Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles

We consider the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity, \begin{document}$ \begin{equation} i\partial_t\psi + \partial_x^2\psi \delta|\psi|^{p-1}\psi = 0, \;\;\;\;\;\;(0.1)\end{equation} $\end{document} where {\delta} {\delta}(x) is delta function supported at origin. In L^2 supercritical setting p>3 , we construct self-similar blow-up solutions belonging to energy space L_x^\infty \cap \dot H_x^1 . This reduced finding outgoing of a certain stationary profile equation. All are obtained by using parabolic cylinder functions (Weber functions) and solving jump condition x 0 imposed \delta term in (0.1). an algebraic involving gamma functions, existence uniqueness intermediate value theorem formulae for digamma function. also compute form these slightly case log Binet formula steepest descent method integral functions.