional integrable systems. II: Solitons, antisolitons, and their bound states. (English) Zbl 0842.58049
It is the second part of the author’s serial work which present an explicit construction of an action-angle map for the non-relativistic Calogero-Moser systems with \(1/sh^2\) and \(-1/ch^2\) pair potentials, and for relativistic generalizations thereof. There are seven chapters and an appendix in the paper.
Chapter 2 is concerned with the determination of the range of variation of the spectrum of the Lax matrix \(L(p)\) as \(p\) varies over the subset \(\Omega^\#_s\) of \(\Omega_s\) where \(L\) has \(\ell\) complex-conjugate pairs of eigenvalues (see paper I).
Chapter 3 is concerned with showing that the date set can be promoted to a symplectic manifold. Here is a crucial step towards solving this problem: it enables to reduce the problem to the special (pure soliton) case \(N_+ N_- = 0\) already handled in paper I.
In Chapter 4 the algebraic and analytic information assembled in chapters 2 and 3 is combined with geometric information to solve the above problem.
The special case \(N_+ N_- =1\) is given in considerable detail at the beginning of chapter 5. Here it also shown that the Lax matrix is not diagonalizable on the exceptional set (separatrix) \(\Omega_e \equiv \Omega/\Omega_s\).
In chapter 6 the extension information is used to study the character and temporal asymptotics of a large class of mutually commuting Hamiltonian flows, containing in particular the flows generated by the Hamiltonian (1.1) and (1.2) (see this paper) in the nonrelativistic and relativistic case, resp. Besides the general case, this chapter also deals with a special situation of considerable physical interest.
[For Part I see the author, Commun. Math. Phys. 115, No. 1, 127-165 (1988; Zbl 0667.58016), for Part III, see the review below].
Chapter 2 is concerned with the determination of the range of variation of the spectrum of the Lax matrix \(L(p)\) as \(p\) varies over the subset \(\Omega^\#_s\) of \(\Omega_s\) where \(L\) has \(\ell\) complex-conjugate pairs of eigenvalues (see paper I).
Chapter 3 is concerned with showing that the date set can be promoted to a symplectic manifold. Here is a crucial step towards solving this problem: it enables to reduce the problem to the special (pure soliton) case \(N_+ N_- = 0\) already handled in paper I.
In Chapter 4 the algebraic and analytic information assembled in chapters 2 and 3 is combined with geometric information to solve the above problem.
The special case \(N_+ N_- =1\) is given in considerable detail at the beginning of chapter 5. Here it also shown that the Lax matrix is not diagonalizable on the exceptional set (separatrix) \(\Omega_e \equiv \Omega/\Omega_s\).
In chapter 6 the extension information is used to study the character and temporal asymptotics of a large class of mutually commuting Hamiltonian flows, containing in particular the flows generated by the Hamiltonian (1.1) and (1.2) (see this paper) in the nonrelativistic and relativistic case, resp. Besides the general case, this chapter also deals with a special situation of considerable physical interest.
[For Part I see the author, Commun. Math. Phys. 115, No. 1, 127-165 (1988; Zbl 0667.58016), for Part III, see the review below].
Reviewer: Sheng Liren (Hefei)
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q51 | Soliton equations |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
References:
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