Introduction to Nonlinear Dispersive Equations

By Felipe Linares, Gustavo Ponce. 2009

574 citations recorded by the source corpus.

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In Mathematical Physics

Citations in the corpus

The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates (2016, methodology): “bal mild solution. 3. TRUNCATED EQUATION AND LOCAL WELLPOSEDNESS This section is devoted to the proof of the local part of Theorem 1.1. In order to transfer the deterministic ﬁxed point argument, see [LP14], Theorems 5.2 and 5.3, to the stochastic setting, we would like to use the Strichartz estimat”
Asymptotic stability of solitons for mKdV (2015, unknown): “decay as x→ +∞. This result was then reﬁned 2For more on the local and global well-posedness and ill-posedness of KdV and generalized KdV equations we refer to the books of Tao [49] and Linares-Ponce [34]. 3Similarly to how it is stated in Theorem 3.1. ASYMPTOTIC STABILITY OF SOLITONS FOR MKDV 3 by ”
Long time behavior of solutions to the mKdV (2014, background): “: R×R→ R, u(0) = u0, where σ= ±1 and u0 is suﬃciently small, smooth and decaying data. The Cauchy problem for (1.1) has been studied extensively. For a summary of known results we refer the reader to [29]. In particular, the mKdV is both locally well-posed [23–25] and globally well-posed [3,12,27] i”
Long‐Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability (2015, background): “Further functional analysis results can be found in Craig, Kappeler and Strauss [22], Cazenave [18], Cazenave and Weissler [19], Kenig, Ponce and Vega [49], Ghidaglia and Saut [38], Linares and Ponce [55], Carroll and Bu [17], Kato [48], Ginibre and Velo [39], Tsutsumi [63], and the references there”
Control of a Korteweg-de Vries equation: A tutorial (2013, unknown): “We refer the interested reader to the books [46] by Tao and [30] by Linares and Ponce, where the KdV equation and other nonlinear dispersive partial differential equations are studied.”
A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation (2007, background): “The initial-value problem posed with initial-data u(x, 0) = u0(x) is locally well-posed in H 1 (see Ginibre-Velo [9]; standard reference texts are Cazenave [3], Linares-Ponce [20], and Tao [29]).”
Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schr\"odinger Equations (2015, background): “1 in [58]) can be used to shorten the proof of the claim, we present a self-contained argument in favor of (3.”
On well posedness for the inhomogeneous nonlinear Schr\"odinger equation (2016, background): “)k Lr1 kD suk Lr2 . Proof. See Christ-Weinstein [7, Proposition 3.1]. The main tool to show the local and global well-posedness are the well-known Strichartz estimates. See for instance Linares-Ponce [22] and Kato [19] (see also Holmer-Roudenko [18] and Guevara [17]). Lemma2.5. The following stateme”
Small data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance (2011, methodology): “ned by ρ= p+1 and 2 r = n 2 − ρ . Proposition 2.2 can be proved easily by using the Strichartz estimate (see Lemma 4.2 and [27] etc) and a ﬁxed point theorem, so we omit the proof in detail (see e.g. [16] and [25] etc). The important thing is that the local existence time T >0 depends only on nan”
Linear and Quasi-Linear Evolution Equations in Hilbert Spaces (2012, methodology): “Moreover, these methods can also be applied to other types of evolution equations, such as the so-called dispersive equations considered in Tao [156], and Linares and Ponce [97]; these include, among others, the Schrödinger and the Kortewegde-Vries equations.”