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In this article we consider nonlinear Schrödinger (NLS) equations in for , , and . We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let be solitary wave solutions of the equation with different speeds . Provided that the relative speeds of the solitary waves are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the is stable for in some suitable sense in . To prove this result, we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear Schrödinger equation. This property is similar to the monotonicity property that has been proved by Martel and Merle for the generalized Korteweg–de Vries (gKdV) equations (see [12, Lem. 16, proof of Prop. 6]) and that was used to prove the stability of the sum of solitons of the gKdV equations by the authors of the present article (see [15, Th. 1(i)]).
We prove that the closure of a complete embedded minimal surface in a Riemannian three-manifold has the structure of a minimal lamination when has positive injectivity radius. When is , we prove that such a surface is properly embedded. Since a complete embedded minimal surface of finite topology in has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi in [5, Corollary 0.7]; a complete embedded minimal surface of finite topology in is proper. More generally, we prove that if is a complete embedded minimal surface of finite topology and has nonpositive sectional curvature (or is the Riemannian product of a Riemannian surface with ), then the closure of has the structure of a minimal lamination
We consider , a solution of which blows up at some time , where , , and . Under a nondegeneracy condition, we show that the mere hypothesis that the blow-up set is continuous and -dimensional implies that it is . In particular, we compute the principal curvatures and directions of . Moreover, a much more refined blow-up behavior is derived for the solution in terms of the newly exhibited geometric objects. Refined regularity for and refined singular behavior of near are linked through a new mechanism of algebraic cancellations that we explain in detail
Let be a closed connected Riemannian manifold, and let be a homotopy class of free loops in . Then, for every compactly supported time-dependent Hamiltonian on the open unit disk cotangent bundle which is sufficiently large over the zero section, we prove the existence of a -periodic orbit whose projection to represents . The proof shows that the Biran-Polterovich-Salamon capacity of the open unit disk cotangent bundle relative to the zero section is finite. If is not simply connected, this leads to an existence result for noncontractible periodic orbits on level hypersurfaces corresponding to a dense set of values of any proper Hamiltonian on bounded from below, whenever the levels enclose . This implies a version of the Weinstein conjecture including multiplicities; we prove existence of closed characteristics—one associated to each nontrivial —on every contact-type hypersurface in enclosing
Let be a set that is definable in an o-minimal structure over . This article shows that in a suitable sense, there are very few rational points of which do not lie on some connected semialgebraic subset of of positive dimension
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