Asymptotics of Multibump Blow-up Self-Similar Solutions of the Nonlinear Schrödinger Equation

This paper examines blow-up self-similar solutions of the cubic nonlinear Schrodinger equation close to the critical dimension $d=2$. It gives a formal asymptotic theory for self-similar solutions with multiple maxima, in which the solution close to each maximum takes the form of a rescaled one-dimensional soliton. As $d \rightarrow 2$, the maxima move to infinity and are centered close to the point $-\log(d-2)/( 2\pi/3 - \sqrt{3/4})$. However, the shape of the solution close to each maxima changes little in this limit, leading to an interesting nonuniform bifurcation. The formulae derived from the asymptotic theory are strongly supported by some numerical calculations.