Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations

By Yvan Martel, F. Merle. 2001

219 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/53d37711e38afdc824ea582325a79cbf60f57455

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Citations in the corpus

Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gKdV Equations (2001, methodology): “From [13] and [15], the L2 norm of the solution at the right of each soliton is almost decreasing in time.”
Asymptotic stability of solitons for mKdV (2015, unknown): “For perturbations in the energy space H(1), definitive asymptotic stability results in front of the solitary wave have been obtained for the whole class of subcritical gKdV equations in a series of papers by Martel-Merle [36, 37, 38].”
Global well-posedness of the KP-I initial-value problem in the energy space (2007, background): “Moreover, the more subtle question of asymptotic stability, as in [ 21 , 22, 18], can also be naturally posed in the energy space.”
Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations (2009, methodology): “Recall that Martel, Merle and Tsai [21] proved stability and asymptotic stability of a sum of N solitons for large time for the subcritical case.”
Asymptotic stability of solitons for the Benjamin-Ono equation (2008, methodology): “These results are reminiscent of similar results for the gKdV equation in [16] and [19], but due to the nonlocal nature of the operator H, the proofs are more involved.”
Stability of cnoidal waves (2006, background): “For solitary waves, stability theory was begun by Benjamin [5] and has since been refined and improved in several ways (see, for example, [4], [8], [11], [22], [25], [26], [37], [38], [39], [40], [43], [49], [50]).”
Classification of radial solutions of the focusing, energy-critical wave equation (2012, methodology): “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”
Why are solitons stable (2008, unknown): “More recently, Martel and Merle [46], [50] were able to consider more general perturbations which were only assumed to be small in the energy norm H1; this generalisation is important for the purposes of understanding how a soliton will collide with another shallow broad soliton, which may have smal”
Universality of blow-up profile for small radial type II blow-up solutions of energy-critical wave equation (2009, methodology): “by classification of compact solutions of the GKdV equation: see Martel Merle [MM00, MM01 , MM02]; - for the mass-critical NLS equation”
Asymptotic stability of solitons of the gKdV equations with general nonlinearity (2007, methodology): “4 Asymptotic stability - Proof of Theorem 1 The proof of the asymptotic stability is based on the nonlinear Liouville property as in [13].”