# Benjamin-Ono equation

**Benjamin-Ono equation**

The Benjamin-Ono equation (BO) Bj1967, On1975, which models one-dimensional internal waves in deep water, is given by

where is the Hilbert transform. This equation is completely integrable (see e.g., AbFs1983, CoiWic1990).

Scaling is and the following results are known:

- LWP in for Ta2004
- Global weak solutions exist for data Sau1979, GiVl1989b, GiVl1991, Tom1990
- Global well-posedness in for Ta2004

## Generalized Benjamin-Ono equation

The *generalized Benjamin-Ono equation* is the scalar equation

where is the positive differentiation operator. When this is KdV; when this is Benjamin-Ono. Both of these two extreme cases are completely integrable, though the intermediate cases are not.

When scaling is and the following results are known:

- LWP in is known for KnKoe2003
- For this is in KnPoVe1994b

- GWP is known when when from the conservation of the Hamiltonian KnPoVe1994b
- The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work MlSauTz2001
- However, this can be salvaged by combining the norm with a weighted Sobolev space, namely where is the energy regularity. CoKnSt2003

## Benjamin-Ono with power nonlinearity

This is the equation

Thus the original Benjamin-Ono equation corresponds to the case The scaling exponent is

- For one has GWP for large data in KnKoe2003 and LWP for small data in MlRi2004
- For small data in LWP was obtained in KnPoVe1994b
- With the addition of a small viscosity term, GWP can also be obtained in by complete integrability methods in FsLu2000, with asymptotics under the additional assumption that the initial data is in
- For the solution map is not MlRi2004

- For LWP for small data in was obtained in KnPoVe1994b.
- For LWP for small data in was obtained in KnPoVe1994b.
- For any and the solution map is not uniformly continuous BiLi2001

## Other generalizations

The KdV-Benjamin Ono equation is formed by combining the linear parts of the KdV and Benjamin-Ono equations together. It is globally well-posed in Li1999, and locally well-posed in KozOgTns2001 (see also HuoGuo2005 where is obtained).

Similarly one can generalize the non-linearity to be k-linear, generating for instance the modified KdV-BO equation, which is locally well-posed in HuoGuo2005. For general gKdV-gBO equations one has local well-posedness in and above GuoTan1992. One can also add damping terms to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping OttSud1970.