# 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

ut + Huxx + uux = 0

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

Scaling is s = − 1 / 2, and the following results are known:

• LWP in Hs for $s \ge 1$ Ta2004
• For s > 9 / 8 this is in KnKoe2003
• For s > 5 / 4 this is in KocTz2003
• For $s \ge 3/2$ this is in Po1991
• For s > 3 / 2 this is in Io1986
• For s > 3 this is in Sau1979
• For no value of s is the solution map uniformly continuous KocTz2005
• Global weak solutions exist for L2 data Sau1979, GiVl1989b, GiVl1991, Tom1990
• Global well-posedness in Hs for $s \ge 1$ Ta2004
• For $s \ge 3/2$ this is in Po1991
• For smooth solutions this is in Sau1979

## Generalized Benjamin-Ono equation

The generalized Benjamin-Ono equation is the scalar equation

$\partial_t u + D_x^{1+a} \partial_x u + u\partial_x u = 0.$

where $D_x = \sqrt{-\Delta}$ is the positive differentiation operator. When a = 1 this is KdV; when a = 0 this is Benjamin-Ono. Both of these two extreme cases are completely integrable, though the intermediate cases 0 < a < 1 are not.

When 0 < a < 1, scaling is s = − 1 / 2 − a, and the following results are known:

• LWP in Hs is known for s > 9 / 8 − 3a / 8 KnKoe2003
• For $s \ge 3/4 (2-a)$ this is in KnPoVe1994b
• GWP is known when $s \ge (a+1)/2$ when a > 4 / 5, 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 Hs norm $|| f ||_{H^s}$ with a weighted Sobolev space, namely $|| xf ||_{H^{s - 2s_*}},$ where s * = (a + 1) / 2 is the energy regularity. CoKnSt2003

## Benjamin-Ono with power nonlinearity

This is the equation

ut + Huxx + (uk)x = 0.

Thus the original Benjamin-Ono equation corresponds to the case k = 2. The scaling exponent is 1 / 2 − 1 / (k − 1).

• For k = 3, one has GWP for large data in H1 KnKoe2003 and LWP for small data in Hs, s > 1 / 2 MlRi2004
• For small data in Hs, s > 1, LWP was obtained in KnPoVe1994b
• With the addition of a small viscosity term, GWP can also be obtained in H1 by complete integrability methods in FsLu2000, with asymptotics under the additional assumption that the initial data is in L1.
• For s < 1 / 2, the solution map is not C3 MlRi2004
• For k = 4, LWP for small data in Hs, s > 5 / 6 was obtained in KnPoVe1994b.
• For k > 4, LWP for small data in Hs, $s \ge 3/4$ was obtained in KnPoVe1994b.
• For any $k \ge 3$ and s < 1 / 2 − 1 / k 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 L2 Li1999, and locally well-posed in H − 3 / 4 + KozOgTns2001 (see also HuoGuo2005 where H − 1 / 8 + 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 H1 / 4 + HuoGuo2005. For general gKdV-gBO equations one has local well-posedness in H3 and above GuoTan1992. One can also add damping terms Hux to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping OttSud1970.