CONSTRUCTION OF MULTIBUBBLE SOLUTIONS FOR THE CRITICAL GKDV EQUATION

We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation partial derivative(t)u+partial derivative(x)(partial derivative(xx)u+u(5)) = 0 containing an arbitrary number K >= 2 of blow up bubbles for any choice of sign and scaling parameters: for any l(1) > l(2) > ... > l(K) > 0 and c(1),..., c(K) is an element of {+/- 1}, there exists an H-1 solution u of the equation such that u(t) - Sigma(K)(k=1) epsilon(k)/lambda(1/2)(k) Q(-x(k)(t)/lambda(k)(t)) -> 0 in H-1 as t down arrow 0 with lambda(k)(t) similar to l(k)t and x(k)(t) similar to -l(k)(-2)t(-1) as t down arrow 0. The construction uses and extends techniques developed mainly in [ Y. Martel and F. Merle, J. Math. Pures Appl. (9), 79 (2000), pp. 339-425] and [Y. Martel, F. Merle, and P. Raphael, Acta Math., 212 (2014), pp. 59-140; J. Eur. Math. Soc. (JEMS), 17 (2015), pp. 1855-1925; Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 14 (2015), pp. 575-631]. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved in [V. Combet and Y. Martel, Bull. Sci. Math., 141 (2017), pp. 20-103].