Riemann - Hilbert Problems in the Continuum Limit of the Toda Lattice
by
Ken McLaughlin
JWB 335, 3:10pm Monday, January 8, 1996
Abstract
The Toda system of differential equations is a Hamiltonian particle
system with Hamiltonian
H = (1/2) \sum_{j = 1}^{n} y_{j}^{2}
+ \sum_{j=1}^{n-1} \exp(x_{j}-x_{j+1}) .
It can be viewed as a chain of n particles, with positions
\{ x_{j} \}_{j=1}^{n},
each linked to its neighbors by "nonlinear springs".
It is a completely integrable system, which means that there exists a
global transformation on the phase space (the transformation is in
fact explicit) such that in the new coordinates, the flow is linear in
time.
A continuum limit of the particle system is obtained by letting the
number of particles in the system tend to infinity. As the ode system
is nonlinear, the continuum limit equations are as well, they are a
hyperbolic system for two unknowns, which may exhibit shock formation.
As the finite dimensional ode system possesses, for arbitrary initial
data, a global solution, one may use the continuum limit of the ode
system as a form of regularization of the hyperbolic pde system,
beyond the time of shock formation. In contrast to the well known
dissipative regularization, what one sees is dispersive
regularization, in which shock fronts produce regions of rapid
oscillations in the solution of the ode system, and hence the
convergence of the ode system to a continuum limit is only in the weak
sense. Moreover the dynamics of the continuum limit is very different
after the time of shock formation.
The procedure that we have used to evaluate this continuum limit
(referred to as the Lax-Levermore method) exploits a representation of
the solution obtained through classical formulae of Orthogonal
Polynomials, and in fact the procedure promises a new direction for
the evaluation of certain asymptotic quantities in Orthogonal
Polynomials.
I will discuss the qualitative behavior of the continuum limit, i.e.
the nonlinear phenomena which is observed numerically, and to what
extent this phenomena has been described rigorously. I will conclude
by describing the Lax-Levermore method, and how a Riemann-Hilbert
factorization problem arises out of such an asymptotic limit.
Requests for preprints and reprints to: rmm@math.utah.edu
Attn: Ken McLaughlin
This source can be found at http://www.math.utah.edu/research/