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/