Non-generic blow-up solutions for the critical focusing NLS in 1-D
نویسندگان
چکیده
منابع مشابه
Non-generic Blow-up Solutions for the Critical Focusing Nls in 1-d
which blows up for t = −ab . Fixing a ∼ 1, b ∼ −1, it is then a natural question to ask whether one may perturb the initial data of (1.2) at time t = 0 such that the corresponding solution exhibits the same type of blow-up behavior. More precisely, the solution should asymptotically behave like √ 1 T−te iΨ(t,x)φ( T−t ) for a bounded function μ(t) and suitable Schwartz function φ, with blow up t...
متن کامل. A P ] 2 9 A ug 2 00 5 NON - GENERIC BLOW - UP SOLUTIONS FOR THE CRITICAL FOCUSING NLS IN 1 -
We consider the critical focusing NLS in 1-d of the form (1.1) i∂ t ψ + ∂ 2 x ψ = −|ψ| 4 ψ, i = √ −1, ψ = ψ(t, x), and ψ complex valued. It is well-known that this equation permits standing wave solutions of the form φ(t, x) = e iαt φ 0 (x, α), α > 0 Indeed, requiring positivity and evenness in x for φ 0 (x, α) implies for example φ 0 (x, α) = α 1 2 (3 2) 1 4 cosh 1 2 (α 2 x) Another remarkable...
متن کاملMinimal blow-up solutions to the mass-critical inhomogeneous NLS equation
We consider the mass-critical focusing nonlinear Schrödinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the crit...
متن کاملSlow Blow-up Solutions for the H(r) Critical Focusing Semi-linear Wave Equation
Given ν > 1 2 and δ > 0 arbitrary, we prove the existence of energy solutions of (0.1) ∂ttu−∆u− u = 0 in R3+1 that blow up exactly at r = t = 0 as t → 0−. These solutions are radial and of the form u = λ(t) 1 2W (λ(t)r)+η(r, t) inside the cone r ≤ t, where λ(t) = t−1−ν , W (r) = (1 + r2/3)− 1 2 is the stationary solution of (0.1), and η is a radiation term with Z [r≤t] ` |∇η(x, t)| + |ηt(x, t)|...
متن کاملExistence and Uniqueness of Minimal Blow-up Solutions to an Inhomogeneous Mass Critical Nls
(1.1) (NLS) { i∂tu = −Δu− k(x)|u|u, (t, x) ∈ R× R, u(t0, x) = u0(x), u0 : R 2 → C, for some smooth bounded inhomogeneity k : R → R+ and some real number t0 < 0. This kind of problem arises naturally in nonlinear optics for the propagation of laser beams. From the mathematical point of view, it is a canonical model to break the large group of symmetries of the k ≡ 1 homogeneous case. From standa...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of the European Mathematical Society
سال: 2009
ISSN: 1435-9855
DOI: 10.4171/jems/143