Theoretical and Numerical Aspects for Global Existence and Blow Up for the Solutions to Boussinesq Paradigm Equation
Abstract
In this paper we prove that the global existence and the blow up of the weak solutions to Boussinesq Paradigm Equation (BPE) depend not only on the initial energy but also on the profiles of the initial data.
The constant d of the critical initial energy which guaranties the above properties of the solution is found explicitly by means of the exact constant of the Sobolev embedding theorem. We demonstrate numerically in the one dimensional case that this constant d is the best possible one for the global existence and a lack of global existence.
In this way we can find the intervals for the velocities c of the solitary wave solutions to BPE in which the solution to BPE with initial data close to the solitons exists globally in time. Thus for different parameters of BPE we give numerically some ranges of the stability of the solitary waves.
 Publication:

Application of Mathematics in Technical and Natural Sciences: 3rd International Conference  AMiTaNS'11
 Pub Date:
 November 2011
 DOI:
 10.1063/1.3659905
 Bibcode:
 2011AIPC.1404...68K
 Keywords:

 solitons;
 solitary waves;
 energy conservation;
 constants;
 mechanics;
 43.25.Rq;
 47.35.Fg;
 45.20.dh;
 06.20.Jr;
 46.05.+b;
 Solitons chaos;
 Solitary waves;
 Energy conservation;
 Determination of fundamental constants;
 General theory of continuum mechanics of solids