Journal ArticleDOI
Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation
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In this paper, the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces, is studied.Abstract:
The main results of this paper are concerned with the "bad" behavior of the KP-I equation with respect to a Picard iteration scheme applied to the associated integral equation, for data in usual or anisotropic Sobolev spaces. This leads to some kind of ill-posedness of the corresponding Cauchy problem: the flow map cannot be of class $C\sp 2$ in any Sobolev space.read more
Citations
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Propagation of ultra-short optical pulses in cubic nonlinear media
T. Schäfer,C. E. Wayne +1 more
TL;DR: In this article, the authors derived a partial differential equation that approximates solutions of Maxwell's equations describing the propagation of ultra-short optical pulses in nonlinear media and extended the prior analysis of Alterman and Rauch.
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Ill-Posedness Issues for the Benjamin--Ono and Related Equations
TL;DR: It is established that the Cauchy problem for the Benjamin--Ono equation and for a rather general class of nonlinear dispersive equations with dispersion slightly weaker than that of the Korteweg--de Vries equation cannot be solved by an iteration scheme based on the Duhamel formula, and the flow map fails to be smooth.
Journal ArticleDOI
An instability property of the nonlinear Schrödinger equation on $S^{d}$
TL;DR: In this paper, the authors consider the NLS on spheres and show that the flow map fails to be uniformly continuous for Sobolev regularity above a threshold suggested by a simple scaling argument.
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Sharp well-posedness results for the BBM equation
Jerry L. Bona,Nikolay Tzvetkov +1 more
TL;DR: In this article, it was shown that the BBM equation cannot be solved by iteration of a bounded mapping leading to a fixed point in a Euclidean space for the given initial data.
Journal ArticleDOI
Global well-posedness of the KP-I initial-value problem in the energy space
TL;DR: In this article, the KP-I initial value problem was shown to be NP-hard, and it was shown that the solution of the problem is polynomial in the number of vertices.
References
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Journal ArticleDOI
Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations
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Commutator estimates and the euler and navier‐stokes equations
Tosio Kato,Gustavo Ponce +1 more
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Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle
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A bilinear estimate with applications to the KdV equation
TL;DR: In this paper, the authors considered the existence problem for the IVP (1.1) problem in H(R) = L(R), where R is the energy space.