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...Ponce established global well-posedness for solutions of the focusing mass-critical gKdV for initial data in Hs (R) with s > 3/4 and mass less than that of the ground state solution. Recently, Farah, =-=[10]-=-, used the I-method of Colliander, Keel, Staffilani, Takaoka, and Tao, [7], to further lower the regularity of the initial data to s > 3/5. In view of the fact that it is both scaling-critical and con...

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...n with derivative (DNLS), refer to [6, 10, 15, 22, 33] on the applications in the context of gKdV equations. The global well-posedness of (1.1)-(1.2) below the energy space H1 was considered by Farah =-=[19]-=-, Fonseca, Linares and Ponce [20]. The authors in [20] proved the global existence in Hs(R) for s > 3/4 by appllying Bourgain’s “Fourier truncation method”. This was improved very recently in [19], wh...

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...ethod. The global well-posedness for the higher order generalized KdV equations is more subtle due to the critical (or supercritical) nature of the problem. However, there are recent results of Farah =-=[13]-=-, Miao et al. [36], and Farah, Linares, and Pastor [14] which address these problems. The generalized KdV equations possess traveling wave solutions called solitons. Indeed, if one assumes that the so...

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...nd Marzuola [20]. Under “sharp smallness condition”, the critical case k = 4 was studied by Fonseca, Linares, and Ponce in [9]. There it was established global well-posedness in Hs(R), s > 3/4. Farah =-=[8]-=- used the I-method of [6], to further lower the regularity of the initial data to s > 3/5. Recently, Miao, Shao, Wu, and Xu [24], improved the latter result to initial data in Hs(R), s > 6/13. Their m...

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...+ · · ·+ ξ2k+2)(ξk+2 + · · ·+ ξ2k+2)]sym, and [m]sym denotes the symmetrization of a multiplier m (see [6]). So far, this precess is rather standard, and it is the same as what in real line case, see =-=[10, 19]-=-, etc.. Now we focus our attention on the term (1.16), and consider the quantity Mk+2 αk+2 . (1.18) If it makes sense, then one may use the identity eiαk+2s = 1 iαk+2 ∂s ( eiαk+2s ) , (1.19) GLOBAL RO...

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...ities to obtain global well-posedness of the Cauchy problem where no conserved quantities are available. This method has been very successful to get sharp global result for several dispersive models, =-=[9, 10, 12]-=- are just a few to mention. The authors in [8] used the so called I operator method and almost conserved quantities to obtain sharp global well-posedness results for the KdV and mKdV equations in real...