Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the high-low method for NLS on the hyperbolic space
58
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we first prove that the cubic, defocusing nonlinear Schrodinger equation on the two dimensional hyperbolic space with radial initial data in $H^s(mathbb{H}^2)$ is globally well-posed and scatters when $s > frac{3}{4}$. Then we extend the result to nonlineraities of order $p>3$. The result is proved by extending the high-low method of Bourgain in the hyperbolic setting and by using a Morawetz type estimate proved by the first author and Ionescu.
rate research
Read More
We prove asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space in the radial case, in space dimension at least 4, and for any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which kind of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavior of radial solutions to the free dynamics.
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the spectral functions. We then concentrate on the physically significant case of $t$-periodic Dirichlet boundary data. After presenting certain heuristic arguments which suggest that the Neumann boundary values become periodic as $ttoinfty$, we show that for the case of the NLS with a sine-wave as Dirichlet data, the asymptotics of the Neumann boundary values can be computed explicitly at least up to third order in a perturbative expansion and indeed at least up to this order are asymptotically periodic.
We study the construction of the Gibbs measures for the {it focusing} mass-critical fractional nonlinear Schrodinger equation on the multi-dimensional torus. We identify the sharp mass threshold for normalizability and non-normalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz-Rose-Speer (1988), Bourgain (1994), and Oh-Sosoe-Tolomeo (2021) on the one-dimensional nonlinear Schrodinger equations. To this purpose, we establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus, which is of independent interest.
We prove global existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension $mgeq3$. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.
We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
