Non-flat conformal blow-up profiles for the 1D critical nonlinear Schr\"odinger equation

For the critical one-dimensional nonlinear Schr\"odinger equation, we construct blow-up solutions that concentrate a soliton at the origin at the conformal blow-up rate, with a non-flat blow-up profile. More precisely, we obtain a blow-up profile that equals $|x|+i\kappa x^2$ near the origin, where $\kappa$ is a universal real constant. Such profile differs from the flat profiles obtained in the same context by Bourgain and Wang [Construction of blowup solutions for the nonlinear Schr\"odinger equation with critical nonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (1997)].