The third order Benjamin–Ono equation on the torus: Well-posedness, traveling waves and stability

Abstract

We consider the third order Benjamin-Ono equation on the torusWe prove that for any t ∈ R, the flow map continuously extends to H s r,0 (T) if s ≥ 0, but does not admit a continuous extension to H −s r,0 (T) if 0 < s < 1 2 . Moreover, we show that the extension is not weakly sequentially continuous in L 2 r,0 (T). We then classify the traveling wave solutions for the third order Benjamin-Ono equation in L 2 r,0 (T) and study their orbital stability.

On the $L^2$ well-posedness and decay estimate of third order Benjamin-Ono equation

“…This yields a bound of 4 , then we use (8.5) to bound χ R φx (x) and χ R φx (y) in L 2 . Denoting x 1 = min{x 0 , y − t}, we integrate by parts to get…”

“…Tanaka [18] showed that a more general third order type Benjamin-Ono type of equations is well-posed in H s (T), s ≥ 5 2 . Gassot [4] proved that for any t ∈ R, the third order Benjamin-Ono flow map continuously extends to H s r,0 (T) if s ≥ 0, but does not admit a continuous extension to H −s r,0 (T) if 0 < s < 1 2 . More generally, people even consider higher order Benjamin-Ono equation, such as the fourth order Benjamin-Ono equations, see for instance [19].…”

Zero-dispersion limit for the Benjamin-Ono equation on the torus with single well initial data

“…In this definition, µ n and κ n are functions of the eigenvalues (λ n (u; ε)) n defined in (10) and in (11).…”

A nonlinear Fourier transform for the Benjamin–Ono equation on the torus and applications

“…For every j ≥ 2, it is possible to study the Hamiltonian evolution associated to the energy E j by using the same nonlinear Fourier transform Φ. In [11], the third-order equation, which corresponds to E 2 , is studied in detail. In particular, though the critical scaling exponent is still s = −1/2, wellposedness is proved to hold in H s if and only if s ≥ 0.…”