Nonlinear wave interactions for the Benjamin-Ono equation.

By Herbert Koch, Nikolay Tzvetkov. 2004

161 citations recorded by the source corpus.

We study the interaction of suitable small and high frequency waves evolving by the flow of the Benjamin-Ono equation. As a consequence, we prove that the flow map of the Benjamin-Ono equation can not be uniformly continuous on bounded sets of H^s(R) for s>0.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d99eac7a79da2468a3fc4f1b45b10996a4ff7e1e

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In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

On the fifth order KdV equation: local well-posedness and lack of uniform continuity of the solution map (2007, methodology): “[14] H. Koch, N. Tzvetkov, On finite energy solutions of the KP-I equation, preprint [15] P.D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure.”
Non-Uniform Dependence for the Periodic CH Equation (2010, methodology): “In this paper, motivated by the work of Koch and Tzvetkov in [ 18 ] and Christ et al.”
Global well-posedness of the Benjamin–Ono equation in low-regularity spaces (2005, background): “We mention that the ﬂow map φ → S σT ( φ ) fails to be uniformly continuous on bounded sets in H σr for any T > 0 and σ > 0; see [13].”
Non-uniform dependence on initial data for the CH equation on the line (2009, methodology): “;2 and 2R. Then for any s0 we have that lim !1 1 2 sk ( x )cos(x )k Hs(R) = 1 p 2 k k L2(R): (4.2) Relation (4.2) is also true if cos is replaced by sin. Although this lemma can be found in [KT], we include its proof here for the convenience of the reader. Proof. Since ( x )cos(x i) b (˘) = 1 ”
Local Well-posedness for dispersion generalized Benjamin-Ono equations in Sobolev spaces (2008, background): “It was proved by Molinet, Saut and Tzvetkov [21] that if 0 α < 1 then Hs assumption alone on the initial data is insufficient for a proof of local well-posedness of (1.1) via Picard iteration by showing the solution mapping fails to be C2 smooth from Hs to C([0, T ]; Hs) at the origin for any s.”
The IVP for the dispersion generalized Benjamin–Ono equation in weighted Sobolev spaces (2011, background): “Actually, in [32] it was proved that this map is not even locally uniformly continuous.”
On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey (2004, background): “It would be very interesting to prove this result (if it is true !) as it was done recently for the Benjamin-Ono equation [22].”
Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems (2016, unknown): “properties for CH type and related nonlinear evolution equations, we refer the reader to [12], [13], [11], [15], [16], [6], [23], [49], [53], [50], [17], [45], [8], [42], [43], [44], [4], [3], [64], [18], [5], [47], [48], [10], [9], [70], and the references therein.”
On well-posedness for the Benjamin–Ono equation (2005, background): “1) fails to be C∞ (Molinet-Saut-Tzvetkov [21]), and even uniformly continuous (Koch-Tzvetkov [18]).”
GLOBAL WELL-POSEDNESS OF THE BENJAMIN–ONO EQUATION IN H1(R) (2003, methodology): “H s 0 1/2 x , as the examples in [ 15 ] already show.”