Abstract:The question is raised as to whether the Benjamin–Ono equation, a nonlinear partial differential integral equation, is a completely integrable Hamiltonian system. The answer is almost certainly ’’yes.’’ Particular solutions suggest the form that general polynomial constants must have. The structure of these and an algorithm to compute them is given. Explicit formulas are given for the first six.
“…As previously recalled, the BO equation has an infinite number of conserved quantities ( [44], the first ones are displayed in [1]). The Hamiltonian flow of those invariants define the aforementioned Benjamin-Ono hierarchy.…”
We survey and compare, mainly in the two-dimensional case, various results obtained by IST and PDE techniques for integrable equations. We also comment on what can be predicted from integrable equations on non integrable ones.
“…As previously recalled, the BO equation has an infinite number of conserved quantities ( [44], the first ones are displayed in [1]). The Hamiltonian flow of those invariants define the aforementioned Benjamin-Ono hierarchy.…”
We survey and compare, mainly in the two-dimensional case, various results obtained by IST and PDE techniques for integrable equations. We also comment on what can be predicted from integrable equations on non integrable ones.
The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on L 2 (T). The paper shows that the Gibbs measures on bounded balls of L 2 satisfy some logarithmic Sobolev inequalities. The space of n-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As n → ∞, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.Date: 1st February 2019.
“…In order to bound S τ (y) for all y e i?, we need the following classical Van der Corput lemma [14] Lemma 8. Suppose that φ £ Cl(R) and φ £ C 2 (R) such that \φ"(ξ)\^l on the support of φ. Then where the constant C is independent of λ, φ and φ.…”
In this paper we study the Cauchy problem for the generalized equation of finite-depth fluidswhere G( ) is a singular integral, and p is an integer larger than 1. We obtain the long time behavior of the fundamental solution of linear problem, and prove that the solutions of the nonlinear problem with small initial data for p > 5/2 -f Vΐΐ/2 are decay in time and freely asymptotic to solutions of the linear problem. In addition we also study some properties of the singular integral G( ) in L q (R) with q > 1.
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