Fractal scatterings in weak solitary wave interactions is analyzed for generalized nonlinear Schrödiger equations (GNLS). Using asymptotic methods, these weak interactions are reduced to a universal second-order map. This map gives the same fractal scattering patterns as those in the GNLS equations both qualitatively and quantitatively. Scaling laws of these fractals are also derived.

Weak interactions of solitary waves in the generalized nonlinear Schrödinger equations are studied. It is first shown that these interactions exhibit similar fractal dependence on initial conditions for different nonlinearities. Then by using the Karpman-Solov'ev method, a universal system of dynamical equations is derived for the velocities, amplitudes, positions and phases of interacting solitary waves. These dynamical equations contain a single parameter, which accounts for the different forms of nonlinearity. When this parameter is zero, these dynamical equations are integrable, and the exact analytical solutions are derived. When this parameter is non-zero, the dynamical equations exhibit fractal structures which match those in the original wave equations both qualitatively and quantitatively. Thus the universal nature of fractal structures in the weak interaction of solitary waves is analytically established. The origin of these fractal structures is also explored. It is shown that these structures bifurcate from the initial conditions where the solutions of the integrable dynamical equations develop finite-time singularities. Based on this observation, an analytical criterion for the existence and locations of fractal structures is obtained. Lastly, these analytical results are applied to the generalized nonlinear Schrödinger equations with various nonlinearities such as the saturable nonlinearity, and predictions on their weak interactions of solitary waves are made.

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We show that the category of finite-length generalized modules for the singlet vertex algebra M(p), p∈Z_+, is equal to the category O_M(p) of C_1-cofinite M(p)-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since O_M(p) includes the uncountably many typical M(p)-modules, which are simple M(p)-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical M(p)-modules. We also introduce a tensor subcategory O_M(p)^T, graded by an algebraic torus T, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of sl_2 at a 2pth root of unity. We compute all tensor products involving simple and projective M(p)-modules, and we prove that both tensor categories O_M(p) and O_M(p)^T are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras W(p) are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.