We report experimental results of a traveling-wave burst and collapse process occurring in convecting binary mixtures in a wide range of the control parameters. Analysis in the framework of the one-dimensional complex Ginzburg Landau (CGI) equation reveals an alternative self-focusing mechanism responsible for this behavior: faster than exponential bursting due to the destabilizing effect of the nonlinearity and collapse due to suppression of the pulse growth at the edges, leading to the destruction of the pulse by compression. The latter effect is associated with the strong nonlinear dispersion of the system. Numerical analysis, based on the CGL equation, closely matches both experimental results and theoretical considerations. The limits of validity of the proposed mechanism are also discussed.