Journal ArticleDOI
Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation
TLDR
In this paper, the authors consider finite time blow up solutions to the critical nonlinear Schrodinger equation, and prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part correspond to the regular part and has a strong L2 limit at blow up time.Abstract:
We consider finite time blow up solutions to the critical nonlinear Schrodinger equation Open image in new window For a suitable class of initial data in the energy space H1, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L2 mass at the blow up point, the second part corresponds to the regular part and has a strong L2 limit at blow up time.read more
Citations
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Journal ArticleDOI
Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation
TL;DR: In this paper, the cubic defocusing nonlinear Schrodinger equa- tion on the two dimensional torus is considered and smooth solutions for which the support of the conserved energy moves to higher Fourier modes are presented.
Journal ArticleDOI
Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation
Frank Merle,Pierre Raphaël +1 more
TL;DR: In this article, the critical nonlinear Schrodinger equation with initial condition u(0, x) = u0 was considered and the initial condition was obtained for the case where x = 0.
Journal ArticleDOI
The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities
TL;DR: In this paper, a comprehensive study of the nonlinear Schrodinger equation where u(t, x) is a complex-valued function in spacetime, λ1 and λ2 are nonzero real constants, and the authors address questions related to local and global wellposedness, finite time blowup, and asymptotic behaviour.
Journal ArticleDOI
Stable blow up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems
Pierre Raphaël,Igor Rodnianski +1 more
TL;DR: In this paper, stable finite time blow up regimes for the energy critical co-rotational Wave Map with the S 2 target in all homotopy classes and for the critical equivariant SO(4) Yang-Mills problem were derived.
Journal ArticleDOI
Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation
TL;DR: In this article, it was shown that the unique radial positive stationary solution of the focusing wave equation is the sum of a rescaled W concentrating at the origin and a small remainder which is continuous with respect to the time variable in the energy space.
References
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Book
The nonlinear Schrödinger equation
TL;DR: In this article, the authors studied several explicit solutions of the Schrödinger equation and derived two important conservation laws for sufficiently regular solutions, which is the first step towards Strichartz' estimates.
Journal ArticleDOI
On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case
J. Ginibre,G. Velo +1 more
TL;DR: In this paper, the authors studied the Cauchy problem for a class of nonlinear Schrodinger equations of the form i( du dt ) = (−Δ + m)u + f(u) in R n with n ⩾ 2, where m is a real constant and f a complex valued nonlinear function.
Journal ArticleDOI
The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation
Frank Merle,Pierre Raphaël +1 more
TL;DR: In this article, the critical nonlinear Schrodinger equation with initial condition u(0, x) = u0 in dimension N = 1 is considered, and local existence in the time of solutions on an interval [0, T] is known, and there exist finite time blowup solutions, that is, u0 such that limt.T 1.
Journal ArticleDOI
On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation
TL;DR: In this paper, the authors considered finite time blow-up solutions to the critical nonlinear Schrodinger equation iut=-Δu-|u|4/Nu with initial condition u0∈H1.
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