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For the critical one-dimensional nonlinear Schr\"odinger equation, weconstruct blow-up solutions that concentrate a soliton at the origin at theconformal blow-up rate, with a non-flat blow-up profile. More precisely, weobtain 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 flatprofiles obtained in the same context by Bourgain and Wang [Construction ofblowup solutions for the nonlinear Schr\"odinger equation with criticalnonlinearity. Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (1997)].

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