A Liouville theorem for the critical generalized Korteweg–de Vries equation

By Yvan Martel, F. Merle. 2000

170 citations recorded by the source corpus.

Abstract We prove in this paper a rigidity theorem on the flow of the critical generalized Korteweg–de Vries equation close to a soliton up to scaling and translation. To prove this result we introduce new tools to understand nonlinear phenomenon. This will give a result of asymptotic completeness.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/01f904099ea0a1ea55966a6ab3a89f30b1448423

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton (2014, background): “[3] Côte, R., Martel, Y. & Merle, F., Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations.”
Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation (2009, methodology): “The only case where such a striking fact was established was for GKdV by Martel and Merle [ MM00 , MM01].”
Minimal-mass blowup solutions of the mass-critical NLS (2006, background): “In the language of Martel and Merle, [19], this reduces matters to establishing a “Liouville theorem” for these equations.”
Blow up for the critical gKdV equation II: minimal mass dynamics (2012, background): “In the series of works [26, 27, 38, 28, 29], Martel and Merle obtain the first qualitative information on the flow for small super critical mass initial data ‖Q‖L2 < ‖u0‖L2 < ‖Q‖L2 < α∗, 0 < α∗ ≪ 1, in particular the existence of finite time blow up solutions for E0 < 0 and the classification of Q a”
Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation (2008, methodology): “For a similar approach of the blow-up problem, see Martel and Merle [11], [10] for the critical KdV equation; see also Merle and Raphaël [12], [13] for the NLS equation.”
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation (2003, background): “in the sequel of papers [17], [18], [19], [20], [21].”
Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations (2001, methodology): “Proof of Lemma 2. Arguing as in [ 15 ], Lemma 17 and Appendix D, and using the orthogonality conditions and (16), (17), Lemma 2 is equivalent to the following property on u. Claim.”
Instability of solitons for the critical generalized Korteweg—de Vries equation (2001, background): “Remark 2. In a forthcoming paper [ MM ], using far more elaborate tools, we will be able to prove the instability result as a consequence of a classication of bounded solutions around Q, which will give a characterization of the solitons.”
Asymptotic stability of solitons for the Benjamin-Ono equation (2008, methodology): “Now, to obtain weak convergence, we need to remove some terms from the equation of wn, following some arguments in [15], Lemma 8 and beginning of proof of Lemma 11.”
Why are solitons stable (2008, unknown): “Then the same Liouville theorem analysis used in the subcritical case can be used to show that ε = 0, which would of course contradict the assumption that M(u) is strictly greater than M(Q); see [44] for details.”
Blow up in finite time and dynamics of blow up solutions for the L^2-critical generalized KdV equation (2002, background): “[9] Y. Martel and F. Merle, Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation, to appear in Ann. of Math.”
On universality of blow-up profile for L2 critical nonlinear Schrödinger equation (2004, result): “This result corresponds in the proof to the Liouville Theorem for critical KdV, see [17].”
Classification of radial solutions of the focusing, energy-critical wave equation (2012, background): “Results of this type for other equations in the global case include those for data close to the soliton for subcritical nonlinearities: see the works of Martel and Merle [27] in the case of generalized KdV equations, of Buslaev and Perelman [4, 3] in the case of one dimensional NLS with specific non”
Blow up for the critical gKdV equation III: exotic regimes (2012, background): “4) the blow up problem has been first studied in a series of works by Martel and Merle [31, 32, 44, 33, 34].”
Asymptotic stability of solitons of the gKdV equations with general nonlinearity (2007, background): “Then, in [12] and [13], we have proved asymptotic stability in the energy space of the solitons of (1.”