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Register a new userTransversality and the Pego-Weinstein theory
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This paper studies the linear stability problem for solitary wave solutions of Hamiltonian PDEs. The linear stability problem is formulated in terms of the Evans function, a complex analytic function denoted by $D(lambda)$, where $lambda$ is the stability exponent. The main result is the introduction of a new factor, denoted $Pi$, in the Pego-Weinstein derivative formula [ D(0) = Pi frac{dI}{dc},, ] where $I$ is the momentum of the solitary wave and $c$ is the speed. Moreover this factor turns out to be related to transversality of the solitary wave, modelled as a homoclinic orbit: the homoclinic orbit is transversely constructed if and only if $Pi eq 0$. The sign of $Pi$ is a symplectic invariant, an intrinsic property of the solitary wave, and is a key new factor affecting the linear stability. A supporting result is the introduction of a new abstract class of Hamiltonian PDEs built on a nonlinear Dirac-type equation, which model a wide range of Hamiltonian PDEs. Examples where the theory applies, other than Dirac operators, are the coupled mode equation in fluid mechanics and optics, the massive Thirring model, and coupled nonlinear wave equations. The new result is already present when the homoclinic orbit representation of the solitary wave lives in a four dimensional phase space, and so the theory is presented for this case, with the generalization to arbitrary dimension sketched.
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In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.
In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps.
In this paper we will develop a general approach which shows that generalized critical relations of families of locally defined holomorphic maps on the complex plane unfold transversally. The main idea is to define a transfer operator, which is a local analogue of the Thurston pullback operator, using holomorphic motions. Assuming a so-called lifting property is satisfied, we obtain information about the spectrum of this transfer operator and thus about transversality. An important new feature of our method is that it is not global: the maps we consider are only required to be defined and holomorphic on a neighbourhood of some finite set. We will illustrate this method by obtaining transversality for a wide class of one-parameter families of interval and circle maps, for example for maps with flat critical points, but also for maps with complex analytic extensions such as certain polynomial-like maps. As in Tsujiis approach cite{Tsu0,Tsu1}, for real maps we obtain {em positive} transversality (where $>0$ holds instead of just $ e 0$), and thus monotonicity of entropy for these families, and also (as an easy application) for the real quadratic family. This method additionally gives results for unimodal families of the form $xmapsto |x|^ell+c$ for $ell>1$ not necessarily an even integer and $c$ real.
In this paper we will give a short and elementary proof that critical relations unfold transversally in the space of rational maps.
We prove the conjectures of Graham-Kumar and Griffeth-Ram concerning the alternation of signs in the structure constants for torus-equivariant K-theory of generalized flag varieties G/P. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant K-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term--the top one--with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary K-theory that brings Kawamata-Viehweg vanishing to bear.
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