TRACE FORMULAS AND CONSERVATION LAWS FOR NONLINEAR EVOLUTION EQUATIONS

Abstract

New trace formulas for linear operators associated with Lax pairs or zero-curvature representations of completely integrable nonlinear evolution equations and their relation to (polynomial) conservation laws are established. We particularly study the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the sine–Gordon equation, and the infinite Toda lattice though our methods apply to any element of the AKNS–ZS class. In the KdV context, we especially extend the range of validity of the infinite sequence of conservation laws to certain long-range situations in which the underlying one-dimensional Schrödinger operator has infinitely many (negative) eigenvalues accumulating at zero. We also generalize inequalities on moments of the eigenvalues of Schrödinger operators to this long-range setting. Moreover, our contour integration approach naturally leads to higher-order Levinson-type theorems for Schrödinger operators on the line.