# 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

${\displaystyle u_{t}+Hu_{xx}+uu_{x}=0}$

where ${\displaystyle H}$ is the Hilbert transform. This equation is completely integrable (see e.g., AbFs1983, CoiWic1990).

Scaling is ${\displaystyle s=-1/2,}$ and the following results are known:

• LWP in ${\displaystyle H^{s}}$ for ${\displaystyle s\geq 1}$ Ta2004
• For ${\displaystyle s>9/8}$ this is in KnKoe2003
• For ${\displaystyle s>5/4}$ this is in KocTz2003
• For ${\displaystyle s\geq 3/2}$ this is in Po1991
• For ${\displaystyle s>3/2}$ this is in Io1986
• For ${\displaystyle s>3}$ this is in Sau1979
• For no value of s is the solution map uniformly continuous KocTz2005
• For ${\displaystyle s<-1/2}$ this is in BiLi2001
• Global weak solutions exist for ${\displaystyle L^{2}}$ data Sau1979, GiVl1989b, GiVl1991, Tom1990
• Global well-posedness in ${\displaystyle H^{s}}$ for ${\displaystyle s\geq 1}$ Ta2004
• For ${\displaystyle s\geq 3/2}$ this is in Po1991
• For smooth solutions this is in Sau1979

## Generalized Benjamin-Ono equation

The generalized Benjamin-Ono equation is the scalar equation

${\displaystyle \partial _{t}u+D_{x}^{1+a}\partial _{x}u+u\partial _{x}u=0.}$

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

When ${\displaystyle 0 scaling is ${\displaystyle s=-1/2-a,}$ and the following results are known:

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

## Benjamin-Ono with power nonlinearity

This is the equation

${\displaystyle u_{t}+Hu_{xx}+(u^{k})_{x}=0.}$

Thus the original Benjamin-Ono equation corresponds to the case ${\displaystyle k=2.}$ The scaling exponent is ${\displaystyle 1/2-1/(k-1).}$

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