scholarly journals Self-similar Langmuir collapse at critical dimension

1991 ◽  
Vol 9 (2) ◽  
pp. 363-370 ◽  
Author(s):  
L. Bergé ◽  
PH. Dousseau ◽  
G. Pelletier ◽  
D. Pesme
Keyword(s):  
Control Parameter ◽  
Linear Time ◽  
Critical Dimension ◽  
Singular Solutions ◽  
Spherically Symmetric ◽  
Scalar Model ◽  
Contraction Rate ◽  
Self Similar ◽  
Similar Solutions ◽  
Time Contraction

Two spherically symmetric versions of a self-similar collapse are investigated within the framework of the Zakharov equations, namely, one relative to a vectorial electric field and the other corresponding to a scalar modeling of the Langmuir field. Singular solutions of both of them depend on a linear time contraction rate Ξ(t) = V(t* – t), where t* and V = – Ξ denote, respectively, the collapse time and the constant collapse velocity. We show that under certain conditions, only the scalar model admits self-similar solutions, varying regularly as a function of the control parameter V from the subsonic (V ≪ 1) to the supersonic (V ≫ 1) regime.

scholarly journals Very singular solutions to a nonlinear parabolic equation with absorption II. Uniqueness

2004 ◽  
Vol 134 (1) ◽  
pp. 39-54 ◽  
Author(s):  
Saïd Benachour ◽  
Herbert Koch ◽  
Philippe Laurençot
Keyword(s):  
Parabolic Equation ◽  
Nonlinear Parabolic Equation ◽  
Singular Solution ◽  
Singular Solutions ◽  
Nonlinear Parabolic ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions ◽  
Very Singular Solution

We prove the uniqueness of the very singular solution to when 1 < p < (N + 2)/(N + 1), thus completing the previous result by Qi and Wang, restricted to self-similar solutions.

scholarly journals ON THE PHYSICAL PROPERTIES OF SPHERICALLY SYMMETRIC SELF-SIMILAR SOLUTIONS

2005 ◽  
Vol 14 (01) ◽  
pp. 73-84 ◽  
Author(s):  
M. SHARIF ◽  
SEHAR AZIZ
Keyword(s):  
Physical Properties ◽  
Radial Direction ◽  
The Self ◽  
Spherically Symmetric ◽  
Self Similar ◽  
Similar Solutions ◽  
Kinematic Properties

In this paper, we are exploring some of the properties of the self-similar solutions of the first kind. In particular, we shall discuss the kinematic properties and also check the singularities of these solutions. We discuss these properties both in co-moving and also in non-co-moving (only in the radial direction) coordinates. Some interesting features of these solutions turn up.

scholarly journals On blow up for the energy super critical defocusing nonlinear Schrödinger equations

2021 ◽  
Author(s):  
Frank Merle ◽  
Pierre Raphaël ◽  
Igor Rodnianski ◽  
Jeremie Szeftel
Keyword(s):  
Blow Up ◽  
Companion Paper ◽  
Spherically Symmetric ◽  
Unique Strong Solution ◽  
Nonlinear Schrödinger ◽  
Degeneracy Condition ◽  
Highly Oscillatory ◽  
Self Similar ◽  
Similar Solutions ◽  
Symmetric Initial Data

AbstractWe consider the energy supercritical defocusing nonlinear Schrödinger equation $$\begin{aligned} i\partial _tu+\Delta u-u|u|^{p-1}=0 \end{aligned}$$ i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension $$d\ge 5$$ d ≥ 5 . In a suitable range of energy supercritical parameters (d, p), we prove the existence of $${\mathcal {C}}^\infty $$ C ∞ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $${\mathcal {C}}^\infty $$ C ∞ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

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