scholarly journals SPHERICALLY SYMMETRIC, COLD COLLAPSE: THE EXACT SOLUTIONS AND A COMPARISON WITH SELF-SIMILAR SOLUTIONS

2017 ◽  
Vol 835 (1) ◽  
pp. 40 ◽  
Author(s):  
Eric R. Coughlin
Keyword(s):  
Exact Solutions ◽  
Spherically Symmetric ◽  
Self Similar ◽  
Similar Solutions

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 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.

scholarly journals Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 511
Author(s):  
Andrei D. Polyanin ◽  
Vsevolod G. Sorokin
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Proportional Delay ◽  
Type Reaction ◽  
Self Similar ◽  
Functional Differential ◽  
Similar Solutions ◽  
Varying Delay

We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0<p<1, 0<q<1). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations).

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