arXiv subjects
Publications and source records attributed to Xavier Carvajal.
Applying an Abstract Interpolation Lemma, we can show persistence of solutions of the initial value problem to higher order nonlinear Schrödinger equation, also called Airy-Schrödinger equation, in weighted Sobolev spaces $\mathcal{X}^{2,θ}$, for $0 \le θ\le 1$.
Using the theory developed by Kenig, Ponce, and Vega, we prove that the Hirota-Satsuma system is locally well-posed in Sobolev spaces $H^s(\mathbb{R}) \times H^{s}(\mathbb{R})$ for $3/4 -3/4$, by establishing new mixed-bilinear estimates involving the two Bourgain-type spaces $X_{s,b}^{-α_-}$ and $X_{s,b}^{-α_+}$ adapted to $\partial_t+α_-\partial_x^3$ and $\partial_t+α_+\partial_x^3$ respectively, where $|α_+|=|α_-|\not = 0$.
Using the theory of almost conserved energies and the ``I-method'' developed by Colliander, Keel, Staffilani, Takaoka and Tao, we prove that the initial value problem for a higher order Schrödinger equation is globally well-posed in Sobolev spaces of order $s>1/4$.
We prove that, the initial value problem associated to u_{t} + i\alphau_{xx} + βu_{xxx} + iγ|u|^{2}u = 0, x,t \in R, is locally well-posed in Sobolev spaces H^{s} for s>-1/4.