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Sohrab Shahshahani - One of the best experts on this subject based on the ideXlab platform.

  • gap Eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space
    Journal of Functional Analysis, 2016
    Co-Authors: Andrew Lawrie, Sungjin Oh, Sohrab Shahshahani
    Abstract:

    In this paper we study k -equivariant wave maps from the hyperbolic plane into the 2-sphere as well as the energy critical equivariant SU(2) S U ( 2 ) Yang–Mills problem on 4-dimensional hyperbolic space. The latter problem bears many similarities to a 2-equivariant wave map into a surface of revolution. As in the case of 1-equivariant wave maps considered in [9] , both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schrodinger operator obtained by linearizing about such a harmonic map admits a simple Positive Eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.

  • gap Eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space
    arXiv: Analysis of PDEs, 2015
    Co-Authors: Andrew Lawrie, Sungjin Oh, Sohrab Shahshahani
    Abstract:

    In this paper we study $k$-equivariant wave maps from the hyperbolic plane into the $2$-sphere as well as the energy critical equivariant $SU(2)$ Yang-Mills problem on $4$-dimensional hyperbolic space. The latter problem bears many similarities to a $2$-equivariant wave map into a surface of revolution. As in the case of $1$-equivariant wave maps considered in~\cite{LOS1}, both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schr\"odinger operator obtained by linearizing about such a harmonic map admits a simple Positive Eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.

Paul M. N. Feehan - One of the best experts on this subject based on the ideXlab platform.

  • Critical-exponent Sobolev norms and the slice theorem for the quotient space of connections
    2015
    Co-Authors: Paul M. N. Feehan
    Abstract:

    Following Taubes, we describe a collection of critical-expo-nent Sobolev norms, discuss their embedding and multiplica-tion properties, and describe optimal Green’s operator esti-mates where the constants depend at most on the first Positive Eigenvalue of the covariant Laplacian of a G connection and the L2 norm of the connection’s curvature, for arbitrary com-pact Lie groups G. Using these critical-exponent norms, we prove a sharp, global analogue of Uhlenbeck’s Coulomb gauge-fixing theorem, where the usual product connection over a ball is replaced by an arbitrary reference connection over the entire manifold. We also prove a quantitative version of the conventional slice theorem for the quotient space of G connec-tions, with an invariant and sharp characterization of those points in the quotient space which are contained in the image of an L4 ball in the Coulomb-gauge slice. Our gauge-fixin

  • critical exponent sobolev norms and the slice theorem for the quotient space of connections
    arXiv: Differential Geometry, 1997
    Co-Authors: Paul M. N. Feehan
    Abstract:

    The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to bound the norm of a Green's operator for a Laplacian, depending on a connection varying in a non-compact family, in terms of minimal data such as the first Positive Eigenvalue of the Laplacian or the L^2 norm of the curvature of the connection. Following Taubes, we describe a collection of critical-exponent Sobolev norms and general Green's operator estimates depending only on first Positive Eigenvalues or the L^2 norm of the connection's curvature. Such estimates are particularly useful in the gluing construction of solutions to non-linear partial differential equations depending on a degenerating parameter, such as the approximate, reference solution in the anti-self-dual or PU(2) monopole equations. We apply them here to prove an optimal slice theorem for the quotient space of connections. The result is optimal in the sense that if a point [A] in the quotient space is known to be just L^2_1-close enough to a reference point [A_0], then the connection A can be placed in Coulomb gauge relative to the connection A_0, with all constants depending at most on the first Positive Eigenvalue of the covariant Laplacian defined by A_0 and the L^2 norm of the curvature of A_0. In this paper we shall for simplicity only consider connections over four-dimensional manifolds, but the methods and results can adapted to manifolds of arbitrary dimension to prove slice theorems which apply when the reference connection is allowed to degenerate.

Miguel J Malacarne - One of the best experts on this subject based on the ideXlab platform.

Andrew Lawrie - One of the best experts on this subject based on the ideXlab platform.

  • gap Eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space
    Journal of Functional Analysis, 2016
    Co-Authors: Andrew Lawrie, Sungjin Oh, Sohrab Shahshahani
    Abstract:

    In this paper we study k -equivariant wave maps from the hyperbolic plane into the 2-sphere as well as the energy critical equivariant SU(2) S U ( 2 ) Yang–Mills problem on 4-dimensional hyperbolic space. The latter problem bears many similarities to a 2-equivariant wave map into a surface of revolution. As in the case of 1-equivariant wave maps considered in [9] , both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schrodinger operator obtained by linearizing about such a harmonic map admits a simple Positive Eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.

  • gap Eigenvalues and asymptotic dynamics of geometric wave equations on hyperbolic space
    arXiv: Analysis of PDEs, 2015
    Co-Authors: Andrew Lawrie, Sungjin Oh, Sohrab Shahshahani
    Abstract:

    In this paper we study $k$-equivariant wave maps from the hyperbolic plane into the $2$-sphere as well as the energy critical equivariant $SU(2)$ Yang-Mills problem on $4$-dimensional hyperbolic space. The latter problem bears many similarities to a $2$-equivariant wave map into a surface of revolution. As in the case of $1$-equivariant wave maps considered in~\cite{LOS1}, both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schr\"odinger operator obtained by linearizing about such a harmonic map admits a simple Positive Eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.

Todd Kapitula - One of the best experts on this subject based on the ideXlab platform.

  • stability criterion for bright solitary waves of the perturbed cubic quintic schroedinger equation
    arXiv: Pattern Formation and Solitons, 1997
    Co-Authors: Todd Kapitula
    Abstract:

    The stability of the bright solitary wave solution to the perturbed cubic-quintic Schroedinger equation is considered. It is shown that in a certain region of parameter space these solutions are unstable, with the instability being manifested as a small Positive Eigenvalue. Furthermore, it is shown that in the complimentary region of parameter space there are no small unstable Eigenvalues. The proof involves a novel calculation of the Evans function, which is of interest in its own right. As a consequence of the Eigenvalue calculation, it is additionally shown that N-bump bright solitary waves bifurcate from the primary wave.