Nonlinear Hamiltonian Waves with Constant Frequency and Surface Waves on Vorticity Discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves, and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasilinear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two-dimensional inviscid, incompressible fluid flows. Thus, the Burgers-Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers-Hilbert and asymptotic equations, and show that the asymptotic equation may be also be derived by means of a near-identity transformation. We derive a semi-classical approximation of the asymptotic equation, and show that spatially periodic, harmonic traveling waves are linearly and modulationaly stable. Numerical solutions of the Burgers-Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small-amplitude smooth solutions of the Burgers-Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. c ⃝ 2000 Wiley Periodicals, Inc.