On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I general theory

1981 ◽  
Vol 3 (1) ◽  
pp. 229-248 ◽  
Author(s):  
E. Horst ◽  
H. Neunzert
Keyword(s):  
General Theory ◽  
Initial Value Problem ◽  
Vlasov Equation ◽  
Classical Solutions ◽  
Initial Value ◽  
Non Linear

Stability of Vlasov equilibria. Part 1. General theory

1982 ◽  
Vol 27 (1) ◽  
pp. 13-24 ◽  
Author(s):  
K. R. Symon ◽  
C. E. Seyler ◽  
H. R. Lewis
Keyword(s):  
Distribution Function ◽  
General Theory ◽  
Linear Stability ◽  
Vlasov Equation ◽  
Stability Problem ◽  
Dispersion Matrix ◽  
Initial Value ◽  
Spectral Matrix ◽  
Concise Representation ◽  
The Stability

We present a general formulation for treating the linear stability of inhomogeneous plasmas for which at least one species is described by the Vlasov equation. Use of Poisson bracket notation and expansion of the perturbation distribution function in terms of eigenfunctions of the unperturbed Liouville operator leads to a concise representation of the stability problem in terms of a symmetric dispersion functional. A dispersion matrix is derived which characterizes the solutions of the linearized initial-value problem. The dispersion matrix is then expressed in terms of a dynamic spectral matrix which characterizes the properties of the unperturbed orbits, in so far as they are relevant to the linear stability of the system.

L∞C(ℝn)-decay of classical solutions for nonlinear Schrödinger equations

1986 ◽  
Vol 104 (3-4) ◽  
pp. 309-327 ◽  
Author(s):  
Nakao Hayashi ◽  
Masayoshi Tsutsumi
Keyword(s):  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Classical Solutions ◽  
Initial Value ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Sign Conditions ◽  
Image Position

SynopsisWe study the initial value problem for the nonlinear Schrödinger equationUnder suitable regularity assumptions on f and ø and growth and sign conditions on f, it is shown that the maximum norms of solutions to (*) decay as t→² ∞ at the same rate as that of solutions to the free Schrödinger equation.

scholarly journals Asymmetric bifurcation

1980 ◽  
Vol 21 (3) ◽  
pp. 297-304 ◽  
Author(s):  
S. Rosenblat
Keyword(s):  
Diffusion Equation ◽  
Initial Value Problem ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Initial Value ◽  
Linear Diffusion ◽  
Non Linear ◽  
Bifurcation And Stability

AbstractA study is made of a non-linear diffusion equation which admits bifurcating solutions in the case where the bifurcation is asymmetric. An analysis of the initial-value problem is made using the method of multiple scales, and the bifurcation and stability characteristics are determined. It is shown that a suitable interpretation of the results can lead to determination of the choice of the bifurcating solution adopted by the system.

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