Solitary Waves Bifurcated from Bloch Band Edges in Two-dimensional Periodic Media

Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined both analytically and numerically in the context of a two-dimensional nonlinear Schr\"odinger equation with a periodic potential. Using multi-scale perturbation methods, envelope equations of solitary waves near Bloch bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of second-order dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch modes, the envelope equations lead to a host of solitary wave structures including reduced-symmetry solitons, dipole-array solitons, vortex-cell solitons, and so on -- many of which have never been reported before. It is also shown analytically that the centers of envelope solutions can only be positioned at four possible locations at or between potential peaks. Numerically, families of these solitary waves are directly computed both near and far away from band edges. Near the band edges, the numerical solutions spread over many lattice sites, and they fully agree with the analytical solutions obtained from envelope equations. Far away from the band edges, solitary waves are strongly localized with intensity and phase profiles characteristic of individual families.Comment: 23 pages, 15 figures. To appear in Phys. Rev.

Do we know how to count powers in pionless and pionful effective field theory?

In this article I summarize recent progress in the effective field theory approach to low energy nuclear systems, with a focus on the power counting issue. In the pionless sector, where the power counting is quite well understood at the nucleon-nucleon (NN) level, I discuss some recent developments toward few- and many-body calculations. In the pionful sector, I focus on the actively debated issue of power counting in the NN sector and some recent developments toward a model-independent NN interaction. Finally, the scenario that the power counting might depend on the number of particles is discussed.Comment: 19 pages, 1 figure. Contributions to the EPJA topical issue: The tower of effective (field) theories and the emergence of nuclear phenomena. Published versio

On the bound of first excursion probability

Method has been developed to improve the lower bound of the first excursion probability that can apply to the problem with either constant or time-dependent barriers. The method requires knowledge of the joint density function of the random process at two arbitrary instants

A nonperturbative parametrization and scenario for EFT renormalization

We present a universal form of the $T$-matrices renormalized in nonperturbative regime and the ensuing notions and properties that fail conventional wisdoms. A universal scale is identified and shown to be renormalization group invariant. The effective range parameters are derived in a nonperturbative scenario with some new predictions within the realm of contact potentials. Some controversies are shown to be due to the failure of conventional wisdoms.Comment: 5 pages, no figure, to appear in Europhys. Let