40,227 research outputs found
The Bounded L2 Curvature Conjecture
This is the main paper in a sequence in which we give a complete proof of the
bounded curvature conjecture. More precisely we show that the time of
existence of a classical solution to the Einstein-vacuum equations depends only
on the -norm of the curvature and a lower bound on the volume radius of
the corresponding initial data set. We note that though the result is not
optimal with respect to the standard scaling of the Einstein equations, it is
nevertheless critical with respect to its causal geometry. Indeed, bounds
on the curvature is the minimum requirement necessary to obtain lower bounds on
the radius of injectivity of causal boundaries. We note also that, while the
first nontrivial improvements for well posedness for quasilinear hyperbolic
systems in spacetime dimensions greater than 1+1 (based on Strichartz
estimates) were obtained in [Ba-Ch1] [Ba-Ch2] [Ta1] [Ta2] [Kl-R1] and optimized
in [Kl-R2] [Sm-Ta], the result we present here is the first in which the full
structure of the quasilinear hyperbolic system, not just its principal part,
plays a crucial role. To achieve our goals we recast the Einstein vacuum
equations as a quasilinear -valued Yang-Mills theory and introduce a
Coulomb type gauge condition in which the equations exhibit a specific new type
of \textit{null structure} compatible with the quasilinear, covariant nature of
the equations. To prove the conjecture we formulate and establish bilinear and
trilinear estimates on rough backgrounds which allow us to make use of that
crucial structure. These require a careful construction and control of
parametrices including error bounds which is carried out in [Sz1]-[Sz4],
as well as a proof of sharp Strichartz estimates for the wave equation on a
rough background which is carried out in \cite{Sz5}.Comment: updated version taking into account the remarks of the refere
Fusion rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds
We show that the set of transfer matrices of an arbitrary fusion type for an
integrable quantum model obey these bilinear functional relations, which are
identified with an integrable dynamical system on a Grassmann manifold (higher
Hirota equation). The bilinear relations were previously known for a particular
class of transfer matrices corresponding to rectangular Young diagrams. We
extend this result for general Young diagrams. A general solution of the
bilinear equations is presented.Comment: LaTex (MPLA macros included) 10 pages, 1 figure, included in the tex
A primal-dual formulation for certifiable computations in Schubert calculus
Formulating a Schubert problem as the solutions to a system of equations in
either Pl\"ucker space or in the local coordinates of a Schubert cell typically
involves more equations than variables. We present a novel primal-dual
formulation of any Schubert problem on a Grassmannian or flag manifold as a
system of bilinear equations with the same number of equations as variables.
This formulation enables numerical computations in the Schubert calculus to be
certified using algorithms based on Smale's \alpha-theory.Comment: 21 page
A novel multi-component generalization of the short pulse equation and its multisoliton solutions
We propose a novel multi-component system of nonlinear equations that
generalizes the short pulse (SP) equation describing the propagation of
ultra-short pulses in optical fibers. By means of the bilinear formalism
combined with a hodograph transformation, we obtain its multi-soliton solutions
in the form of a parametric representation. Notably, unlike the determinantal
solutions of the SP equation, the proposed system is found to exhibit solutions
expressed in terms of pfaffians. The proof of the solutions is performed within
the framework of an elementary theory of determinants. The reduced 2-component
system deserves a special consideration. In particular, we show by establishing
a Lax pair that the system is completely integrable. The properties of
solutions such as loop solitons and breathers are investigated in detail,
confirming their solitonic behavior. A variant of the 2-component system is
also discussed with its multisoliton solutions.Comment: Minor correction
Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies
In this paper, we give a procedure of how to discretize the recursion
operators by considering unified bilinear forms of integrable hierarchies. As
two illustrative examples, the unified bilinear forms of the AKNS hierarchy and
the KdV hierarchy are presented from their recursion operators. Via the
compatibility between soliton equations and their auto-B\"acklund
transformations, the bilinear integrable hierarchies are discretized and the
discrete recursion operators are obtained. The discrete recursion operators
converge to the original continuous forms after a standard limit.Comment: 11Page
N-Dark-Dark Solitons in the Generally Coupled Nonlinear Schroedinger Equations
N-dark-dark solitons in the generally coupled integrable NLS equations are
derived by the KP-hierarchy reduction method. These solitons exist when
nonlinearities are all defocusing, or both focusing and defocusing
nonlinearities are mixed. When these solitons collide with each other, energies
in both components of the solitons completely transmit through. This behavior
contrasts collisions of bright-bright solitons in similar systems, where
polarization rotation and soliton reflection can take place. It is also shown
that in the mixed-nonlinearity case, two dark-dark solitons can form a
stationary bound state.Comment: 26 pages, 3 figure
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