BLOW-UP FOR THE 1D NONLINEAR SCHRODINGER EQUATION WITH POINT NONLINEARITY II: SUPERCRITICAL BLOW-UP PROFILES

We consider the 1D nonlinear Schrodinger equation (NLS) with focusing point nonlinearity, i partial derivative(t)psi + partial derivative(2)(x)psi + delta vertical bar psi vertical bar(p-1)psi = 0, (0.1) where delta = delta(x) is the delta function supported at the origin. In the L-2 supercritical setting p > 3, we construct self-similar blow-up solutions belonging to the energy space L-x(infinity) boolean AND (H) over dot(x)(1). This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at x = 0 imposed by the delta term in (0, 1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case 0 < p - 3 << 1 using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.

Automated summary

The research focuses on self-similar blow-up solutions of the 1D nonlinear Schrodinger equation with focusing point nonlinearity. By utilizing parabolic cylinder functions and solving the stationary profile equation, all outgoing solutions are obtained. The jump condition involving gamma functions is solved using the intermediate value theorem and formulae for the digamma function to establish existence and uniqueness of solutions.