Modulated waves in a semiclassical continuum limit of an integrable NLS chain

A one‐dimensional integrable lattice system of ODEs for complex functions Q_n(τ) that exhibits dispersive phenomena in the phase is studied. We consider wave solutions of the local form Q_n(τ) ∼ qexp(i(kn + ωτ + c)), in which q, k, and ω modulate on long time and long space scales t = ετ and x = εn. Such solutions arise from initial data of the form Q_n(0) = q(nε) exp(iϕ(nε)/ε), the phase derivative ϕ′ 0 giving the local value of the phase difference k. Formal asymptotic analysis as ε → 0 yields a first‐order system of PDEs for q and ϕ′ as functions of x and t. A certain finite subchain of the discrete system is solvable by an inverse spectral transform. We propose formulae for the asymptotic spectral data and use them to study the limiting behavior of the solution in the case of initial data |Q_n| < 1, which yield hyperbolic PDEs in the formal limit. We show that the hyperbolic case is amenable to Lax‐Levermore theory. The associated maximization problem in the spectral domain is solved by means of a scalar Riemann‐Hilbert problem for a special class of data for all times before breaking of the formal PDEs. Under certain assumptions on asymptotic behaviors, the phase and amplitude modulation of the discrete systems is shown to be governed by the formal PDEs. Modulation equations after breaking time are not studied. Full details of the WKB theory and numerical results are left to a future exposition. © 2000 John Wiley & Sons, Inc.