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

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

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 $H^{s}$ for $s\geq 1$ Ta2004
  • For $s>9/8$ this is in KnKoe2003
  • For $s>5/4$ this is in KocTz2003
  • For $s\geq 3/2$ 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
    • For $s<-1/2$ this is in BiLi2001
• Global weak solutions exist for $L^{2}$ data Sau1979, GiVl1989b, GiVl1991, Tom1990
• Global well-posedness in $H^{s}$ for $s\geq 1$ Ta2004
  • For $s\geq 3/2$ this is in Po1991
  • For smooth solutions this is in Sau1979

Generalized Benjamin-Ono equation

The generalized Benjamin-Ono equation is the scalar equation

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

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 $H^{s}$ is known for $s>9/8-3a/8$ KnKoe2003
  • For $s\geq 3/4(2-a)$ this is in KnPoVe1994b
• GWP is known when $s\geq (a+1)/2$ 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 $H^{s}$ 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

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

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