Waves of maximal height for a class of nonlocal equations with homogeneous symbols

Abstract

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order´r, where r ą 1. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol mpkq " k´2. Thereby we recover its unique highest 2π-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.

Symmetry of periodic traveling waves for nonlocal dispersive equations

“…for some constant of integration B ∈ R. The present work is not devoted to the existence of solutions of (1.2), but rather to the symmetry of solutions whenever they exist. Results on the existence of solutions of (1.2) in the periodic setting can be found for instance in [5,17,18] and references therein. Our main result reads:…”

“…However, if φ = c 2 at some point, then φ may exhibit a singularity in the form of a cusp or peak, cf. [5,18] so that φ looses its smoothness property. Such a cusp or corner singularity can be compared to a stagnation point on the surface for solutions of the water wave problem, similar as the appearance of a peak in the extremal Stokes wave.…”

Waves of maximal height for a class of nonlocal equations with inhomogeneous symbols

“…Building on the integral kernel K, we prove the existence and regularity properties of a highest, 2π-periodic traveling-wave solution of equation (1.2). Our most direct influence are the works by Ehrnström and Wahlén [12] and by Bruell and Dhara [5]. In the former work, the investigation started with the Whitham equation…”

“…The main result is the existence of a highest, cusped, P -periodic travelingwave solution, which is even, strictly decreasing, smooth on each half-period, and belongs to the Hölder space C 1 2 (R). The proof is based on the regularity and monotonicity properties of the convolution kernel induced by m. On the other hand, Bruell and Dhara [5] considered the fractional KdV equation as in equation (1.1), where L is the Fourier multiplier operator given by the homogeneous symbol…”

“…In the present paper, we will combine the approach in [12] to treat inhomogeneous symbols, and the derivation for an estimate of a priori traveling-wave solution to nonlocal equations as in [5]. We recall that the symbol m for the fractional KdV equation (1.1) is Bessel potential function, which is inhomogeneous and of order strictly smaller than −1.…”