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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Journal ArticleDOI
L2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity
Frank Merle,Yoshio Tsutsumi +1 more
TL;DR: In this article, the authors considered the case of the critical power and proved that u(t) has no limit in L2 as t → T. In particular, they further showed a phenomenon of L2 concentration at the origin.
Journal Article
Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity
Jean Bourgain,Wei-Min Wang +1 more
Journal ArticleDOI
Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation
Yvan Martel,Frank Merle +1 more
TL;DR: In this article, the critical generalized KdV equation was considered and it was shown that the two conservation laws do not imply a bound in H 1 uniform in time for all H 1 solutions and thus global existence.
Journal ArticleDOI
On the Formation of Singularities in Solutions of the Critical Nonlinear Schrodinger Equation
TL;DR: For the one-dimensional nonlinear Schrodinger equation with critical power nonlinearity, the Cauchy problem with initial data close to a soliton is considered in this paper, and it is shown that for a certain class of initial perturbations the solution develops a self-similar singularity infinite time T*, the profile being given by the ground state solitary wave and the limiting self-focusing law being of the form¶¶\( \lambda(t) \sim (ln \mid ln(T^* -t)\mid)^{ 1/
Journal ArticleDOI
Stability of the log-log bound for blow up solutions to the critical non linear Schrödinger equation
TL;DR: In this paper, the authors considered finite time blow up solutions to the critical nonlinear Schrodinger equation and established the stability in energy space H 1 of the log-log upper bound exhibited in the negative energy case, and a sharp lower bound on blow up rate in the other regime which corresponds to known explicit blow-up solutions.
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