| Abstract: |
| The fractional KdV equation \(u_t + u u_x - |D|^{\alpha} u_x = 0\) contains the KdV (\(\alpha = 2\)), the Benjamin--Ono (\(\alpha = 1\)), the Burgers (an exceptional case, (\(\alpha = 0\))), the Burgers--Hilbert (\(\alpha = -1\)) and the reduced Ostrovskii equation (\(\alpha = -2\)), and has been proposed as a family of dispersive model equations suitable to the study of the balance between dispersive and nonlinear effects. Classical solutions exist globally in time for not too small values of \(\alpha > 0\), whereas wave-breaking comes into play as the dispersion gets weaker (\(\alpha\) negative). Hunter and Ifrim showed that, for \(alpha = -1\), the time of existence of classical solutions in the Burgers-Hilbert equation can be extended from \(1/\varepsilon\) to \(1/\varepsilon^2\) when the initial data is of size \(\varepsilon\). Using a normal-form transformation inspired by theirs, but working almost completely in Fourier space, we extend this result to all \(\alpha \in (-1,0) \cup (0,1)\). |
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