Solitons supported by singular spatial modulation of the Kerr nonlinearity

We introduce a setting based on the one-dimensional (1D) nonlinear Schroedinger equation (NLSE) with the self-focusing (SF) cubic term modulated by a singular function of the coordinate, |x|^{-a}. It may be additionally combined with the uniform self-defocusing (SDF) nonlinear background, and with a similar singular repulsive linear potential. The setting, which can be implemented in optics and BEC, aims to extend the general analysis of the existence and stability of solitons in NLSEs. Results for fundamental solitons are obtained analytically and verified numerically. The solitons feature a quasi-cuspon shape, with the second derivative diverging at the center, and are stable in the entire existence range, which is 0 < a < 1. Dipole (odd) solitons are found too. They are unstable in the infinite domain, but stable in the semi-infinite one. In the presence of the SDF background, there are two subfamilies of fundamental solitons, one stable and one unstable, which exist together above a threshold value of the norm (total power of the soliton). The system which additionally includes the singular repulsive linear potential emulates solitons in a uniform space of the fractional dimension, 0 < D < 1. A two-dimensional extension of the system, based on the quadratic nonlinearity, is formulated too.Comment: Physical Review A, in pres

Blow up dynamics for smooth equivariant solutions to the energy critical Schr\"odinger map

We consider the energy critical Schr\"odinger map problem with the 2-sphere target for equivariant initial data of homotopy index $k=1$. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blow up solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy

Wave envelopes with second-order spatiotemporal dispersion : I. Bright Kerr solitons and cnoidal waves

We propose a simple scalar model for describing pulse phenomena beyond the conventional slowly-varying envelope approximation. The generic governing equation has a cubic nonlinearity and we focus here mainly on contexts involving anomalous group-velocity dispersion. Pulse propagation turns out to be a problem firmly rooted in frames-of-reference considerations. The transformation properties of the new model and its space-time structure are explored in detail. Two distinct representations of exact analytical solitons and their associated conservation laws (in both integral and algebraic forms) are presented, and a range of new predictions is made. We also report cnoidal waves of the governing nonlinear equation. Crucially, conventional pulse theory is shown to emerge as a limit of the more general formulation. Extensive simulations examine the role of the new solitons as robust attractors

On Singularity formation for the L^2-critical Boson star equation

We prove a general, non-perturbative result about finite-time blowup solutions for the $L^2$-critical boson star equation $i\partial_t u = \sqrt{-\Delta+m^2} \, u - (|x|^{-1} \ast |u|^2) u$ in 3 space dimensions. Under the sole assumption that the solution blows up in $H^{1/2}$ at finite time, we show that $u(t)$ has a unique weak limit in $L^2$ and that $|u(t)|^2$ has a unique weak limit in the sense of measures. Moreover, we prove that the limiting measure exhibits minimal mass concentration. A central ingredient used in the proof is a "finite speed of propagation" property, which puts a strong rigidity on the blowup behavior of $u$. As the second main result, we prove that any radial finite-time blowup solution $u$ converges strongly in $L^2$ away from the origin. For radial solutions, this result establishes a large data blowup conjecture for the $L^2$-critical boson star equation, similar to a conjecture which was originally formulated by F. Merle and P. Raphael for the $L^2$-critical nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704]. We also discuss some extensions of our results to other $L^2$-critical theories of gravitational collapse, in particular to critical Hartree-type equations.Comment: 24 pages. Accepted in Nonlinearit