A numerical approach to Blow-up issues for dispersive perturbations of   Burgers' equation
  We provide a detailed numerical study of various issues pertaining to the dynamics of the Burgers equation perturbed by a weak dispersive term: blow-up in finite time versus global existence, nature of the blow-up, existence for "long" times, and the decomposition of the initial data into solitary waves plus radiation. We numerically construct solitons for fractionary Korteweg-de Vries equations. 
C. Klein, J.-C. Saut
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