The transition from diffusion to blow-up for a nonlinear Schrödinger equation in dimension 1

Spectral splitting method for nonlinear Schrödinger equations with singular potential

“…We recall also that (Theorem 1 in [2]) if the initial data w 0 belongs to H 1 ðRÞ then Eq. (1) admits a unique local solution…”

“…The phenomenon of blow-up has been investigated extensively in the case of the free nonlinear Schrö dinger equation (see, e.g., [5]) and recently new results have been obtained for NLS with Dirac's delta interaction [2]. In such a problem the definition of blow-up is the same as in the standard case: that is, let w t 2 H 1 ðRÞ be the unique maximal solution of the Cauchy problem (1) with initial data w 0 2 H 1 ðRÞ.…”

“…Usually, existence of global solution when < 0 is proved by means of some a priori estimates on the norm of the gradient of w t based on the energy conservation and on the Gagliardo-Niremberg inequality (see Appendix in [2]). In particular we have that…”

“…Such a basic model has recently attracted an increasing interest, from both theoretical [2,10,12,20] and numerical [11,[14][15][16] points of view; in fact, scattering of solitons by point defects is a natural model extensively studied in physics (see, e.g., [9]). …”

The Nonlinear Schrödinger Equation for the Delta-Comb Potential: Quasi-Classical Chaos and Bifurcations of Periodic Stationary Solutions

“…The first derivative of the wave function ψ is discontinuous at x = 2πn (cf. the studies [2,20,54,56,67] of a NLSE for a single delta-potential) 6) whereas the wave function itself is continuous. The discontinuity of the delta-comb potential does not affect the area-preserving quality of the flow (ψ(…”