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In this paper we consider the slightly $L^2$-supercritical gKdV equations$\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity$5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . In the previous work of theauthor we know that there exists an stable self-similar blow-up dynamics forslightly $L^2$-supercritical gKdV equations. Such solution can be viewed assolutions with single blow-up point. In this paper we will prove the existenceof solutions with multiple blow-up points, and give a description of theformation of the singularity near the blow-up time.

Blow-up solutions for $L^2$-supercritical gKdV equations with exactly $k$ blow-up points