Benjamin-Ono equation

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Benjamin-Ono equation

[Thanks to Nikolay Tzvetkov and Felipe Linares for help with this section - Ed]

The generalized Benjamin-Ono equation BO_a is the scalar equation

u_t + D_x^{1+a} u_x + uu_x = 0

where D_x = sqrt{-Delta} is the positive differentiation operator. When a=1 this is [#kdv KdV]; when a=0 this is the Benjamin-Ono equation (BO) [[references.html#Bj1967 Bj1967]], references.html#On1975 On1975, which models one-dimensional internal waves in deep water. Both of these equations are completely integrable (see e.g. references.html#AbFs1983 AbFs1983, references.html#CoiWic1990 CoiWic1990), though the intermediate cases 0 < a < 1 are not.

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

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

One can replace the quadratic non-linearity uu_x by higher powers u^{k-1} u_x, in analogy with KdV and gKdV, giving rise to the gBO-k equations (let us take a=0 for sake of discussion).The scaling exponent is 1/2 - 1/(k-1).

  • For k=3, one has GWP for large data in H^1 [KnKoe-p] and LWP for small data in H^s, s > ½ [MlRi-p]
    • For small data in H^s, s>1, LWP was obtained in references.html#KnPoVe1994b 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 < ½, the solution map is not C^3 [MlRi-p]
  • For k=4, LWP for small data in H^s, s > 5/6 was obtained in references.html#KnPoVe1994b KnPoVe1994b.
  • For k>4, LWP for small data in H^s, s >=3/4 was obtained in references.html#KnPoVe1994b KnPoVe1994b.
  • For any k >= 3 and s < 1/2 - 1/k the solution map is not uniformly continuous [BiLi-p]

The KdV-Benjamin Ono (KdV-BO) equation is formed by combining the linear parts of the KdV and Benjamin-Ono equations together.It is globally well-posed in L^2 references.html#Li1999 Li1999, and locally well-posed in H^{-3/4+} [KozOgTns] (see also [HuoGuo-p] 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+} [HuoGuo-p].For general gKdV-gBO equations one has local well-posednessin H^3 and above references.html#GuoTan1992 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 references.html#OttSud1982 OttSud1982.