7,704 research outputs found
Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies
Combining algebro-geometric methods and factorization techniques for finite
difference expressions we provide a complete and self-contained treatment of
all real-valued quasi-periodic finite-gap solutions of both the Toda and
Kac-van Moerbeke hierarchies. In order to obtain our principal new result, the
algebro-geometric finite-gap solutions of the Kac-van Moerbeke hierarchy, we
employ particular commutation methods in connection with Miura-type
transformations which enable us to transfer whole classes of solutions (such as
finite-gap solutions) from the Toda hierarchy to its modified counterpart, the
Kac-van Moerbeke hierarchy, and vice versa.Comment: LaTeX, to appear in Memoirs of the Amer. Math. So
High prices for rare species can drive large populations extinct: the anthropogenic Allee effect revisited
Consumer demand for plant and animal products threatens many populations with
extinction. The anthropogenic Allee effect (AAE) proposes that such extinctions
can be caused by prices for wildlife products increasing with species rarity.
This price-rarity relationship creates financial incentives to extract the last
remaining individuals of a population, despite higher search and harvest costs.
The AAE has become a standard approach for conceptualizing the threat of
economic markets on endangered species. Despite its potential importance for
conservation, AAE theory is based on a simple graphical model with limited
analysis of possible population trajectories. By specifying a general class of
functions for price-rarity relationships, we show that the classic theory can
understate the risk of species extinction. AAE theory proposes that only
populations below a critical Allee threshold will go extinct due to increasing
price-rarity relationships. Our analysis shows that this threshold can be much
higher than the original theory suggests, depending on initial harvest effort.
More alarmingly, even species with population sizes above this Allee threshold,
for which AAE predicts persistence, can be destined to extinction. Introducing
even a minimum price for harvested individuals, close to zero, can cause large
populations to cross the classic anthropogenic Allee threshold on a trajectory
towards extinction. These results suggest that traditional AAE theory may give
a false sense of security when managing large harvested populations
Operator splitting for the Benjamin-Ono equation
In this paper we analyze operator splitting for the Benjamin-Ono equation,
u_t = uu_x + Hu_xx, where H denotes the Hilbert transform. If the initial data
are sufficiently regular, we show the convergence of both Godunov and Strang
splitting.Comment: 18 Page
The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy
We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik
hierarchy with complex-valued initial data and prove unique solvability
globally in time for a set of initial (Dirichlet divisor) data of full measure.
To this effect we develop a new algorithm for constructing stationary
complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy,
which is of independent interest as it solves the inverse algebro-geometric
spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators,
starting from a suitably chosen set of initial divisors of full measure.
Combined with an appropriate first-order system of differential equations with
respect to time (a substitute for the well-known Dubrovin-type equations), this
yields the construction of global algebro-geometric solutions of the
time-dependent Ablowitz-Ladik hierarchy.
The treatment of general (non-unitary) Lax operators associated with general
coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties
that, to the best of our knowledge, are successfully overcome here for the
first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but
applies generally to (1+1)-dimensional completely integrable soliton equations
of differential-difference type.Comment: 47 page
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